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TD-R/Ma feuille d'exercice2 - Bibm@th.net.html
Félix MARQUET 0eb717651e feat: add exercise sheets for Bernoulli, integrals, binomial distribution, and cardinality 
- Added exercise sheets for Bernoulli experiments including probability of success, failure, and simulation of multiple experiments.
- Added exercise sheets for calculating integrals using different methods such as rectangles, Simpson's rule, and the integrate function.
- Added exercise sheets for binomial distribution including probability calculations and cumulative probability.
- Added exercise sheets for calculating the cardinality of sets including unique elements and empty sets.
2024-12-20 08:26:12 +01:00

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Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Loi d'un dé truqué<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1253"> [Signaler une erreur]</a><a id="exo1" href="javascript:ajouteexo(1253,1,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1253&#39;,&#39;imageenonce1253&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce1253"></a><div class="inner" id="innerenonce1253">On considère un dé cubique truqué dont les faces sont numérotés de 1 à 6 et on note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-2-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4" class="mjx-mrow"><span id="MJXc-Node-5" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-2">X</script> la variable aléatoire donnée par le numéro de la face du dessus. On suppose que le dé est truqué de sorte que la probabilité d'obtenir une face est proportionnelle au numéro inscrit sur cette face.
<ol class="enumeratechiffre">
<li> Déterminer la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-3-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7" class="mjx-mrow"><span id="MJXc-Node-8" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-3">X</script>, calculer son espérance.
</li><li> On pose <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-4-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10" class="mjx-mrow"><span id="MJXc-Node-11" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-12" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-13" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-14" class="mjx-texatom"><span id="MJXc-Node-15" class="mjx-mrow"><span id="MJXc-Node-16" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">/</span></span></span></span><span id="MJXc-Node-17" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-4">Y=1/X</script>. Déterminer la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-5-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-18" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-19" class="mjx-mrow"><span id="MJXc-Node-20" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-5">Y</script>, et son espérance.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1253&#39;,&#39;imageindication1253&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication1253"></a><div class="inner" id="innerindication1253" style="display:none;"><ol class="enumeratechiffre">
<li> Utiliser que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-6-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/munderover&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-21" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-22" class="mjx-mrow"><span id="MJXc-Node-23" class="mjx-munderover"><span class="mjx-base"><span id="MJXc-Node-24" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">∑</span></span></span><span class="mjx-stack" style="vertical-align: -0.31em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.383em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-30" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-25" class="mjx-texatom" style=""><span id="MJXc-Node-26" class="mjx-mrow"><span id="MJXc-Node-27" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-28" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-29" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span><span id="MJXc-Node-31" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-32" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-33" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-34" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-35" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-36" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-37" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-38" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mn>6</mn></munderover><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-6">\sum_{k=1}^6 P(X=k)=1</script>, pour déterminer la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-7-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-39" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-40" class="mjx-mrow"><span id="MJXc-Node-41" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-7">X</script>.
</li><li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-8-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-42" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-43" class="mjx-mrow"><span id="MJXc-Node-44" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-45" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-46" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-47" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-48" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-49" class="mjx-texatom"><span id="MJXc-Node-50" class="mjx-mrow"><span id="MJXc-Node-51" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-52" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-53" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-54" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-55" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-56" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-57" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-58" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-59" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-60" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-8">P(Y=1/k)=P(X=k)</script>.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1253&#39;,&#39;imagecorrige1253&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas(1).png" id="imagecorrige1253"></a><div class="inner" id="innercorrige1253" style=""><ol class="enumeratechiffre">
<li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-9-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-61" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-62" class="mjx-mrow"><span id="MJXc-Node-63" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-9">X</script> prend ses valeurs dans <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-10-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-64" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-65" class="mjx-mrow"><span id="MJXc-Node-66" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-67" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-68" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-69" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-70" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-71" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-72" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>6</mn><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-10">\{1,\dots,6\}</script>. Par hypothèse, il existe un réel <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-11-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-73" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-74" class="mjx-mrow"><span id="MJXc-Node-75" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-11">a</script> tel que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-12-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-76" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-77" class="mjx-mrow"><span id="MJXc-Node-78" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-79" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-80" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-81" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-82" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-83" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-84" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-85" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-86" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>k</mi><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-12">P(X=k)=ka</script>. Maintenant,
puisque <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-13-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-87" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-88" class="mjx-mrow"><span id="MJXc-Node-89" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-90" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-91" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>X</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-13">P_X</script> est une loi de probabilité, on a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-14-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/munderover&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27FA;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27F9;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;21.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-92" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-93" class="mjx-mrow"><span id="MJXc-Node-94" class="mjx-munderover"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.236em; padding-top: 0.141em; padding-left: 0.771em;"><span id="MJXc-Node-101" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-op"><span id="MJXc-Node-95" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.122em;"><span id="MJXc-Node-96" class="mjx-texatom" style=""><span id="MJXc-Node-97" class="mjx-mrow"><span id="MJXc-Node-98" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-99" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-100" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span id="MJXc-Node-102" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-103" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-104" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-105" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-106" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-107" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-108" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-109" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-110" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-111" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟺</span></span><span id="MJXc-Node-112" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-113" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-114" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.422em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.422em; top: -1.378em;"><span id="MJXc-Node-115" class="mjx-mrow"><span id="MJXc-Node-116" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-117" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-118" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span></span><span class="mjx-denominator" style="width: 2.422em; bottom: -0.711em;"><span id="MJXc-Node-119" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.422em;"></span></span><span class="mjx-vsize" style="height: 2.088em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-120" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-121" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-122" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-123" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟹</span></span><span id="MJXc-Node-124" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-125" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-126" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-127" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-128" class="mjx-texatom"><span id="MJXc-Node-129" class="mjx-mrow"><span id="MJXc-Node-130" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-131" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mn>6</mn></munderover><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mi>a</mi><mfrac><mrow><mn>6</mn><mo>×</mo><mn>7</mn></mrow><mn>2</mn></mfrac><mo>=</mo><mn>1</mn><mspace width="thickmathspace"></mspace><mo stretchy="false">⟹</mo><mspace width="thickmathspace"></mspace><mi>a</mi><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>21.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-14">\sum_{k=1}^6 P(X=k)=1\iff a\frac{6\times 7}{2}=1\implies a=1/21.</script>
On a donc :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-15-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;center center center center center center center&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot; rowlines=&quot;solid&quot; columnlines=&quot;solid solid solid solid solid solid&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-132" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-133" class="mjx-mrow"><span id="MJXc-Node-134" class="mjx-mtable" style="vertical-align: -1.361em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-135" class="mjx-mtr" style="height: 1.421em;"><span id="MJXc-Node-136" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.471em 0px 0.4em; border-bottom: 1.3px solid; width: 4.236em;"><span id="MJXc-Node-137" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-138" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-139" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.471em 0px; border-bottom: 1.3px solid; width: 1.089em;"><span id="MJXc-Node-140" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-141" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-142" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.471em 0px; border-bottom: 1.3px solid; width: 1.089em;"><span id="MJXc-Node-143" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-144" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-145" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.471em 0px; border-bottom: 1.3px solid; width: 1.089em;"><span id="MJXc-Node-146" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-147" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-148" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.471em 0px; border-bottom: 1.3px solid; width: 1.089em;"><span id="MJXc-Node-149" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-150" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-151" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.471em 0px; border-bottom: 1.3px solid; width: 1.089em;"><span id="MJXc-Node-152" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-153" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-154" class="mjx-mtd" style="padding: 0.221em 0.4em 0px 0.471em; border-bottom: 1.3px solid; width: 1.089em;"><span id="MJXc-Node-155" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-156" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-157" class="mjx-mtr" style="height: 1.801em;"><span id="MJXc-Node-158" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.471em 0px 0.4em;"><span id="MJXc-Node-159" class="mjx-mrow" style="margin-top: -0.052em;"><span id="MJXc-Node-160" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-161" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-162" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-163" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-164" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-165" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-166" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.471em 0px;"><span id="MJXc-Node-167" class="mjx-mrow" style="margin-top: -0.052em;"><span id="MJXc-Node-168" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.318em;"><span id="MJXc-Node-169" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.611em;"><span id="MJXc-Node-170" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-171" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.471em 0px;"><span id="MJXc-Node-172" class="mjx-mrow" style="margin-top: -0.052em;"><span id="MJXc-Node-173" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.318em;"><span id="MJXc-Node-174" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.611em;"><span id="MJXc-Node-175" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-176" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.471em 0px;"><span id="MJXc-Node-177" class="mjx-mrow" style="margin-top: -0.052em;"><span id="MJXc-Node-178" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.339em;"><span id="MJXc-Node-179" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.611em;"><span id="MJXc-Node-180" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.379em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-181" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.471em 0px;"><span id="MJXc-Node-182" class="mjx-mrow" style="margin-top: -0.052em;"><span id="MJXc-Node-183" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.329em;"><span id="MJXc-Node-184" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.611em;"><span id="MJXc-Node-185" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.372em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-186" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.471em 0px;"><span id="MJXc-Node-187" class="mjx-mrow" style="margin-top: -0.052em;"><span id="MJXc-Node-188" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.34em;"><span id="MJXc-Node-189" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.611em;"><span id="MJXc-Node-190" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.38em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-191" class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.471em;"><span id="MJXc-Node-192" class="mjx-mrow" style="margin-top: -0.052em;"><span id="MJXc-Node-193" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.34em;"><span id="MJXc-Node-194" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.611em;"><span id="MJXc-Node-195" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.38em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="center center center center center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid" columnlines="solid solid solid solid solid solid"><mtr><mtd><mi>k</mi></mtd><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd><mtd><mn>3</mn></mtd><mtd><mn>4</mn></mtd><mtd><mn>5</mn></mtd><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></mtd><mtd><mfrac><mn>1</mn><mn>21</mn></mfrac></mtd><mtd><mfrac><mn>2</mn><mn>21</mn></mfrac></mtd><mtd><mfrac><mn>3</mn><mn>21</mn></mfrac></mtd><mtd><mfrac><mn>4</mn><mn>21</mn></mfrac></mtd><mtd><mfrac><mn>5</mn><mn>21</mn></mfrac></mtd><mtd><mfrac><mn>6</mn><mn>21</mn></mfrac></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-15">\begin{array}{c|c|c|c|c|c|c}
k&1&2&3&4&5&6\\
\hline
P(X=k)&\frac{1}{21}&\frac{2}{21}&\frac{3}{21}&\frac{4}{21}&\frac{5}{21}&\frac{6}{21}
\end{array}</script>
On vérifie aisément en appliquant la formule <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-16-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/munderover&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-196" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-197" class="mjx-mrow"><span id="MJXc-Node-198" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-199" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-200" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-201" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-202" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-203" class="mjx-munderover MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-204" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">∑</span></span></span><span class="mjx-stack" style="vertical-align: -0.31em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.383em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-210" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-205" class="mjx-texatom" style=""><span id="MJXc-Node-206" class="mjx-mrow"><span id="MJXc-Node-207" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-208" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-209" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span><span id="MJXc-Node-211" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-212" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-213" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-214" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-215" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-216" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-217" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mn>6</mn></munderover><mi>k</mi><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-16">E(X)=\sum_{k=1}^6 kP(X=k)</script> que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-17-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-218" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-219" class="mjx-mrow"><span id="MJXc-Node-220" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-221" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-222" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-223" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-224" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-225" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.34em;"><span id="MJXc-Node-226" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.632em;"><span id="MJXc-Node-227" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.395em; vertical-align: -0.447em;"></span></span><span id="MJXc-Node-228" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>13</mn><mn>3</mn></mfrac><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-17">E(X)=\frac{13}{3}.</script>
</li><li> On a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-18-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27FA;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-229" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-230" class="mjx-mrow"><span id="MJXc-Node-231" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-232" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-233" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-234" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-235" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟺</span></span><span id="MJXc-Node-236" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-237" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-238" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-239" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-240" class="mjx-texatom"><span id="MJXc-Node-241" class="mjx-mrow"><span id="MJXc-Node-242" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-243" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mi>k</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mi>Y</mi><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-18">X=k\iff Y=1/k</script>. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-19-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-244" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-245" class="mjx-mrow"><span id="MJXc-Node-246" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-19">Y</script> prend donc ses valeurs dans <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-20-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-247" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-248" class="mjx-mrow"><span id="MJXc-Node-249" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-250" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-251" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-252" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-253" class="mjx-texatom"><span id="MJXc-Node-254" class="mjx-mrow"><span id="MJXc-Node-255" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-256" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-257" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-258" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-259" class="mjx-texatom"><span id="MJXc-Node-260" class="mjx-mrow"><span id="MJXc-Node-261" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-262" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-263" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-264" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-265" class="mjx-texatom"><span id="MJXc-Node-266" class="mjx-mrow"><span id="MJXc-Node-267" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-268" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-269" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-270" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-271" class="mjx-texatom"><span id="MJXc-Node-272" class="mjx-mrow"><span id="MJXc-Node-273" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-274" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-275" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-276" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-277" class="mjx-texatom"><span id="MJXc-Node-278" class="mjx-mrow"><span id="MJXc-Node-279" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-280" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-281" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo>,</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn><mo>,</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>4</mn><mo>,</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>5</mn><mo>,</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>6</mn><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-20">\{1,1/2,1/3,1/4,1/5,1/6\}</script>, et la loi est donnée par :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-21-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;center center center center center center center&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot; rowlines=&quot;solid&quot; columnlines=&quot;solid solid solid solid solid solid&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-282" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-283" class="mjx-mrow"><span id="MJXc-Node-284" class="mjx-mtable" style="vertical-align: -1.398em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-285" class="mjx-mtr" style="height: 1.496em;"><span id="MJXc-Node-286" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.471em 0px 0.4em; border-bottom: 1.3px solid; width: 5.147em;"><span id="MJXc-Node-287" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-288" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-289" class="mjx-texatom"><span id="MJXc-Node-290" class="mjx-mrow"><span id="MJXc-Node-291" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-292" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-293" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.471em 0px; border-bottom: 1.3px solid; width: 1.5em;"><span id="MJXc-Node-294" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-295" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-296" class="mjx-texatom"><span id="MJXc-Node-297" class="mjx-mrow"><span id="MJXc-Node-298" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-299" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-300" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.471em 0px; border-bottom: 1.3px solid; width: 1.5em;"><span id="MJXc-Node-301" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-302" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-303" class="mjx-texatom"><span id="MJXc-Node-304" class="mjx-mrow"><span id="MJXc-Node-305" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-306" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-307" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.471em 0px; border-bottom: 1.3px solid; width: 1.5em;"><span id="MJXc-Node-308" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-309" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-310" class="mjx-texatom"><span id="MJXc-Node-311" class="mjx-mrow"><span id="MJXc-Node-312" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-313" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-314" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.471em 0px; border-bottom: 1.3px solid; width: 1.5em;"><span id="MJXc-Node-315" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-316" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-317" class="mjx-texatom"><span id="MJXc-Node-318" class="mjx-mrow"><span id="MJXc-Node-319" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-320" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-321" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.471em 0px; border-bottom: 1.3px solid; width: 1.5em;"><span id="MJXc-Node-322" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-323" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-324" class="mjx-texatom"><span id="MJXc-Node-325" class="mjx-mrow"><span id="MJXc-Node-326" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-327" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-328" class="mjx-mtd" style="padding: 0.221em 0.4em 0px 0.471em; border-bottom: 1.3px solid; width: 1.5em;"><span id="MJXc-Node-329" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-330" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-331" class="mjx-texatom"><span id="MJXc-Node-332" class="mjx-mrow"><span id="MJXc-Node-333" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-334" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-335" class="mjx-mtr" style="height: 1.801em;"><span id="MJXc-Node-336" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.471em 0px 0.4em;"><span id="MJXc-Node-337" class="mjx-mrow" style="margin-top: -0.052em;"><span id="MJXc-Node-338" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-339" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-340" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-341" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-342" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-343" class="mjx-texatom"><span id="MJXc-Node-344" class="mjx-mrow"><span id="MJXc-Node-345" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-346" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-347" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-348" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.471em 0px;"><span id="MJXc-Node-349" class="mjx-mrow" style="margin-top: -0.052em;"><span id="MJXc-Node-350" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.318em;"><span id="MJXc-Node-351" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.611em;"><span id="MJXc-Node-352" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-353" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.471em 0px;"><span id="MJXc-Node-354" class="mjx-mrow" style="margin-top: -0.052em;"><span id="MJXc-Node-355" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.318em;"><span id="MJXc-Node-356" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.611em;"><span id="MJXc-Node-357" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-358" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.471em 0px;"><span id="MJXc-Node-359" class="mjx-mrow" style="margin-top: -0.052em;"><span id="MJXc-Node-360" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.339em;"><span id="MJXc-Node-361" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.611em;"><span id="MJXc-Node-362" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.379em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-363" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.471em 0px;"><span id="MJXc-Node-364" class="mjx-mrow" style="margin-top: -0.052em;"><span id="MJXc-Node-365" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.329em;"><span id="MJXc-Node-366" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.611em;"><span id="MJXc-Node-367" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.372em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-368" class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.471em 0px;"><span id="MJXc-Node-369" class="mjx-mrow" style="margin-top: -0.052em;"><span id="MJXc-Node-370" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.34em;"><span id="MJXc-Node-371" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.611em;"><span id="MJXc-Node-372" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.38em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-373" class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.471em;"><span id="MJXc-Node-374" class="mjx-mrow" style="margin-top: -0.052em;"><span id="MJXc-Node-375" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.34em;"><span id="MJXc-Node-376" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.611em;"><span id="MJXc-Node-377" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.38em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="center center center center center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid" columnlines="solid solid solid solid solid solid"><mtr><mtd><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>k</mi></mtd><mtd><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>1</mn></mtd><mtd><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mtd><mtd><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn></mtd><mtd><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>4</mn></mtd><mtd><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>5</mn></mtd><mtd><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>6</mn></mtd></mtr><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>k</mi><mo stretchy="false">)</mo></mtd><mtd><mfrac><mn>1</mn><mn>21</mn></mfrac></mtd><mtd><mfrac><mn>2</mn><mn>21</mn></mfrac></mtd><mtd><mfrac><mn>3</mn><mn>21</mn></mfrac></mtd><mtd><mfrac><mn>4</mn><mn>21</mn></mfrac></mtd><mtd><mfrac><mn>5</mn><mn>21</mn></mfrac></mtd><mtd><mfrac><mn>6</mn><mn>21</mn></mfrac></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-21">\begin{array}{c|c|c|c|c|c|c}
1/k&1/1&1/2&1/3&1/4&1/5&1/6\\
\hline
P(Y=1/k)&\frac{1}{21}&\frac{2}{21}&\frac{3}{21}&\frac{4}{21}&\frac{5}{21}&\frac{6}{21}
\end{array}</script>
Le calcul de l'espérance n'est pas plus difficile, et donne :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-22-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-378" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-379" class="mjx-mrow"><span id="MJXc-Node-380" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-381" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-382" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-383" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-384" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-385" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-386" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.74em;"><span id="MJXc-Node-387" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.108em; vertical-align: -0.74em;"></span></span><span id="MJXc-Node-388" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>2</mn><mn>7</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-22">E(Y)=\frac{2}{7}.</script>
Attention à l'erreur suivante : ce n'est pas parce que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-23-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-389" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-390" class="mjx-mrow"><span id="MJXc-Node-391" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-392" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-393" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-394" class="mjx-texatom"><span id="MJXc-Node-395" class="mjx-mrow"><span id="MJXc-Node-396" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-397" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-23">Y=1/X</script> que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-24-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-398" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-399" class="mjx-mrow"><span id="MJXc-Node-400" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-401" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-402" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-403" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-404" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-405" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-406" class="mjx-texatom"><span id="MJXc-Node-407" class="mjx-mrow"><span id="MJXc-Node-408" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-409" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-410" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-411" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-412" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-413" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-414" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-415" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>!</mo><mo>!</mo><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-24">E(Y)=1/E(X)!!!</script>.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>2</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - En plein dans le mille!<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2257"> [Signaler une erreur]</a><a id="exo2" href="javascript:ajouteexo(2257,2,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2257&#39;,&#39;imageenonce2257&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2257"></a><div class="inner" id="innerenonce2257">Un joueur tire sur une cible de 10cm de rayon, constituée de couronnes concentriques,
délimitées par des cercles de rayons 1,2, ..., 10 cm, et numérotées respectivement de 10 à 1. La
probabilité d’atteindre la couronne <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-25-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-416" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-417" class="mjx-mrow"><span id="MJXc-Node-418" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-25">k</script> est proportionnelle à l’aire de cette couronne, et on suppose que
le joueur atteint sa cible à chaque lancer. Soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-26-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-419" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-420" class="mjx-mrow"><span id="MJXc-Node-421" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-26">X</script> la variable aléatoire qui à chaque lancer associe le numéro de la cible.
<ol class="enumeratechiffre">
<li> Quelle est la loi de probabilité de X ?
</li><li> Le joueur gagne <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-27-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-422" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-423" class="mjx-mrow"><span id="MJXc-Node-424" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-27">k</script> euros s’il atteint la couronne numérotée <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-28-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-425" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-426" class="mjx-mrow"><span id="MJXc-Node-427" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-28">k</script> pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-29-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-428" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-429" class="mjx-mrow"><span id="MJXc-Node-430" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-29">k</script> compris entre 6 et 10, tandis qu’il perd 2 euros s’il atteint l’une des couronnes périphériques numérotées de 1 à 5. Le jeu est-il
favorable au joueur ?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2257&#39;,&#39;imageindication2257&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2257"></a><div class="inner" id="innerindication2257" style="display:none;"><ol class="enumeratechiffre">
<li> Calculer l'aire de chaque couronne.
</li><li> Introduire la variable aléatoire égale au gain algébrique du joueur, puis calculer l'espérance de cette variable aléatoire.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2257&#39;,&#39;imagecorrige2257&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2257"></a><div class="inner" id="innercorrige2257" style="display:none;"><ol class="enumeratechiffre">
<li> On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-30-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-431" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-432" class="mjx-mrow"><span id="MJXc-Node-433" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-434" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-435" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>k</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-30">A_k</script> l'aire de la couronne <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-31-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-436" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-437" class="mjx-mrow"><span id="MJXc-Node-438" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-31">k</script>, et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-32-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-439" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-440" class="mjx-mrow"><span id="MJXc-Node-441" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-32">A</script> l'aire totale. Par les hypothèses d'équiprobabilité faites dans l'énoncé, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-33-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-442" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-443" class="mjx-mrow"><span id="MJXc-Node-444" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-445" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-446" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-447" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-448" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-449" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-450" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-451" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.354em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.354em; top: -1.418em;"><span id="MJXc-Node-452" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-453" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-454" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span><span class="mjx-denominator" style="width: 1.354em; bottom: -0.758em;"><span id="MJXc-Node-455" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.354em;"></span></span><span class="mjx-vsize" style="height: 2.176em; vertical-align: -0.758em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><msub><mi>A</mi><mi>k</mi></msub><mi>A</mi></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-33">P(X=k)=\frac{A_k}{A}</script>
pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-34-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-456" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-457" class="mjx-mrow"><span id="MJXc-Node-458" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-34">k</script> compris entre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-35-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-459" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-460" class="mjx-mrow"><span id="MJXc-Node-461" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-35">1</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-36-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-462" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-463" class="mjx-mrow"><span id="MJXc-Node-464" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math></span></span><script type="math/tex" id="MathJax-Element-36">10</script>. Pour la couronne <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-37-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-465" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-466" class="mjx-mrow"><span id="MJXc-Node-467" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-37">k</script>, le cercle extérieur est de rayon <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-38-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;11&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-468" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-469" class="mjx-mrow"><span id="MJXc-Node-470" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">11</span></span><span id="MJXc-Node-471" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-472" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn><mo>−</mo><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-38">11-k</script> et le cercle intérieur de rayon <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-39-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-473" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-474" class="mjx-mrow"><span id="MJXc-Node-475" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-476" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-477" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>−</mo><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-39">10-k</script> (un petit dessin pourra aider pour faire attention à l'inversion entre l'ordre des cercles et les rayons, et il faut aussi bien faire attention à ce que pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-40-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-478" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-479" class="mjx-mrow"><span id="MJXc-Node-480" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-481" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-482" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-40">k=1</script>, c'est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-41-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;11&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-483" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-484" class="mjx-mrow"><span id="MJXc-Node-485" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">11</span></span><span id="MJXc-Node-486" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-487" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-488" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-489" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn><mo>−</mo><mi>k</mi><mo>=</mo><mn>10</mn></math></span></span><script type="math/tex" id="MathJax-Element-41">11-k=10</script> qui donne le rayon de la cible). On a donc, en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-42-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-490" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-491" class="mjx-mrow"><span id="MJXc-Node-492" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span><span id="MJXc-Node-493" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-494" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-495" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><msup><mi>m</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-42">cm^2</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-43-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;11&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-496" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-497" class="mjx-mrow"><span id="MJXc-Node-498" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-499" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-500" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-501" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-502" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-503" class="mjx-texatom"><span id="MJXc-Node-504" class="mjx-mrow"><span id="MJXc-Node-505" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-506" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-507" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">11</span></span><span id="MJXc-Node-508" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-509" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-510" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-511" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-512" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-513" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-514" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-515" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-516" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-517" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-518" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-519" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-520" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-521" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-522" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-523" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-524" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-525" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span><span id="MJXc-Node-526" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-527" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-528" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-529" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>k</mi></msub><mo>=</mo><mi>π</mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo stretchy="false">(</mo><mn>11</mn><mo>−</mo><mi>k</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><mo stretchy="false">(</mo><mn>10</mn><mo>−</mo><mi>k</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mi>π</mi><mo stretchy="false">(</mo><mn>21</mn><mo>−</mo><mn>2</mn><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-43">A_k=\pi\big((11-k)^2-(10-k)^2)=\pi(21-2k)</script>. On en déduit que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-44-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mn&gt;100&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-530" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-531" class="mjx-mrow"><span id="MJXc-Node-532" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-533" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-534" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-535" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-536" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-537" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-538" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-539" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.443em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.443em; top: -1.396em;"><span id="MJXc-Node-540" class="mjx-mrow"><span id="MJXc-Node-541" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span><span id="MJXc-Node-542" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-543" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-544" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="mjx-denominator" style="width: 3.443em; bottom: -0.733em;"><span id="MJXc-Node-545" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">100</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.443em;"></span></span><span class="mjx-vsize" style="height: 2.128em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-546" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>21</mn><mo>−</mo><mn>2</mn><mi>k</mi></mrow><mn>100</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-44">P(X=k)=\frac{21-2k}{100}.</script>
</li><li> Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-45-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-547" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-548" class="mjx-mrow"><span id="MJXc-Node-549" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-45">Y</script> la variable aléatoire correspondant au gain algébrique du joueur. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-46-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-550" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-551" class="mjx-mrow"><span id="MJXc-Node-552" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-46">Y</script> prend ses valeurs dans <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-47-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-553" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-554" class="mjx-mrow"><span id="MJXc-Node-555" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-556" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-557" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-558" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-559" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-560" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-561" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-562" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-563" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-564" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-565" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-566" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>9</mn><mo>,</mo><mn>10</mn></math></span></span><script type="math/tex" id="MathJax-Element-47">-2,6,7,8,9,10</script>. L'événement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-48-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;&amp;quot;&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;&amp;quot;&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-567" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-568" class="mjx-mrow"><span id="MJXc-Node-569" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.295em;">"</span></span><span id="MJXc-Node-570" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-571" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-572" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-573" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-574" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.295em;">"</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>"</mo><mi>Y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mo>"</mo></math></span></span><script type="math/tex" id="MathJax-Element-48">"Y=-2"</script> est égal à l'événement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-49-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;&amp;quot;&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;&amp;quot;&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-575" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-576" class="mjx-mrow"><span id="MJXc-Node-577" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.295em;">"</span></span><span id="MJXc-Node-578" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-579" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-580" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-581" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.295em;">"</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>"</mo><mi>X</mi><mo>≤</mo><mn>5</mn><mo>"</mo></math></span></span><script type="math/tex" id="MathJax-Element-49">"X\leq 5"</script>, et donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-50-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;75&lt;/mn&gt;&lt;mn&gt;100&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-582" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-583" class="mjx-mrow"><span id="MJXc-Node-584" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-585" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-586" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-587" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-588" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-589" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-590" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-591" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-592" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-593" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-594" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-595" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-596" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-597" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-598" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-599" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-600" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-601" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-602" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-603" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-604" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-605" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-606" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-607" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-608" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-609" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-610" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-611" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-612" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-613" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-614" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-615" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-616" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-617" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-618" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-619" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-620" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-621" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-622" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-623" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-624" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-625" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-626" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-627" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.7em; top: -1.378em;"><span id="MJXc-Node-628" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">75</span></span></span><span class="mjx-denominator" style="width: 1.7em; bottom: -0.733em;"><span id="MJXc-Node-629" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">100</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.7em;"></span></span><span class="mjx-vsize" style="height: 2.11em; vertical-align: -0.733em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>4</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>5</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>75</mn><mn>100</mn></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-50">P(Y=-2)=P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)=\frac{75}{100}</script>
d'après le résultat de la question précédente.
D'autre part, pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-51-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-630" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-631" class="mjx-mrow"><span id="MJXc-Node-632" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-633" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-634" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-635" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-636" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-637" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-638" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>6</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>10</mn></math></span></span><script type="math/tex" id="MathJax-Element-51">k=6,\dots,10</script>, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-52-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-639" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-640" class="mjx-mrow"><span id="MJXc-Node-641" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-642" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-643" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-644" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-645" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-646" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-647" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-648" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-649" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-650" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-651" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-652" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-653" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-654" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-52">P(Y=k)=P(X=k).</script>
On en déduit le calcul de l'espérance de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-53-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-655" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-656" class="mjx-mrow"><span id="MJXc-Node-657" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-53">Y</script> :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-54-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;75&lt;/mn&gt;&lt;mn&gt;100&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;100&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-658" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-659" class="mjx-mrow"><span id="MJXc-Node-660" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-661" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-662" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-663" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-664" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-665" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-666" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-667" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-668" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.7em; top: -1.378em;"><span id="MJXc-Node-669" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">75</span></span></span><span class="mjx-denominator" style="width: 1.7em; bottom: -0.733em;"><span id="MJXc-Node-670" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">100</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.7em;"></span></span><span class="mjx-vsize" style="height: 2.11em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-671" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-672" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 16.702em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 16.702em; top: -1.378em;"><span id="MJXc-Node-673" class="mjx-mrow"><span id="MJXc-Node-674" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-675" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-676" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-677" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-678" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-679" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-680" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-681" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-682" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-683" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-684" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-685" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-686" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-687" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-688" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-689" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-690" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-691" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-692" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="width: 16.702em; bottom: -0.733em;"><span id="MJXc-Node-693" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">100</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 16.702em;"></span></span><span class="mjx-vsize" style="height: 2.11em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-694" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-695" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.367em;"><span id="MJXc-Node-696" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.733em;"><span id="MJXc-Node-697" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.099em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-698" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>2</mn><mo>×</mo><mfrac><mn>75</mn><mn>100</mn></mfrac><mo>+</mo><mfrac><mrow><mn>6</mn><mo>×</mo><mn>9</mn><mo>+</mo><mn>7</mn><mo>×</mo><mn>7</mn><mo>+</mo><mn>8</mn><mo>×</mo><mn>5</mn><mo>+</mo><mn>9</mn><mo>×</mo><mn>3</mn><mo>+</mo><mn>10</mn><mo>×</mo><mn>1</mn></mrow><mn>100</mn></mfrac><mo>=</mo><mfrac><mn>3</mn><mn>10</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-54">E(Y)=-2\times\frac{75}{100}+\frac{6\times 9+7\times 7+8\times 5+9\times 3+10\times 1}{100}=\frac{3}{10}.</script>
L'espérance est positive. Le jeu est favorable au joueur. En moyenne, il peut espérer gagne 0,3 euros par partie.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>3</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Plus grand nombre tiré<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2253"> [Signaler une erreur]</a><a id="exo3" href="javascript:ajouteexo(2253,3,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2253&#39;,&#39;imageenonce2253&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2253"></a><div class="inner" id="innerenonce2253">On lance deux dés parfaitement équilibrés. On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-55-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-699" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-700" class="mjx-mrow"><span id="MJXc-Node-701" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-55">X</script> le plus grand des numéros obtenus. Déterminer la loi de la variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-56-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-702" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-703" class="mjx-mrow"><span id="MJXc-Node-704" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-56">X</script>.</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2253&#39;,&#39;imageindication2253&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2253"></a><div class="inner" id="innerindication2253" style="display:none;">Définir un univers adapté à l'expérience aléatoire, et écrire les événements <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-57-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-705" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-706" class="mjx-mrow"><span id="MJXc-Node-707" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-708" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-709" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-57">X=k</script> en fonction des événements élémentaires.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2253&#39;,&#39;imagecorrige2253&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2253"></a><div class="inner" id="innercorrige2253" style="display:none;">On choisit pour univers <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-58-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x03A9;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;msup&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-710" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-711" class="mjx-mrow"><span id="MJXc-Node-712" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">Ω</span></span><span id="MJXc-Node-713" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-714" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-715" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-716" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-717" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-718" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-719" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-720" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-721" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-722" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Ω</mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>6</mn><msup><mo fence="false" stretchy="false">}</mo><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-58">\Omega=\{1,\dots,6\}^2</script> que l'on munit de la probabilité uniforme puisque les deux dés sont équilibrés. Remarquons que le cardinal de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-59-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x03A9;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-723" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-724" class="mjx-mrow"><span id="MJXc-Node-725" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">Ω</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Ω</mi></math></span></span><script type="math/tex" id="MathJax-Element-59">\Omega</script> vaut <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-60-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-726" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-727" class="mjx-mrow"><span id="MJXc-Node-728" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>36</mn></math></span></span><script type="math/tex" id="MathJax-Element-60">36</script>. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-61-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-729" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-730" class="mjx-mrow"><span id="MJXc-Node-731" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-61">X</script> prend ses valeurs dans <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-62-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-732" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-733" class="mjx-mrow"><span id="MJXc-Node-734" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-735" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-736" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-737" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-738" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-739" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-740" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>6</mn><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-62">\{1,\dots,6\}</script>. L'événement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-63-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-741" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-742" class="mjx-mrow"><span id="MJXc-Node-743" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-744" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-745" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-63">X=1</script> est égal à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-64-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-746" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-747" class="mjx-mrow"><span id="MJXc-Node-748" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-749" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-750" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-751" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-752" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-753" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-754" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-64">\{(1,1)\}</script> et donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-65-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-755" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-756" class="mjx-mrow"><span id="MJXc-Node-757" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-758" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-759" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-760" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-761" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-762" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-763" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-764" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.368em;"><span id="MJXc-Node-765" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.733em;"><span id="MJXc-Node-766" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-767" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>36</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-65">P(X=1)=\frac{1}{36}.</script>
De même, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-66-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0 thickmathspace&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;11&lt;/mn&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-768" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-769" class="mjx-mrow"><span id="MJXc-Node-770" class="mjx-mtable" style="vertical-align: -5.588em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-771" class="mjx-mtr" style="height: 2.228em;"><span id="MJXc-Node-772" class="mjx-mtd" style="padding: 0px; text-align: right; width: 4.215em;"><span id="MJXc-Node-773" class="mjx-mrow" style="margin-top: 0.368em;"><span id="MJXc-Node-774" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-775" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-776" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-777" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-778" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-779" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-780" class="mjx-mtd" style="padding: 0px 0.139em 0px 0px; width: 1.056em;"><span id="MJXc-Node-781" class="mjx-mrow" style="margin-top: 0.368em;"><span id="MJXc-Node-782" class="mjx-mi"></span><span id="MJXc-Node-783" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-784" class="mjx-mtd" style="padding: 0px 0px 0px 0.139em; text-align: left; width: 34.615em;"><span id="MJXc-Node-785" class="mjx-mrow"><span id="MJXc-Node-786" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-787" class="mjx-texatom"><span id="MJXc-Node-788" class="mjx-mrow"><span id="MJXc-Node-789" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-790" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-791" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-792" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-793" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-794" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-795" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-796" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-797" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-798" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-799" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-800" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-801" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-802" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-803" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-804" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-805" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-806" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-807" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-808" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span><span id="MJXc-Node-809" class="mjx-texatom"><span id="MJXc-Node-810" class="mjx-mrow"><span id="MJXc-Node-811" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span><span id="MJXc-Node-812" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-813" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.368em;"><span id="MJXc-Node-814" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.711em;"><span id="MJXc-Node-815" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">12</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-816" class="mjx-mtr" style="height: 2.4em;"><span id="MJXc-Node-817" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-818" class="mjx-mrow" style="margin-top: 0.368em;"><span id="MJXc-Node-819" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-820" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-821" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-822" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-823" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-824" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-825" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-826" class="mjx-mrow" style="margin-top: 0.368em;"><span id="MJXc-Node-827" class="mjx-mi"></span><span id="MJXc-Node-828" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-829" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-830" class="mjx-mrow"><span id="MJXc-Node-831" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-832" class="mjx-texatom"><span id="MJXc-Node-833" class="mjx-mrow"><span id="MJXc-Node-834" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-835" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-836" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-837" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-838" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-839" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-840" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-841" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-842" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-843" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-844" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-845" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-846" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-847" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-848" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-849" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-850" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-851" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-852" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-853" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-854" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-855" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-856" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-857" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-858" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-859" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-860" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-861" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-862" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-863" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-864" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-865" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span><span id="MJXc-Node-866" class="mjx-texatom"><span id="MJXc-Node-867" class="mjx-mrow"><span id="MJXc-Node-868" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span><span id="MJXc-Node-869" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-870" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.368em;"><span id="MJXc-Node-871" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.733em;"><span id="MJXc-Node-872" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-873" class="mjx-mtr" style="height: 2.41em;"><span id="MJXc-Node-874" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-875" class="mjx-mrow" style="margin-top: 0.378em;"><span id="MJXc-Node-876" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-877" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-878" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-879" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-880" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-881" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-882" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-883" class="mjx-mrow" style="margin-top: 0.378em;"><span id="MJXc-Node-884" class="mjx-mi"></span><span id="MJXc-Node-885" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-886" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-887" class="mjx-mrow"><span id="MJXc-Node-888" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-889" class="mjx-texatom"><span id="MJXc-Node-890" class="mjx-mrow"><span id="MJXc-Node-891" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-892" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-893" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-894" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-895" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-896" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-897" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-898" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-899" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-900" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-901" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-902" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-903" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-904" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-905" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-906" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-907" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-908" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-909" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-910" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-911" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-912" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-913" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-914" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-915" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-916" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-917" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-918" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-919" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-920" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-921" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-922" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-923" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-924" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-925" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-926" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-927" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-928" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-929" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-930" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-931" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-932" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-933" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-934" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span><span id="MJXc-Node-935" class="mjx-texatom"><span id="MJXc-Node-936" class="mjx-mrow"><span id="MJXc-Node-937" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span><span id="MJXc-Node-938" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-939" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.378em;"><span id="MJXc-Node-940" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.733em;"><span id="MJXc-Node-941" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.11em; vertical-align: -0.733em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-942" class="mjx-mtr" style="height: 2.387em;"><span id="MJXc-Node-943" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-944" class="mjx-mrow" style="margin-top: 0.368em;"><span id="MJXc-Node-945" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-946" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-947" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-948" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-949" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-950" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-951" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-952" class="mjx-mrow" style="margin-top: 0.368em;"><span id="MJXc-Node-953" class="mjx-mi"></span><span id="MJXc-Node-954" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-955" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-956" class="mjx-mrow"><span id="MJXc-Node-957" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-958" class="mjx-texatom"><span id="MJXc-Node-959" class="mjx-mrow"><span id="MJXc-Node-960" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-961" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-962" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-963" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-964" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-965" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-966" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-967" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-968" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-969" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-970" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-971" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-972" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-973" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-974" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-975" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-976" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-977" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-978" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-979" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-980" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-981" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-982" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-983" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-984" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-985" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-986" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-987" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-988" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-989" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-990" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-991" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-992" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-993" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-994" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-995" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-996" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-997" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-998" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-999" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-1000" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1001" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-1002" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1003" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-1004" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1005" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-1006" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1007" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1008" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1009" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-1010" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1011" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-1012" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1013" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1014" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1015" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span><span id="MJXc-Node-1016" class="mjx-texatom"><span id="MJXc-Node-1017" class="mjx-mrow"><span id="MJXc-Node-1018" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span><span id="MJXc-Node-1019" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1020" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-1021" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.719em;"><span id="MJXc-Node-1022" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.087em; vertical-align: -0.719em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-1023" class="mjx-mtr" style="height: 2.25em;"><span id="MJXc-Node-1024" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-1025" class="mjx-mrow" style="margin-top: 0.368em;"><span id="MJXc-Node-1026" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1027" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1028" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1029" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1030" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1031" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-1032" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-1033" class="mjx-mrow" style="margin-top: 0.368em;"><span id="MJXc-Node-1034" class="mjx-mi"></span><span id="MJXc-Node-1035" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-1036" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-1037" class="mjx-mrow"><span id="MJXc-Node-1038" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1039" class="mjx-texatom"><span id="MJXc-Node-1040" class="mjx-mrow"><span id="MJXc-Node-1041" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-1042" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-1043" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1044" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1045" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1046" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1047" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1048" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-1049" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1050" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1051" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1052" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1053" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1054" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-1055" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1056" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-1057" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1058" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1059" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1060" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-1061" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1062" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1063" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1064" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1065" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1066" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-1067" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1068" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-1069" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1070" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1071" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1072" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-1073" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1074" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1075" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1076" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1077" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1078" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-1079" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1080" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1081" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1082" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-1083" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1084" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-1085" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1086" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1087" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1088" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1089" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1090" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-1091" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1092" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1093" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1094" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-1095" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1096" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-1097" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1098" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1099" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1100" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1101" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1102" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.12em; padding-bottom: 0.528em;">;</span></span><span id="MJXc-Node-1103" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1104" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1105" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1106" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1107" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1108" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span><span id="MJXc-Node-1109" class="mjx-texatom"><span id="MJXc-Node-1110" class="mjx-mrow"><span id="MJXc-Node-1111" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span><span id="MJXc-Node-1112" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1113" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.368em;"><span id="MJXc-Node-1114" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">11</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.733em;"><span id="MJXc-Node-1115" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-1116" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo fence="false" stretchy="false">{</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>12</mn></mfrac></mtd></mtr><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo fence="false" stretchy="false">{</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>=</mo><mfrac><mn>5</mn><mn>36</mn></mfrac></mtd></mtr><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>4</mn><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo fence="false" stretchy="false">{</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>=</mo><mfrac><mn>7</mn><mn>36</mn></mfrac></mtd></mtr><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>5</mn><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo fence="false" stretchy="false">{</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>6</mn><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo fence="false" stretchy="false">{</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>6</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>6</mn><mo>,</mo><mn>6</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>6</mn><mo>,</mo><mn>5</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>6</mn><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>6</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>6</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>;</mo><mo stretchy="false">(</mo><mn>6</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>=</mo><mfrac><mn>11</mn><mn>36</mn></mfrac><mo>.</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-66">\begin{eqnarray*}
P(X=2)&=&P\big(\{(1,2);(2,1);(2,2)\}\big)=\frac 1{12}\\
P(X=3)&=&P\big(\{(1,3);(2,3);(3,3);(3,1);(3,2)\}\big)=\frac 5{36}\\
P(X=4)&=&P\big(\{(1,4);(2,4);(3,4);(4,4);(4,3);(4,2);(4,1)\}\big)=\frac7{36}\\
P(X=5)&=&P\big(\{(1,5);(2,5);(3,5);(4,5);(5,5);(5,4);(5,3);(5,2);(5,1)\}\big)=\frac 14\\
P(X=6)&=&P\big(\{(1,6);(2,6);(3,6);(4,6);(5,6);(6,6);(6,5);(6,4);(6,3);(6,2);(6,1)\}\big)=\frac{11}{36}.
\end{eqnarray*}</script></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>4</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Garagiste<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1254"> [Signaler une erreur]</a><a id="exo4" href="javascript:ajouteexo(1254,4,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1254&#39;,&#39;imageenonce1254&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce1254"></a><div class="inner" id="innerenonce1254">Un garagiste dispose de deux voitures de location. Chacune est utilisable en moyenne 4 jours sur 5. Il loue les voitures avec une marge brute de 300 euros par jour et par voiture.
On considère <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-67-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1117" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1118" class="mjx-mrow"><span id="MJXc-Node-1119" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-67">X</script> la variable aléatoire égale au nombre de clients se présentant chaque jour pour louer une voiture. On suppose que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-68-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x03A9;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1120" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1121" class="mjx-mrow"><span id="MJXc-Node-1122" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1123" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-1124" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">Ω</span></span><span id="MJXc-Node-1125" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-1126" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-1127" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">{</span></span><span id="MJXc-Node-1128" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-1129" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-1130" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-1131" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-1132" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span><span id="MJXc-Node-1133" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-1134" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span><span id="MJXc-Node-1135" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-68">X(\Omega)=\{0,1,2,3\}</script> avec
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-69-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1136" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1137" class="mjx-mrow"><span id="MJXc-Node-1138" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1139" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-1140" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1141" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-1142" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-1143" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-1144" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-1145" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-1146" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-1147" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-1148" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.256em; padding-bottom: 0.328em;">&nbsp;</span></span><span id="MJXc-Node-1149" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.256em; padding-bottom: 0.328em;">&nbsp;</span></span><span id="MJXc-Node-1150" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1151" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-1152" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1153" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-1154" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-1155" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-1156" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-1157" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-1158" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-1159" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span><span id="MJXc-Node-1160" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.256em; padding-bottom: 0.328em;">&nbsp;</span></span><span id="MJXc-Node-1161" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.256em; padding-bottom: 0.328em;">&nbsp;</span></span><span id="MJXc-Node-1162" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1163" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-1164" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1165" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-1166" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span><span id="MJXc-Node-1167" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-1168" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-1169" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-1170" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-1171" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span><span id="MJXc-Node-1172" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.256em; padding-bottom: 0.328em;">&nbsp;</span></span><span id="MJXc-Node-1173" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.256em; padding-bottom: 0.328em;">&nbsp;</span></span><span id="MJXc-Node-1174" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1175" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-1176" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1177" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-1178" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span><span id="MJXc-Node-1179" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-1180" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-1181" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-1182" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-1183" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mtext>&nbsp;</mtext><mtext>&nbsp;</mtext><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>3</mn><mtext>&nbsp;</mtext><mtext>&nbsp;</mtext><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>4</mn><mtext>&nbsp;</mtext><mtext>&nbsp;</mtext><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>2.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-69">P(X=0)=0,1\ \ P(X=1)=0,3\ \ P(X=2)=0,4\ \ P(X=3)=0,2.</script>
<ol class="enumeratechiffre">
<li> On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-70-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1184" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1185" class="mjx-mrow"><span id="MJXc-Node-1186" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.04em;">Z</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math></span></span><script type="math/tex" id="MathJax-Element-70">Z</script> le nombre de voitures disponibles par jour. Déterminer la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-71-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1187" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1188" class="mjx-mrow"><span id="MJXc-Node-1189" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.04em;">Z</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math></span></span><script type="math/tex" id="MathJax-Element-71">Z</script>. On pourra considérer dans la suite que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-72-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1190" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1191" class="mjx-mrow"><span id="MJXc-Node-1192" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-72">X</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-73-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1193" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1194" class="mjx-mrow"><span id="MJXc-Node-1195" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.04em;">Z</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math></span></span><script type="math/tex" id="MathJax-Element-73">Z</script> sont indépendantes.
</li><li> On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-74-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1196" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1197" class="mjx-mrow"><span id="MJXc-Node-1198" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-74">Y</script> la variable aléatoire : " nombre de clients satisfaits par jour". Déterminer la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-75-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1199" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1200" class="mjx-mrow"><span id="MJXc-Node-1201" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-75">Y</script>.
</li><li> Calculer la marge brute moyenne par jour.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1254&#39;,&#39;imageindication1254&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication1254"></a><div class="inner" id="innerindication1254" style="display:none;"><ol class="enumeratechiffre">
<li> Quel est l'ensemble des valeurs prises par <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-76-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1202" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1203" class="mjx-mrow"><span id="MJXc-Node-1204" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math></span></span><script type="math/tex" id="MathJax-Element-76">Z</script>. En déduire la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-77-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1205" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1206" class="mjx-mrow"><span id="MJXc-Node-1207" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math></span></span><script type="math/tex" id="MathJax-Element-77">Z</script> par un simple calcul de probabilité.
</li><li> Calculer la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-78-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1208" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1209" class="mjx-mrow"><span id="MJXc-Node-1210" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-78">Y</script> en utilisant la formule des probabilités totales.
</li><li> On cherche simplement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-79-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;300&lt;/mn&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1211" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1212" class="mjx-mrow"><span id="MJXc-Node-1213" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">300</span></span><span id="MJXc-Node-1214" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-1215" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1216" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-1217" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-79">300E(Y)</script>.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1254&#39;,&#39;imagecorrige1254&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige1254"></a><div class="inner" id="innercorrige1254" style="display:none;"><ol class="enumeratechiffre">
<li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-80-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1218" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1219" class="mjx-mrow"><span id="MJXc-Node-1220" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math></span></span><script type="math/tex" id="MathJax-Element-80">Z</script> est élément de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-81-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1221" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1222" class="mjx-mrow"><span id="MJXc-Node-1223" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-1224" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1225" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1226" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1227" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1228" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1229" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-81">\{0,1,2\}</script>. On a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-82-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;mn&gt;25&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1230" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1231" class="mjx-mrow"><span id="MJXc-Node-1232" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1233" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1234" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-1235" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1236" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1237" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1238" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1239" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.379em;"><span id="MJXc-Node-1240" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-1241" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.111em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-1242" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1243" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.379em;"><span id="MJXc-Node-1244" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-1245" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.111em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-1246" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1247" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.368em;"><span id="MJXc-Node-1248" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">16</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.733em;"><span id="MJXc-Node-1249" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">25</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>4</mn><mn>5</mn></mfrac><mo>×</mo><mfrac><mn>4</mn><mn>5</mn></mfrac><mo>=</mo><mfrac><mn>16</mn><mn>25</mn></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-82">P(Z=2)=\frac{4}{5}\times\frac{4}{5}=\frac{16}{25}</script>
(les deux voitures sont disponibles). D'autre part,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-83-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;25&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1250" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1251" class="mjx-mrow"><span id="MJXc-Node-1252" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1253" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1254" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-1255" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1256" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1257" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1258" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1259" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-1260" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-1261" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-1262" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1263" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-1264" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-1265" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-1266" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1267" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.368em;"><span id="MJXc-Node-1268" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.733em;"><span id="MJXc-Node-1269" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">25</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mo>×</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>25</mn></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-83">P(Z=0)=\frac{1}{5}\times\frac{1}{5}=\frac{1}{25}</script>
(les deux voitures sont simultanément indisponibles). Enfin, on obtient :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-84-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mn&gt;25&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1270" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1271" class="mjx-mrow"><span id="MJXc-Node-1272" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1273" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1274" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-1275" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1276" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1277" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1278" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1279" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1280" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1281" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1282" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1283" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-1284" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1285" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1286" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1287" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1288" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1289" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1290" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-1291" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1292" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1293" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1294" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1295" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.368em;"><span id="MJXc-Node-1296" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.733em;"><span id="MJXc-Node-1297" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">25</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-1298" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>8</mn><mn>25</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-84">P(Z=1)=1-P(Z=0)-P(Z=2)=\frac{8}{25}.</script>
On aurait aussi pu directement remarquer que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-85-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1299" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1300" class="mjx-mrow"><span id="MJXc-Node-1301" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math></span></span><script type="math/tex" id="MathJax-Element-85">Z</script> suit une loi binomiale <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-86-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1302" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1303" class="mjx-mrow"><span id="MJXc-Node-1304" class="mjx-texatom"><span id="MJXc-Node-1305" class="mjx-mrow"><span id="MJXc-Node-1306" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.47em; padding-bottom: 0.353em; padding-right: 0.007em;">B</span></span></span></span><span id="MJXc-Node-1307" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1308" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1309" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1310" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1311" class="mjx-texatom"><span id="MJXc-Node-1312" class="mjx-mrow"><span id="MJXc-Node-1313" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-1314" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-1315" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">B</mi></mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>5</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-86">\mathcal B(2,4/5)</script>.
</li><li> Remarquons que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-87-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1316" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1317" class="mjx-mrow"><span id="MJXc-Node-1318" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-87">Y</script> est à valeurs dans <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-88-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1319" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1320" class="mjx-mrow"><span id="MJXc-Node-1321" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-1322" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1323" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1324" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1325" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1326" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1327" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-88">\{0,1,2\}</script>. On calcule sa loi en utilisant la formule des probabilités totales.
L'événement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-89-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1328" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1329" class="mjx-mrow"><span id="MJXc-Node-1330" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-1331" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1332" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-89">Y=0</script> se produit si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-90-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1333" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1334" class="mjx-mrow"><span id="MJXc-Node-1335" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1336" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1337" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-90">X=0</script> ou bien si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-91-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1338" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1339" class="mjx-mrow"><span id="MJXc-Node-1340" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1341" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-1342" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>≥</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-91">X\geq 1</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-92-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1343" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1344" class="mjx-mrow"><span id="MJXc-Node-1345" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-1346" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1347" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-92">Z=0</script>. Ces deux événements étant disjoints, on a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-93-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1348" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1349" class="mjx-mrow"><span id="MJXc-Node-1350" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1351" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1352" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-1353" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1354" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1355" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1356" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1357" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1358" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1359" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1360" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1361" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1362" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1363" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1364" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1365" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1366" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1367" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-1368" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1369" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-1370" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-1371" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1372" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1373" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1374" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1375" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1376" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1377" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1378" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1379" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1380" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1381" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1382" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1383" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1384" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1385" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-1386" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1387" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1388" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1389" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1390" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-1391" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1392" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1393" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mn>1</mn><mo>∩</mo><mi>Z</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mn>1</mn><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-93">P(Y=0)=P(X=0)+P(X\geq 1\cap Z=0)=P(X=0)+P(X\geq 1)P(Z=0)</script>
(la disponibilité des voitures étant supposée indépendante de l'arrivée des clients).
D'où :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-94-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;136.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1394" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1395" class="mjx-mrow"><span id="MJXc-Node-1396" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1397" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1398" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-1399" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1400" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1401" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1402" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1403" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1404" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1405" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1406" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1407" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1408" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1409" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-1410" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1411" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-1412" class="mjx-mrow"><span id="MJXc-Node-1413" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-1414" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-1415" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-1416" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-1417" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1418" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-1419" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1420" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1421" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1422" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">136.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>+</mo><mn>0</mn><mo>,</mo><mn>9</mn><mo>×</mo><msup><mrow><mo>(</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>,</mo><mn>136.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-94">P(Y=0)=0,1+0,9\times\left(\frac{1}{5}\right)^2= 0,136.</script>
De même, l'événement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-95-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1423" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1424" class="mjx-mrow"><span id="MJXc-Node-1425" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-1426" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1427" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-95">Y=1</script> se produit si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-96-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1428" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1429" class="mjx-mrow"><span id="MJXc-Node-1430" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1431" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1432" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-96">X=1</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-97-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1433" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1434" class="mjx-mrow"><span id="MJXc-Node-1435" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-1436" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-1437" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo>≥</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-97">Z\geq 1</script> ou bien si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-98-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1438" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1439" class="mjx-mrow"><span id="MJXc-Node-1440" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1441" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-1442" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>≥</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-98">X\geq 2</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-99-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1443" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1444" class="mjx-mrow"><span id="MJXc-Node-1445" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-1446" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1447" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-99">Z=1</script>. On en déduit :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-100-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;48.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1448" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1449" class="mjx-mrow"><span id="MJXc-Node-1450" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1451" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1452" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-1453" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1454" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1455" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1456" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1457" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1458" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1459" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1460" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1461" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1462" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1463" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1464" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1465" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-1466" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-1467" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1468" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1469" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1470" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1471" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1472" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1473" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-1474" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1475" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1476" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1477" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1478" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-1479" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1480" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1481" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1482" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1483" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1484" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1485" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">48.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>≥</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mn>2</mn><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>48.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-100">P(Y=1)=P(X=1)P(Z\geq 1)+P(X\geq 2)P(Z=1)= 0,48.</script>
Enfin, l'événement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-101-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1486" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1487" class="mjx-mrow"><span id="MJXc-Node-1488" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-1489" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1490" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-101">Y=2</script> est réalisé si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-102-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1491" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1492" class="mjx-mrow"><span id="MJXc-Node-1493" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1494" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-1495" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>≥</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-102">X\geq 2</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-103-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1496" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1497" class="mjx-mrow"><span id="MJXc-Node-1498" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-1499" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1500" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo>=</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-103">Z=2</script>. Ceci donne :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-104-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;384.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1501" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1502" class="mjx-mrow"><span id="MJXc-Node-1503" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1504" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1505" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-1506" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1507" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1508" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1509" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1510" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1511" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1512" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1513" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-1514" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1515" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1516" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1517" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1518" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-1519" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1520" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1521" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1522" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1523" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1524" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1525" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1526" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1527" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-1528" class="mjx-mrow"><span id="MJXc-Node-1529" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-1530" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.379em;"><span id="MJXc-Node-1531" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-1532" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.111em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-1533" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1534" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-1535" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1536" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1537" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1538" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">384.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mn>2</mn><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>6</mn><mo>×</mo><msup><mrow><mo>(</mo><mfrac><mn>4</mn><mn>5</mn></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>,</mo><mn>384.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-104">P(Y=2)=P(X\geq 2)P(Z=2)=0,6\times\left(\frac{4}{5}\right)^2= 0,384.</script>
</li><li> La marge brute vaut <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-105-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;300&lt;/mn&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1539" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1540" class="mjx-mrow"><span id="MJXc-Node-1541" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">300</span></span><span id="MJXc-Node-1542" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-105">300Y</script>. La marge brute moyenne par jour est en euros :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-106-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;300&lt;/mn&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;300&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;136&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;48&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;384&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;374&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1543" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1544" class="mjx-mrow"><span id="MJXc-Node-1545" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-1546" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1547" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">300</span></span><span id="MJXc-Node-1548" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-1549" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1550" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1551" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">300</span></span><span id="MJXc-Node-1552" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1553" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1554" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1555" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1556" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1557" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">136</span></span><span id="MJXc-Node-1558" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1559" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1560" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1561" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1562" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1563" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">48</span></span><span id="MJXc-Node-1564" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1565" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1566" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1567" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-1568" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1569" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">384</span></span><span id="MJXc-Node-1570" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1571" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1572" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">374</span></span><span id="MJXc-Node-1573" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1574" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><mn>300</mn><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>300</mn><mo stretchy="false">(</mo><mn>0</mn><mo>×</mo><mn>0</mn><mo>,</mo><mn>136</mn><mo>+</mo><mn>1</mn><mo>×</mo><mn>0</mn><mo>,</mo><mn>48</mn><mo>+</mo><mn>2</mn><mo>×</mo><mn>0</mn><mo>,</mo><mn>384</mn><mo stretchy="false">)</mo><mo>=</mo><mn>374</mn><mo>,</mo><mn>4.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-106">E(300Y)= 300(0\times 0,136+1\times 0,48+2\times 0,384)=374,4.</script>
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>5</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Analyse de sang<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2295"> [Signaler une erreur]</a><a id="exo5" href="javascript:ajouteexo(2295,5,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2295&#39;,&#39;imageenonce2295&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2295"></a><div class="inner" id="innerenonce2295">On cherche à dépister une maladie détectable à l'aide d'un examen sanguin. On suppose que dans notre population, il y a une proportion <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-107-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1575" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1576" class="mjx-mrow"><span id="MJXc-Node-1577" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-107">p</script> de personnes qui n'ont pas cette maladie.
<ol class="enumeratechiffre">
<li> On analyse le sang de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-108-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1578" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1579" class="mjx-mrow"><span id="MJXc-Node-1580" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">r</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-108">r</script> personnes de la population, avec <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-109-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1581" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1582" class="mjx-mrow"><span id="MJXc-Node-1583" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">r</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-109">r</script> entier au moins égal à 2. On suppose que l'effectif de la population est suffisamment grand pour que le choix de ces <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-110-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1584" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1585" class="mjx-mrow"><span id="MJXc-Node-1586" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">r</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-110">r</script> personnes s'apparente à un tirage avec remise. Quelle est la probabilité qu'aucune de ces personnes ne soit atteinte de la maladie?
</li><li> On regroupe le sang de ces <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-111-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1587" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1588" class="mjx-mrow"><span id="MJXc-Node-1589" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">r</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-111">r</script> personnes, puis on procède à l'analyse de sang. Si l'analyse est négative, aucune de ces personnes n'est malade et on arrête. Si l'analyse est positive, on fait toutes les analyses individuelles (on avait pris soin de conserver une partie du sang recueilli avant l'analyse groupée). On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-112-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1590" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1591" class="mjx-mrow"><span id="MJXc-Node-1592" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-112">Y</script> la variable aléatoire qui donne le nombre d'analyses de sang effectuées. Donner la loi de probabilité de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-113-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1593" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1594" class="mjx-mrow"><span id="MJXc-Node-1595" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-113">Y</script> et calculer son espérance en fonction de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-114-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1596" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1597" class="mjx-mrow"><span id="MJXc-Node-1598" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">r</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-114">r</script> et de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-115-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1599" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1600" class="mjx-mrow"><span id="MJXc-Node-1601" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-115">p</script>.
</li><li> On s'intéresse à une population de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-116-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1602" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1603" class="mjx-mrow"><span id="MJXc-Node-1604" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-116">n</script> personnes, et on effectue des analyses collectives après avoir mélangé les prélèvements par groupe de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-117-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1605" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1606" class="mjx-mrow"><span id="MJXc-Node-1607" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">r</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-117">r</script> personnes, où <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-118-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1608" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1609" class="mjx-mrow"><span id="MJXc-Node-1610" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">r</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-118">r</script> est un diviseur de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-119-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1611" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1612" class="mjx-mrow"><span id="MJXc-Node-1613" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-119">n</script>. Montrer que le nombre d'analyses que l'on peut espérer économiser, par rapport à la démarche consistant à tester immédiatement toutes les personnes, est égal à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-120-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1614" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1615" class="mjx-mrow"><span id="MJXc-Node-1616" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-1617" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1618" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1619" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">r</span></span></span></span><span id="MJXc-Node-1620" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-1621" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.566em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.8em; top: -1.128em;"><span id="MJXc-Node-1622" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.8em; bottom: -0.524em;"><span id="MJXc-Node-1623" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">r</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.291em; width: 0.566em;"></span></span><span class="mjx-vsize" style="height: 1.168em; vertical-align: -0.37em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><msup><mi>p</mi><mi>r</mi></msup><mo>−</mo><mfrac><mi>n</mi><mi>r</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-120">np^r-\frac nr</script>.
</li><li> Dans cette question, on suppose que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-121-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1624" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1625" class="mjx-mrow"><span id="MJXc-Node-1626" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span><span id="MJXc-Node-1627" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-1628" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-1629" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-1630" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">9</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>9</mn></math></span></span><script type="math/tex" id="MathJax-Element-121">p=0,9</script> et on admet qu'il existe un réel <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-122-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1631" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1632" class="mjx-mrow"><span id="MJXc-Node-1633" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span><span id="MJXc-Node-1634" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">&gt;</span></span><span id="MJXc-Node-1635" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>&gt;</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-122">a>1</script> de sorte que la fonction <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-123-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x21A6;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1636" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1637" class="mjx-mrow"><span id="MJXc-Node-1638" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-1639" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.198em; padding-bottom: 0.328em;">↦</span></span><span id="MJXc-Node-1640" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-1641" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1642" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span></span><span id="MJXc-Node-1643" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-1644" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.546em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.772em; top: -1.341em;"><span id="MJXc-Node-1645" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.772em; bottom: -0.524em;"><span id="MJXc-Node-1646" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.291em; width: 0.546em;"></span></span><span class="mjx-vsize" style="height: 1.319em; vertical-align: -0.37em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo stretchy="false">↦</mo><msup><mi>p</mi><mi>x</mi></msup><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-123">x\mapsto p^x-\frac{1}x</script> est croissante sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-124-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1647" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1648" class="mjx-mrow"><span id="MJXc-Node-1649" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-1650" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-1651" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-1652" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span><span id="MJXc-Node-1653" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mi>a</mi><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-124">[1,a]</script> et décroissante sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-125-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1654" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1655" class="mjx-mrow"><span id="MJXc-Node-1656" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-1657" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span><span id="MJXc-Node-1658" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-1659" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-1660" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.133em; padding-bottom: 0.328em;">∞</span></span><span id="MJXc-Node-1661" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">[</mo></math></span></span><script type="math/tex" id="MathJax-Element-125">[a,+\infty[</script>. Écrire un algorithme permettant de déterminer
pour quelle valeur de l'entier <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-126-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1662" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1663" class="mjx-mrow"><span id="MJXc-Node-1664" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">r</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-126">r</script> le nombre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-127-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1665" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1666" class="mjx-mrow"><span id="MJXc-Node-1667" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1668" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1669" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">r</span></span></span></span><span id="MJXc-Node-1670" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-1671" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.341em;"><span id="MJXc-Node-1672" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.524em;"><span id="MJXc-Node-1673" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">r</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.291em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.319em; vertical-align: -0.37em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mi>r</mi></msup><mo>−</mo><mfrac><mn>1</mn><mi>r</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-127">p^r-\frac 1r</script> est maximal.
</li></ol>
</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2295&#39;,&#39;imageindication2295&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2295"></a><div class="inner" id="innerindication2295" style="display:none;"><ol class="enumeratechiffre">
<li>
</li><li> Quelles sont les valeurs prises par <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-128-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1674" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1675" class="mjx-mrow"><span id="MJXc-Node-1676" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-128">Y</script>?
</li><li> Utiliser la valeur de l'espérance calculée à la question précédente.
</li><li>
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2295&#39;,&#39;imagecorrige2295&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2295"></a><div class="inner" id="innercorrige2295" style="display:none;"><ol class="enumeratechiffre">
<li> On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-129-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1677" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1678" class="mjx-mrow"><span id="MJXc-Node-1679" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1680" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1681" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-129">A_i</script> l'événement "la <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-130-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1682" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1683" class="mjx-mrow"><span id="MJXc-Node-1684" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-130">i</script>-ème personne choisie n'est pas malade". On cherche la probabilité de l'événement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-131-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mo&gt;&amp;#x22EF;&lt;/mo&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1685" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1686" class="mjx-mrow"><span id="MJXc-Node-1687" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1688" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1689" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-1690" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-1691" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-1692" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1693" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-1694" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-1695" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋯</span></span><span id="MJXc-Node-1696" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-1697" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-1698" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1699" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mn>1</mn></msub><mo>∩</mo><msub><mi>A</mi><mn>2</mn></msub><mo>∩</mo><mo>⋯</mo><mo>∩</mo><msub><mi>A</mi><mi>r</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-131">A_1\cap A_2\cap\dots\cap A_r</script>. L'énoncé nous dit que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-132-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1700" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1701" class="mjx-mrow"><span id="MJXc-Node-1702" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1703" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1704" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1705" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1706" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span id="MJXc-Node-1707" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1708" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1709" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-132">P(A_i)=p</script> (proportion <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-133-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1710" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1711" class="mjx-mrow"><span id="MJXc-Node-1712" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-133">p</script> de personnes qui n'ont pas la maladie), et que les événements <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-134-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1713" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1714" class="mjx-mrow"><span id="MJXc-Node-1715" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1716" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1717" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-1718" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1719" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-1720" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1721" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-1722" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1723" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>A</mi><mi>r</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-134">A_1,\dots,A_r</script> sont mutuellement indépendants (le choix de ces <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-135-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1724" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1725" class="mjx-mrow"><span id="MJXc-Node-1726" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-135">r</script> personnes s'apparente à un tirage avec remise). On en déduit donc que la probabilité qu'aucune des <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-136-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1727" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1728" class="mjx-mrow"><span id="MJXc-Node-1729" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-136">r</script> personnes ne soit atteinte de la maladie est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-137-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1730" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1731" class="mjx-mrow"><span id="MJXc-Node-1732" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1733" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1734" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mi>r</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-137">p^r</script>.
</li><li> La variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-138-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1735" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1736" class="mjx-mrow"><span id="MJXc-Node-1737" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-138">Y</script> peut prendre deux valeurs : la valeur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-139-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1738" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1739" class="mjx-mrow"><span id="MJXc-Node-1740" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-139">1</script> (l'analyse collective est négative), et la valeur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-140-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1741" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1742" class="mjx-mrow"><span id="MJXc-Node-1743" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-1744" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1745" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>+</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-140">r+1</script> (l'analyse collective est positive, et il faut donc faire également toutes les analyses individuelles). On a de plus <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-141-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1746" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1747" class="mjx-mrow"><span id="MJXc-Node-1748" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1749" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1750" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-1751" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1752" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1753" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1754" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1755" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-1756" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1757" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mi>p</mi><mi>r</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-141">P(Y=1)=p^r</script> (l'événement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-142-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1758" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1759" class="mjx-mrow"><span id="MJXc-Node-1760" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-1761" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-1762" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1763" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1764" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-142">\{Y=1\}</script> est aussi l'événement aucune des <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-143-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1765" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1766" class="mjx-mrow"><span id="MJXc-Node-1767" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-143">r</script> personnes n'est malade) et donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-144-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1768" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1769" class="mjx-mrow"><span id="MJXc-Node-1770" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-1771" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1772" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-1773" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1774" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-1775" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1776" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1777" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1778" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1779" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1780" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1781" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-1782" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1783" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>p</mi><mi>r</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-144">P(Y=r+1)=1-p^r</script>. On en déduit que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-145-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1784" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1785" class="mjx-mrow"><span id="MJXc-Node-1786" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-1787" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1788" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-1789" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1790" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1791" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-1792" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1793" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span id="MJXc-Node-1794" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1795" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1796" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-1797" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1798" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1799" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1800" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1801" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1802" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1803" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-1804" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1805" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span id="MJXc-Node-1806" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1807" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1808" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1809" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-1810" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1811" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1812" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1813" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1814" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-1815" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1816" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1817" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span id="MJXc-Node-1818" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>p</mi><mi>r</mi></msup><mo>+</mo><mo stretchy="false">(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>p</mi><mi>r</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>r</mi><msup><mi>p</mi><mi>r</mi></msup><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-145">E(Y)=p^r+(r+1)(1-p^r)=(r+1)-rp^r.</script>
</li><li> Si on analyse immédiatement toutes les personnes, on réalise <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-146-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1819" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1820" class="mjx-mrow"><span id="MJXc-Node-1821" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-146">n</script> analyses. En les regroupant par groupe de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-147-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1822" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1823" class="mjx-mrow"><span id="MJXc-Node-1824" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-147">r</script> personnes, notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-148-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1825" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1826" class="mjx-mrow"><span id="MJXc-Node-1827" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math></span></span><script type="math/tex" id="MathJax-Element-148">Z</script> le nombre d'analyses effectuées, et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-149-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1828" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1829" class="mjx-mrow"><span id="MJXc-Node-1830" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.182em;"><span id="MJXc-Node-1831" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1832" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-1833" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1834" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-1835" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-1836" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.182em;"><span id="MJXc-Node-1837" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.275em; padding-right: 0.071em;"><span id="MJXc-Node-1838" class="mjx-texatom" style=""><span id="MJXc-Node-1839" class="mjx-mrow"><span id="MJXc-Node-1840" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-1841" class="mjx-texatom"><span id="MJXc-Node-1842" class="mjx-mrow"><span id="MJXc-Node-1843" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-1844" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>Y</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>r</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-149">Y_1,\dots,Y_{n/r}</script> le nombre d'analyses effectués sur chacun des <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-150-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1845" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1846" class="mjx-mrow"><span id="MJXc-Node-1847" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-1848" class="mjx-texatom"><span id="MJXc-Node-1849" class="mjx-mrow"><span id="MJXc-Node-1850" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-1851" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-150">n/r</script> groupes constitués. Alors d'après la question précédente, on sait que l'espérance de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-151-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1852" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1853" class="mjx-mrow"><span id="MJXc-Node-1854" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.182em;"><span id="MJXc-Node-1855" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1856" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-151">Y_i</script> est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-152-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1857" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1858" class="mjx-mrow"><span id="MJXc-Node-1859" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1860" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-1861" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1862" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1863" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1864" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1865" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-1866" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1867" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1868" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span id="MJXc-Node-1869" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>r</mi><msup><mi>p</mi><mi>r</mi></msup><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-152">(r+1)-rp^r.</script> Le nombre moyen d'analyses que l'on peut espérer réaliser est donc égal à
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-153-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;&amp;#x22EF;&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1870" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1871" class="mjx-mrow"><span id="MJXc-Node-1872" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-1873" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1874" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.182em;"><span id="MJXc-Node-1875" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1876" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-1877" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1878" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1879" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋯</span></span><span id="MJXc-Node-1880" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1881" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-1882" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1883" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.182em;"><span id="MJXc-Node-1884" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.275em; padding-right: 0.071em;"><span id="MJXc-Node-1885" class="mjx-texatom" style=""><span id="MJXc-Node-1886" class="mjx-mrow"><span id="MJXc-Node-1887" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-1888" class="mjx-texatom"><span id="MJXc-Node-1889" class="mjx-mrow"><span id="MJXc-Node-1890" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-1891" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span></span></span><span id="MJXc-Node-1892" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1893" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1894" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.8em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.8em; top: -1.144em;"><span id="MJXc-Node-1895" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-denominator" style="width: 0.8em; bottom: -0.722em;"><span id="MJXc-Node-1896" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.8em;"></span></span><span class="mjx-vsize" style="height: 1.865em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-1897" class="mjx-texatom"><span id="MJXc-Node-1898" class="mjx-mrow"><span id="MJXc-Node-1899" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-1900" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1901" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-1902" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1903" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1904" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1905" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1906" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-1907" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1908" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1909" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span id="MJXc-Node-1910" class="mjx-texatom"><span id="MJXc-Node-1911" class="mjx-mrow"><span id="MJXc-Node-1912" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span><span id="MJXc-Node-1913" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><msub><mi>Y</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo>⋯</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mi>Y</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>r</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>n</mi><mi>r</mi></mfrac><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo stretchy="false">(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>r</mi><msup><mi>p</mi><mi>r</mi></msup><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-153">E(Y_1)+\dots+E(Y_{n/r})=\frac nr\big((r+1)-rp^r\big).</script>
Le nombre d'analyses que l'on peut espérer économiser vaut donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-154-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1914" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1915" class="mjx-mrow"><span id="MJXc-Node-1916" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-1917" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1918" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.8em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.8em; top: -1.144em;"><span id="MJXc-Node-1919" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-denominator" style="width: 0.8em; bottom: -0.722em;"><span id="MJXc-Node-1920" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.8em;"></span></span><span class="mjx-vsize" style="height: 1.865em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-1921" class="mjx-texatom"><span id="MJXc-Node-1922" class="mjx-mrow"><span id="MJXc-Node-1923" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-1924" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1925" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-1926" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1927" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1928" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1929" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1930" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-1931" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1932" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1933" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span id="MJXc-Node-1934" class="mjx-texatom"><span id="MJXc-Node-1935" class="mjx-mrow"><span id="MJXc-Node-1936" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span><span id="MJXc-Node-1937" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1938" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-1939" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-1940" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-1941" class="mjx-mrow"><span id="MJXc-Node-1942" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1943" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1944" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span id="MJXc-Node-1945" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1946" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-1947" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.722em;"><span id="MJXc-Node-1948" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.722em;"></span></span></span><span id="MJXc-Node-1949" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span><span id="MJXc-Node-1950" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>n</mi><mo>−</mo><mfrac><mi>n</mi><mi>r</mi></mfrac><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo stretchy="false">(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>r</mi><msup><mi>p</mi><mi>r</mi></msup><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>=</mo><mi>n</mi><mrow><mo>(</mo><mrow><msup><mi>p</mi><mi>r</mi></msup><mo>−</mo><mfrac><mn>1</mn><mi>r</mi></mfrac></mrow><mo>)</mo></mrow><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-154">n-\frac nr\big((r+1)-rp^r\big)=n\left(p^r-\frac 1r\right).</script>
</li><li> On nous demande donc de donner un algorithme qui détermine le maximum d'une suite croissante, puis décroissante. Voici un algorithme sous Python qui fonctionne.
<p></p><div class="codebox"><div style="font-family: &#39;Courier New&#39;;font-size:10pt" dir="ltr">
<br>from math import *
<br>
<br>def maximum():
<br>&nbsp;&nbsp;&nbsp;&nbsp;r=1
<br>&nbsp;&nbsp;&nbsp;&nbsp;val=-1
<br>&nbsp;&nbsp;&nbsp;&nbsp;p=0.9
<br>&nbsp;&nbsp;&nbsp;&nbsp;while ((pow(p,r)-1/r)&gt;val):
<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;val=(pow(p,r)-1/r)
<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;r=r+1
<br>&nbsp;&nbsp;&nbsp;&nbsp;return r-1
<br></div></div><p>
La variable val contient la plus grande valeur atteinte jusqu'au rang présent. Elle est initialisée à un nombre négatif afin que l'on rentre dans la boucle. Il faut bien afficher <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-155-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1951" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1952" class="mjx-mrow"><span id="MJXc-Node-1953" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-1954" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1955" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-155">r-1</script> et non <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-156-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1956" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1957" class="mjx-mrow"><span id="MJXc-Node-1958" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-156">r</script>, car quand on sort de la boucle, on a dépassé le maximum.
</p></li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>6</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Trouver le paramètre d'une loi uniforme connaissant son espérance<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2262"> [Signaler une erreur]</a><a id="exo6" href="javascript:ajouteexo(2262,6,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2262&#39;,&#39;imageenonce2262&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2262"></a><div class="inner" id="innerenonce2262">Soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-157-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1959" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1960" class="mjx-mrow"><span id="MJXc-Node-1961" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-157">X</script> une variable aléatoire suivant une loi uniforme sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-158-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1962" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1963" class="mjx-mrow"><span id="MJXc-Node-1964" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">{</span></span><span id="MJXc-Node-1965" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-1966" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-1967" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-1968" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-1969" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.328em;">…</span></span><span id="MJXc-Node-1970" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-1971" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span><span id="MJXc-Node-1972" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>a</mi><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-158">\{0,1,\dots,a\}</script>, où <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-159-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1973" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1974" class="mjx-mrow"><span id="MJXc-Node-1975" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span><span id="MJXc-Node-1976" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-1977" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-1978" class="mjx-mrow"><span id="MJXc-Node-1979" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-159">a\in\mathbb N</script>. On suppose que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-160-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1980" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1981" class="mjx-mrow"><span id="MJXc-Node-1982" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-1983" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-1984" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1985" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-1986" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-1987" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">6</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-160">E(X)=6</script>. Déterminer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-161-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1988" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1989" class="mjx-mrow"><span id="MJXc-Node-1990" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-161">a</script>.</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2262&#39;,&#39;imageindication2262&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2262"></a><div class="inner" id="innerindication2262" style="display:none;">Calculer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-162-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1991" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1992" class="mjx-mrow"><span id="MJXc-Node-1993" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-1994" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1995" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-1996" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-162">E(X)</script> en fonction de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-163-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-1997" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1998" class="mjx-mrow"><span id="MJXc-Node-1999" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-163">a</script>.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2262&#39;,&#39;imagecorrige2262&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2262"></a><div class="inner" id="innercorrige2262" style="display:none;">On a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-164-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;&amp;#x22EF;&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2000" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2001" class="mjx-mrow"><span id="MJXc-Node-2002" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-2003" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2004" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2005" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2006" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2007" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.419em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.007em; top: -1.318em;"><span id="MJXc-Node-2008" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.007em; bottom: -0.693em;"><span id="MJXc-Node-2009" class="mjx-mrow" style=""><span id="MJXc-Node-2010" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-2011" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2012" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.419em;"></span></span><span class="mjx-vsize" style="height: 1.422em; vertical-align: -0.49em;"></span></span><span id="MJXc-Node-2013" class="mjx-texatom"><span id="MJXc-Node-2014" class="mjx-mrow"><span id="MJXc-Node-2015" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-2016" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-2017" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2018" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2019" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2020" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋯</span></span><span id="MJXc-Node-2021" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2022" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-2023" class="mjx-texatom"><span id="MJXc-Node-2024" class="mjx-mrow"><span id="MJXc-Node-2025" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span><span id="MJXc-Node-2026" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2027" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.515em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.729em; top: -1.103em;"><span id="MJXc-Node-2028" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.729em; bottom: -0.611em;"><span id="MJXc-Node-2029" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.515em;"></span></span><span class="mjx-vsize" style="height: 1.212em; vertical-align: -0.432em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mn>0</mn><mo>+</mo><mn>1</mn><mo>+</mo><mo>⋯</mo><mo>+</mo><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>=</mo><mfrac><mi>a</mi><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-164">E(X)=\frac1{a+1}\big(0+1+\dots+a\big)=\frac a2</script>. On en déduit donc que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-165-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2030" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2031" class="mjx-mrow"><span id="MJXc-Node-2032" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-2033" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2034" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">12</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>12</mn></math></span></span><script type="math/tex" id="MathJax-Element-165">a=12</script>.</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>7</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Avion<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1257"> [Signaler une erreur]</a><a id="exo7" href="javascript:ajouteexo(1257,7,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1257&#39;,&#39;imageenonce1257&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce1257"></a><div class="inner" id="innerenonce1257"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-166-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2035" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2036" class="mjx-mrow"><span id="MJXc-Node-2037" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-166">A</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-167-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2038" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2039" class="mjx-mrow"><span id="MJXc-Node-2040" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-167">B</script> sont deux avions ayant respectivement 4 et 2 moteurs. Les moteurs sont supposés indépendants les uns des autres, et ils ont une probabilité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-168-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2041" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2042" class="mjx-mrow"><span id="MJXc-Node-2043" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-168">p</script> de tomber en panne.
Chaque avion arrive à destination si strictement moins de la moitié de ses moteurs tombe en panne. Quel avion choisissez-vous? (on discutera en fonction de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-169-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2044" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2045" class="mjx-mrow"><span id="MJXc-Node-2046" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-169">p</script>).</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1257&#39;,&#39;imageindication1257&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication1257"></a><div class="inner" id="innerindication1257" style="display:none;">Noter <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-170-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2047" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2048" class="mjx-mrow"><span id="MJXc-Node-2049" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-170">X</script> la variable aléatoire du nombre de moteurs de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-171-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2050" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2051" class="mjx-mrow"><span id="MJXc-Node-2052" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-171">A</script> qui tombent en panne, et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-172-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2053" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2054" class="mjx-mrow"><span id="MJXc-Node-2055" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-172">Y</script> la variable aléatoire du nombre de moteurs de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-173-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2056" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2057" class="mjx-mrow"><span id="MJXc-Node-2058" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-173">B</script> qui tombent en panne.
On arrive à destination si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-174-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2059" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2060" class="mjx-mrow"><span id="MJXc-Node-2061" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2062" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2063" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-2064" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2065" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-174">X=0,1</script> ou si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-175-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2066" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2067" class="mjx-mrow"><span id="MJXc-Node-2068" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-2069" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2070" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-175">Y=0</script>. </div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1257&#39;,&#39;imagecorrige1257&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige1257"></a><div class="inner" id="innercorrige1257" style="display:none;">On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-176-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2071" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2072" class="mjx-mrow"><span id="MJXc-Node-2073" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-176">X</script> la variable aléatoire du nombre de moteurs de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-177-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2074" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2075" class="mjx-mrow"><span id="MJXc-Node-2076" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-177">A</script> qui tombent en panne, et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-178-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2077" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2078" class="mjx-mrow"><span id="MJXc-Node-2079" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-178">Y</script> la variable aléatoire du nombre de moteurs de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-179-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2080" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2081" class="mjx-mrow"><span id="MJXc-Node-2082" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-179">B</script> qui tombent en panne.
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-180-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2083" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2084" class="mjx-mrow"><span id="MJXc-Node-2085" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-180">X</script> suit une loi binomiale <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-181-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2086" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2087" class="mjx-mrow"><span id="MJXc-Node-2088" class="mjx-texatom"><span id="MJXc-Node-2089" class="mjx-mrow"><span id="MJXc-Node-2090" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.47em; padding-bottom: 0.353em; padding-right: 0.007em;">B</span></span></span></span><span id="MJXc-Node-2091" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2092" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-2093" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2094" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2095" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">B</mi></mrow><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-181">\mathcal{B}(4,p)</script>. En particulier, on a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-182-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2096" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2097" class="mjx-mrow"><span id="MJXc-Node-2098" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2099" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2100" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2101" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2102" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-2103" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2104" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2105" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2106" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2107" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2108" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2109" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2110" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2111" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2112" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-2113" class="mjx-TeXmathchoice"><span id="MJXc-Node-2114" class="mjx-texatom"><span id="MJXc-Node-2115" class="mjx-mrow"><span id="MJXc-Node-2116" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-2117" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.5em;"><span class="mjx-numerator" style="width: 0.5em; top: -1.379em;"><span id="MJXc-Node-2118" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="width: 0.5em; bottom: -0.733em;"><span id="MJXc-Node-2119" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="mjx-vsize" style="height: 2.111em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-2120" class="mjx-TeXmathchoice"><span id="MJXc-Node-2121" class="mjx-texatom"><span id="MJXc-Node-2122" class="mjx-mrow"><span id="MJXc-Node-2123" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-2124" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2125" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2126" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-2127" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2128" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2129" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2130" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2131" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2132" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2133" class="mjx-texatom" style=""><span id="MJXc-Node-2134" class="mjx-mrow"><span id="MJXc-Node-2135" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-2136" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2137" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span></span><span id="MJXc-Node-2138" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2139" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-2140" class="mjx-TeXmathchoice"><span id="MJXc-Node-2141" class="mjx-texatom"><span id="MJXc-Node-2142" class="mjx-mrow"><span id="MJXc-Node-2143" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-2144" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.5em;"><span class="mjx-numerator" style="width: 0.5em; top: -1.379em;"><span id="MJXc-Node-2145" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="width: 0.5em; bottom: -0.711em;"><span id="MJXc-Node-2146" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-2147" class="mjx-TeXmathchoice"><span id="MJXc-Node-2148" class="mjx-texatom"><span id="MJXc-Node-2149" class="mjx-mrow"><span id="MJXc-Node-2150" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-2151" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2152" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2153" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-2154" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2155" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2156" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2157" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2158" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2159" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2160" class="mjx-texatom" style=""><span id="MJXc-Node-2161" class="mjx-mrow"><span id="MJXc-Node-2162" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-2163" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2164" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-2165" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2166" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2167" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2168" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2169" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2170" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2171" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2172" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span id="MJXc-Node-2173" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2174" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-2175" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2176" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2177" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2178" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2179" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2180" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2181" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2182" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-2183" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>0</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><msup><mi>p</mi><mn>0</mn></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>4</mn><mo>−</mo><mn>0</mn></mrow></msup><mo>+</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>1</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><msup><mi>p</mi><mn>1</mn></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>4</mn><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mn>4</mn></msup><mo>+</mo><mn>4</mn><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mn>3</mn></msup><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-182">P(X=0)+P(X=1)=\binom 4 0p^0(1-p)^{4-0}+\binom 4 1p^1(1-p)^{4-1}=(1-p)^4+4p(1-p)^3.</script>
D'autre part, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-183-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2184" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2185" class="mjx-mrow"><span id="MJXc-Node-2186" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-183">Y</script> suit une loi binomiale <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-184-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2187" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2188" class="mjx-mrow"><span id="MJXc-Node-2189" class="mjx-texatom"><span id="MJXc-Node-2190" class="mjx-mrow"><span id="MJXc-Node-2191" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.47em; padding-bottom: 0.353em; padding-right: 0.007em;">B</span></span></span></span><span id="MJXc-Node-2192" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2193" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-2194" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2195" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2196" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">B</mi></mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-184">\mathcal{B}(2,p)</script>. En particulier,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-185-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2197" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2198" class="mjx-mrow"><span id="MJXc-Node-2199" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2200" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2201" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-2202" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2203" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-2204" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2205" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2206" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2207" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2208" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2209" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2210" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2211" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2212" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-2213" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-185">P(Y=0)=(1-p)^2.</script>
On a intérêt à prendre l'avion <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-186-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2214" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2215" class="mjx-mrow"><span id="MJXc-Node-2216" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-186">A</script> si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-187-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2217" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2218" class="mjx-mrow"><span id="MJXc-Node-2219" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2220" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2221" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2222" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2223" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-2224" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2225" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2226" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2227" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2228" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2229" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2230" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2231" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2232" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-2233" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2234" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2235" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-2236" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2237" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-2238" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≥</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-187">P(X=0)+P(X=1)\geq P(Y=0)</script>. Ceci donne :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-188-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;0.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2239" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2240" class="mjx-mrow"><span id="MJXc-Node-2241" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2242" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2243" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2244" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2245" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2246" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2247" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2248" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-2249" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2250" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-2251" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2252" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-2253" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2254" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2255" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-2256" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo stretchy="false">(</mo><mn>2</mn><mo>−</mo><mn>3</mn><mi>p</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-188">p(1-p)^2(2-3p)\geq 0.</script>
Donc, si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-189-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2257" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2258" class="mjx-mrow"><span id="MJXc-Node-2259" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-2260" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-2261" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2262" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&lt;</span></span><span id="MJXc-Node-2263" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-2264" class="mjx-texatom"><span id="MJXc-Node-2265" class="mjx-mrow"><span id="MJXc-Node-2266" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-2267" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>p</mi><mo>&lt;</mo><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-189">0\leq p<2/3</script> (cas que l'on espère être celui du monde réel), il est préférable de choisir <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-190-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2268" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2269" class="mjx-mrow"><span id="MJXc-Node-2270" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-190">A</script>. Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-191-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2271" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2272" class="mjx-mrow"><span id="MJXc-Node-2273" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2274" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2275" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-2276" class="mjx-texatom"><span id="MJXc-Node-2277" class="mjx-mrow"><span id="MJXc-Node-2278" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-2279" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-191">p=2/3</script>, le choix est indifférent,
et si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-192-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2280" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2281" class="mjx-mrow"><span id="MJXc-Node-2282" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2283" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-2284" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-2285" class="mjx-texatom"><span id="MJXc-Node-2286" class="mjx-mrow"><span id="MJXc-Node-2287" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-2288" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>&gt;</mo><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-192">p>2/3</script>, il vaut mieux choisir <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-193-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2289" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2290" class="mjx-mrow"><span id="MJXc-Node-2291" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-193">B</script>.</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>8</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Lancer de pièces<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1262"> [Signaler une erreur]</a><a id="exo8" href="javascript:ajouteexo(1262,8,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1262&#39;,&#39;imageenonce1262&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce1262"></a><div class="inner" id="innerenonce1262">On lance <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-194-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2292" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2293" class="mjx-mrow"><span id="MJXc-Node-2294" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-194">n</script> fois une pièce parfaitement équilibrée. Quelle est la probabilité d'obtenir strictement plus de piles que de faces.</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1262&#39;,&#39;imageindication1262&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication1262"></a><div class="inner" id="innerindication1262" style="display:none;">Utiliser la symétrie entre les piles et les faces! On pourra distinguer les cas où <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-195-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2295" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2296" class="mjx-mrow"><span id="MJXc-Node-2297" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-195">n</script> est pair et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-196-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2298" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2299" class="mjx-mrow"><span id="MJXc-Node-2300" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-196">n</script> est impair.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1262&#39;,&#39;imagecorrige1262&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige1262"></a><div class="inner" id="innercorrige1262" style="display:none;">Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-197-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2301" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2302" class="mjx-mrow"><span id="MJXc-Node-2303" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-197">A</script> l'événément "obtenir strictement plus de piles que de faces" et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-198-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2304" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2305" class="mjx-mrow"><span id="MJXc-Node-2306" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-198">B</script> l'événément "obtenir strictement plus de faces que de piles". La pièce étant parfaitement équilibrée, par symétrie, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-199-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2307" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2308" class="mjx-mrow"><span id="MJXc-Node-2309" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2310" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2311" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2312" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2313" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2314" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2315" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2316" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2317" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-199">P(A)=P(B)</script>.
On distingue alors deux cas :
<ul class="rien">
<li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-200-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2318" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2319" class="mjx-mrow"><span id="MJXc-Node-2320" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-200">n</script> est impair. Dans ce cas, le couple <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-201-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2321" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2322" class="mjx-mrow"><span id="MJXc-Node-2323" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2324" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2325" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2326" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2327" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-201">(A,B)</script> est un système complet d'événements et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-202-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2328" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2329" class="mjx-mrow"><span id="MJXc-Node-2330" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2331" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2332" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2333" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2334" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2335" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-2336" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-2337" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-202">P(A)=\frac 12</script>.
</li><li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-203-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2338" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2339" class="mjx-mrow"><span id="MJXc-Node-2340" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-2341" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2342" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-2343" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-203">n=2r</script> est pair. Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-204-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2344" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2345" class="mjx-mrow"><span id="MJXc-Node-2346" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-204">X</script> la variable aléatoire du nombre de piles obtenues. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-205-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2347" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2348" class="mjx-mrow"><span id="MJXc-Node-2349" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-205">X</script> suit une loi binomiale <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-206-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2350" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2351" class="mjx-mrow"><span id="MJXc-Node-2352" class="mjx-texatom"><span id="MJXc-Node-2353" class="mjx-mrow"><span id="MJXc-Node-2354" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.47em; padding-bottom: 0.353em; padding-right: 0.007em;">B</span></span></span></span><span id="MJXc-Node-2355" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2356" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-2357" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2358" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2359" class="mjx-texatom"><span id="MJXc-Node-2360" class="mjx-mrow"><span id="MJXc-Node-2361" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-2362" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-2363" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">B</mi></mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-206">\mathcal B(n,1/2)</script>. On a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-207-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;et&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2364" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2365" class="mjx-mrow"><span id="MJXc-Node-2366" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2367" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2368" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2369" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2370" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-2371" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-2372" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2373" class="mjx-texatom"><span id="MJXc-Node-2374" class="mjx-mrow"><span id="MJXc-Node-2375" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">&nbsp;et&nbsp;</span></span></span></span><span id="MJXc-Node-2376" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2377" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2378" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2379" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2380" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&lt;</span></span><span id="MJXc-Node-2381" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-2382" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2383" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>A</mi><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mi>r</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;et&nbsp;</mtext></mrow><mi>B</mi><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>&lt;</mo><mi>r</mi><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-207">A=(X>r)\textrm{ et }B=(X<r).</script>
On a donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-208-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27F9;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2384" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2385" class="mjx-mrow"><span id="MJXc-Node-2386" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2387" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2388" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2389" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2390" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2391" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2392" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2393" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2394" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2395" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2396" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2397" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2398" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2399" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2400" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-2401" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2402" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2403" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2404" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-2405" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟹</span></span><span id="MJXc-Node-2406" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-2407" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2408" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2409" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2410" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2411" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2412" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-2413" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-2414" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-2415" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2416" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-2417" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-2418" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-2419" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2420" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2421" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2422" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2423" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-2424" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2425" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2426" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-2427" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-2428" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-2429" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2430" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-2431" class="mjx-TeXmathchoice"><span id="MJXc-Node-2432" class="mjx-texatom"><span id="MJXc-Node-2433" class="mjx-mrow"><span id="MJXc-Node-2434" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-2435" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.951em;"><span class="mjx-numerator" style="width: 0.951em; top: -1.368em;"><span id="MJXc-Node-2436" class="mjx-mrow"><span id="MJXc-Node-2437" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-2438" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span class="mjx-denominator" style="width: 0.951em; bottom: -0.722em;"><span id="MJXc-Node-2439" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-2440" class="mjx-TeXmathchoice"><span id="MJXc-Node-2441" class="mjx-texatom"><span id="MJXc-Node-2442" class="mjx-mrow"><span id="MJXc-Node-2443" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-2444" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2445" class="mjx-mrow"><span id="MJXc-Node-2446" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-2447" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-2448" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-2449" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-2450" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2451" class="mjx-texatom" style=""><span id="MJXc-Node-2452" class="mjx-mrow"><span id="MJXc-Node-2453" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-2454" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-2455" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2456" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-2457" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mspace width="thickmathspace"></mspace><mo stretchy="false">⟹</mo><mspace width="thickmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mrow><mn>2</mn><mi>r</mi></mrow><mi>r</mi></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><msup><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-208">P(A)+P(B)+P(X=r)=1\implies P(A)=\frac 12-\frac 12P(X=r)=\frac 12-\binom {2r} r \left(\frac 12\right)^{2r+1}.</script>
</li></ul></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>9</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Tirage avec remises<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1274"> [Signaler une erreur]</a><a id="exo9" href="javascript:ajouteexo(1274,9,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1274&#39;,&#39;imageenonce1274&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce1274"></a><div class="inner" id="innerenonce1274">Une urne contient <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-209-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2458" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2459" class="mjx-mrow"><span id="MJXc-Node-2460" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-209">N</script> boules numérotées de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-210-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2461" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2462" class="mjx-mrow"><span id="MJXc-Node-2463" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-210">1</script> à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-211-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2464" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2465" class="mjx-mrow"><span id="MJXc-Node-2466" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-211">N</script>. On en tire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-212-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2467" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2468" class="mjx-mrow"><span id="MJXc-Node-2469" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-212">n</script> en effectuant des tirages avec remise. On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-213-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2470" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2471" class="mjx-mrow"><span id="MJXc-Node-2472" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-213">X</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-214-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2473" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2474" class="mjx-mrow"><span id="MJXc-Node-2475" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-214">Y</script> le plus petit et le plus grand des nombres obtenus. Déterminer la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-215-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2476" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2477" class="mjx-mrow"><span id="MJXc-Node-2478" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-215">X</script> et la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-216-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2479" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2480" class="mjx-mrow"><span id="MJXc-Node-2481" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-216">Y</script>.</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1274&#39;,&#39;imageindication1274&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication1274"></a><div class="inner" id="innerindication1274" style="display:none;">Calculer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-217-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2482" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2483" class="mjx-mrow"><span id="MJXc-Node-2484" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2485" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2486" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2487" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-2488" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2489" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-217">P(X\geq k)</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-218-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2490" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2491" class="mjx-mrow"><span id="MJXc-Node-2492" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2493" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2494" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-2495" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-2496" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2497" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-218">P(Y\leq k)</script>.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1274&#39;,&#39;imagecorrige1274&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige1274"></a><div class="inner" id="innercorrige1274" style="display:none;">Il n'est pas facile de calculer directement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-219-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2498" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2499" class="mjx-mrow"><span id="MJXc-Node-2500" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2501" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2502" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2503" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2504" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2505" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-219">P(X=k)</script>. En revanche, il est beaucoup plus facile de calculer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-220-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2506" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2507" class="mjx-mrow"><span id="MJXc-Node-2508" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2509" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2510" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2511" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-2512" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2513" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-220">P(X\geq k)</script>. En effet, on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-221-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2514" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2515" class="mjx-mrow"><span id="MJXc-Node-2516" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2517" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-2518" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>≥</mo><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-221">X\geq k</script> si et seulement si tous les tirages ont amené un nombre supérieur ou égal à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-222-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2519" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2520" class="mjx-mrow"><span id="MJXc-Node-2521" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-222">k</script>. La probabilité qu'un tirage amène un nombre supérieur ou égal à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-223-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2522" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2523" class="mjx-mrow"><span id="MJXc-Node-2524" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-223">k</script> valant
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-224-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2525" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2526" class="mjx-mrow"><span id="MJXc-Node-2527" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.592em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 3.665em; top: -1.428em;"><span id="MJXc-Node-2528" class="mjx-mrow" style=""><span id="MJXc-Node-2529" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-2530" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2531" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2532" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2533" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 3.665em; bottom: -0.628em;"><span id="MJXc-Node-2534" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.592em;"></span></span><span class="mjx-vsize" style="height: 1.454em; vertical-align: -0.444em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>N</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mi>N</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-224">\frac{N-k+1}N</script> et les tirages étant indépendants, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-225-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2535" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2536" class="mjx-mrow"><span id="MJXc-Node-2537" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2538" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2539" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2540" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-2541" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2542" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2543" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2544" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 5.792em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 5.792em; top: -1.542em;"><span id="MJXc-Node-2545" class="mjx-mrow"><span id="MJXc-Node-2546" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2547" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-2548" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2549" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2550" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2551" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2552" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2553" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2554" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="mjx-denominator" style="width: 5.792em; bottom: -0.725em;"><span id="MJXc-Node-2555" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.085em;"><span id="MJXc-Node-2556" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0.227em; padding-right: 0.071em;"><span id="MJXc-Node-2557" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 5.792em;"></span></span><span class="mjx-vsize" style="height: 2.268em; vertical-align: -0.725em;"></span></span><span id="MJXc-Node-2558" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>n</mi></msup></mrow><msup><mi>N</mi><mi>n</mi></msup></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-225">P(X\geq k)=\frac{(N-k+1)^n}{N^n}.</script>
On déduit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-226-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2559" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2560" class="mjx-mrow"><span id="MJXc-Node-2561" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2562" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2563" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2564" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2565" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2566" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-226">P(X=k)</script> par la formule
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-227-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2567" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2568" class="mjx-mrow"><span id="MJXc-Node-2569" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2570" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2571" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2572" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2573" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2574" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2575" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2576" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2577" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2578" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2579" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-2580" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2581" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2582" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2583" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2584" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2585" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2586" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-2587" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2588" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2589" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2590" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2591" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2592" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 10.883em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 10.883em; top: -1.542em;"><span id="MJXc-Node-2593" class="mjx-mrow"><span id="MJXc-Node-2594" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2595" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-2596" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2597" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2598" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2599" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2600" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2601" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2602" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-2603" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2604" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2605" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-2606" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2607" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2608" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2609" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2610" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="mjx-denominator" style="width: 10.883em; bottom: -0.725em;"><span id="MJXc-Node-2611" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.085em;"><span id="MJXc-Node-2612" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0.227em; padding-right: 0.071em;"><span id="MJXc-Node-2613" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 10.883em;"></span></span><span class="mjx-vsize" style="height: 2.268em; vertical-align: -0.725em;"></span></span><span id="MJXc-Node-2614" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mi>k</mi><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>n</mi></msup><mo>−</mo><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mi>k</mi><msup><mo stretchy="false">)</mo><mi>n</mi></msup></mrow><msup><mi>N</mi><mi>n</mi></msup></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-227">P(X=k)=P(X\geq k)-P(X\geq k+1)=\frac{(N-k+1)^n-(N-k)^n}{N^n}.</script>
La démarche est similaire pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-228-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2615" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2616" class="mjx-mrow"><span id="MJXc-Node-2617" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-228">Y</script>, mais cette fois on calcule <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-229-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2618" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2619" class="mjx-mrow"><span id="MJXc-Node-2620" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2621" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2622" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-2623" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-2624" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2625" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-229">P(Y\leq k)</script> qui vaut
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-230-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2626" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2627" class="mjx-mrow"><span id="MJXc-Node-2628" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2629" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2630" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-2631" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-2632" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2633" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2634" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2635" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.623em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.623em; top: -1.396em;"><span id="MJXc-Node-2636" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2637" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2638" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span><span class="mjx-denominator" style="width: 1.623em; bottom: -0.725em;"><span id="MJXc-Node-2639" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.085em;"><span id="MJXc-Node-2640" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0.227em; padding-right: 0.071em;"><span id="MJXc-Node-2641" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.623em;"></span></span><span class="mjx-vsize" style="height: 2.121em; vertical-align: -0.725em;"></span></span><span id="MJXc-Node-2642" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><msup><mi>k</mi><mi>n</mi></msup><msup><mi>N</mi><mi>n</mi></msup></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-230">P(Y\leq k)=\frac{k^n}{N^n}.</script>
Il vient
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-231-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2643" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2644" class="mjx-mrow"><span id="MJXc-Node-2645" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2646" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2647" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-2648" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2649" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2650" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2651" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2652" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2653" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2654" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-2655" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-2656" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2657" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2658" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2659" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2660" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2661" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-2662" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-2663" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2664" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2665" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2666" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2667" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2668" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 5.884em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 5.884em; top: -1.542em;"><span id="MJXc-Node-2669" class="mjx-mrow"><span id="MJXc-Node-2670" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2671" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2672" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-2673" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2674" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2675" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2676" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2677" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2678" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2679" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2680" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="mjx-denominator" style="width: 5.884em; bottom: -0.725em;"><span id="MJXc-Node-2681" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.085em;"><span id="MJXc-Node-2682" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0.227em; padding-right: 0.071em;"><span id="MJXc-Node-2683" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 5.884em;"></span></span><span class="mjx-vsize" style="height: 2.268em; vertical-align: -0.725em;"></span></span><span id="MJXc-Node-2684" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>k</mi><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mi>k</mi><mi>n</mi></msup><mo>−</mo><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>n</mi></msup></mrow><msup><mi>N</mi><mi>n</mi></msup></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-231">P(Y=k)=P(Y\leq k)-P(Y\leq k-1)=\frac{k^n -(k-1)^n }{N^n}.</script></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>10</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Le restaurateur<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2569"> [Signaler une erreur]</a><a id="exo10" href="javascript:ajouteexo(2569,10,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2569&#39;,&#39;imageenonce2569&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2569"></a><div class="inner" id="innerenonce2569">Un restaurateur accueille chaque soir 70 clients. Il sait qu'en moyenne, deux clients sur cinq prennent une crème brûlée. Il pense que s'il prépare 30 crèmes brûlées, dans plus de 70\% des cas, la demande sera satisfaite.
<ol class="enumeratechiffre">
<li> A-t-il raison?
</li><li> Combien de crèmes brûlées doit-il fabriquer au minimum pour que la demande soit satisfaite dans au moins 90\% des cas.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2569&#39;,&#39;imageindication2569&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2569"></a><div class="inner" id="innerindication2569" style="display:none;">Introduire la variable aléatoire égale au nombre de crèmes brûlées commandées chaque soir.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2569&#39;,&#39;imagecorrige2569&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2569"></a><div class="inner" id="innercorrige2569" style="display:none;">Soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-232-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2685" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2686" class="mjx-mrow"><span id="MJXc-Node-2687" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-232">X</script> la variable aléatoire comptant le nombre de crèmes brûlées commandées chaque soir. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-233-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2688" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2689" class="mjx-mrow"><span id="MJXc-Node-2690" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-233">X</script> compte le nombre de succès (=crème brûlée commandée) dans la répétition indépendante de 70 épreuves de Bernoulli dont chacune a une probabilité de succès égale à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-234-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0.4&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2691" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2692" class="mjx-mrow"><span id="MJXc-Node-2693" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">0.4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.4</mn></math></span></span><script type="math/tex" id="MathJax-Element-234">0.4</script>. Ainsi, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-235-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2694" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2695" class="mjx-mrow"><span id="MJXc-Node-2696" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-235">X</script> suit une loi binomiale de paramètre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-236-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;70&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2697" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2698" class="mjx-mrow"><span id="MJXc-Node-2699" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">70</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>70</mn></math></span></span><script type="math/tex" id="MathJax-Element-236">70</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-237-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0.4&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2700" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2701" class="mjx-mrow"><span id="MJXc-Node-2702" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">0.4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.4</mn></math></span></span><script type="math/tex" id="MathJax-Element-237">0.4</script>.
<ol class="enumeratechiffre">
<li> La demande est satisfaite si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-238-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2703" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2704" class="mjx-mrow"><span id="MJXc-Node-2705" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2706" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-2707" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">30</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>≤</mo><mn>30</mn></math></span></span><script type="math/tex" id="MathJax-Element-238">X\leq 30</script>. On doit donc déterminer si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-239-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;0.7&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2708" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2709" class="mjx-mrow"><span id="MJXc-Node-2710" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2711" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2712" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2713" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-2714" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">30</span></span><span id="MJXc-Node-2715" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2716" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-2717" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">0.7</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mn>30</mn><mo stretchy="false">)</mo><mo>≥</mo><mn>0.7</mn></math></span></span><script type="math/tex" id="MathJax-Element-239">P(X\leq 30)\geq 0.7</script>. A l'aide d'une calculatrice ou d'un tableur, on trouve facilement que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-240-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2243;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;73&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2718" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2719" class="mjx-mrow"><span id="MJXc-Node-2720" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2721" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2722" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2723" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-2724" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">30</span></span><span id="MJXc-Node-2725" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2726" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.295em;">≃</span></span><span id="MJXc-Node-2727" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-2728" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2729" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">73</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mn>30</mn><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn><mo>,</mo><mn>73</mn></math></span></span><script type="math/tex" id="MathJax-Element-240">P(X\leq 30)\simeq 0,73</script> : le restaurateur a raison.
</li><li> On cherche le plus petit entier <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-241-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2730" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2731" class="mjx-mrow"><span id="MJXc-Node-2732" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-241">k</script> tel que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-242-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2733" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2734" class="mjx-mrow"><span id="MJXc-Node-2735" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2736" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2737" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2738" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-2739" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-2740" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2741" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-2742" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-2743" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2744" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn><mo>,</mo><mn>9</mn></math></span></span><script type="math/tex" id="MathJax-Element-242">P(X\leq k)\geq 0,9</script>. Toujours à l'aide d'un tableur, on constate que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-243-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;32&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2243;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;86&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2745" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2746" class="mjx-mrow"><span id="MJXc-Node-2747" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2748" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2749" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2750" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-2751" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">32</span></span><span id="MJXc-Node-2752" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2753" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.295em;">≃</span></span><span id="MJXc-Node-2754" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-2755" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2756" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">86</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mn>32</mn><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn><mo>,</mo><mn>86</mn></math></span></span><script type="math/tex" id="MathJax-Element-243">P(X\leq 32)\simeq 0,86</script> et que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-244-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;33&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2243;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;91&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2757" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2758" class="mjx-mrow"><span id="MJXc-Node-2759" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2760" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2761" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2762" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-2763" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">33</span></span><span id="MJXc-Node-2764" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2765" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.295em;">≃</span></span><span id="MJXc-Node-2766" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-2767" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2768" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">91</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mn>33</mn><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn><mo>,</mo><mn>91</mn></math></span></span><script type="math/tex" id="MathJax-Element-244">P(X\leq 33)\simeq 0,91</script>. Il suffit de préparer 33 crèmes brûlées pour satisfaire la clientèle dans 90\% des cas. Ca vaut peut-être le coup de fabriquer 3 crèmes supplémentaires!
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>11</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Code de la route!<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2160"> [Signaler une erreur]</a><a id="exo11" href="javascript:ajouteexo(2160,11,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2160&#39;,&#39;imageenonce2160&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2160"></a><div class="inner" id="innerenonce2160">L'examen du code de la route se compose de 40 questions. Pour chaque question, on a le choix entre 4 réponses possibles. Une seule de ces réponses est correcte. Un candidat se présente à l'examen. Il arrive qu’il connaisse la réponse
à certaines questions. Il répond alors à coup sûr. S’il ignore la réponse, il
choisit au hasard entre les 4 réponses proposées. On suppose toutes les questions
indépendantes et que pour chacune de ces questions, la probabilité que
le candidat connaisse la vraie réponse est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-245-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2769" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2770" class="mjx-mrow"><span id="MJXc-Node-2771" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-245">p</script>. On note, pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-246-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;40&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2772" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2773" class="mjx-mrow"><span id="MJXc-Node-2774" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-2775" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.458em;">≤</span></span><span id="MJXc-Node-2776" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">i</span></span><span id="MJXc-Node-2777" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.458em;">≤</span></span><span id="MJXc-Node-2778" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">40</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mn>40</mn></math></span></span><script type="math/tex" id="MathJax-Element-246">1\leq i\leq 40</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-247-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2779" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2780" class="mjx-mrow"><span id="MJXc-Node-2781" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2782" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2783" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-247">A_i</script> l'événement : "le candidat donne la bonne réponse à la <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-248-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2784" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2785" class="mjx-mrow"><span id="MJXc-Node-2786" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-248">i</script>-ème
question". On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-249-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2787" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2788" class="mjx-mrow"><span id="MJXc-Node-2789" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.032em;">S</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></span></span><script type="math/tex" id="MathJax-Element-249">S</script> la variable aléatoire égale au nombre total de bonnes réponses.
<ol class="enumeratechiffre">
<li> Calculer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-250-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2790" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2791" class="mjx-mrow"><span id="MJXc-Node-2792" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2793" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-2794" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2795" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2796" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">i</span></span></span></span><span id="MJXc-Node-2797" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-250">P(A_i)</script>.
</li><li> Quelle est la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-251-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2798" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2799" class="mjx-mrow"><span id="MJXc-Node-2800" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.032em;">S</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></span></span><script type="math/tex" id="MathJax-Element-251">S</script> (justifier!)?
</li><li> A quelle condition sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-252-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2801" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2802" class="mjx-mrow"><span id="MJXc-Node-2803" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-252">p</script> le candidat donnera en moyenne au moins 36 bonnes réponses?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2160&#39;,&#39;imageindication2160&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2160"></a><div class="inner" id="innerindication2160" style="display:none;"><ol class="enumeratechiffre">
<li> Utiliser la formule des probabilités totales.
</li><li> Reconnaitre une loi classique.
</li><li> C'est une question d'espérance...
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2160&#39;,&#39;imagecorrige2160&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2160"></a><div class="inner" id="innercorrige2160" style="display:none;"><ol class="enumeratechiffre">
<li> Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-253-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2804" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2805" class="mjx-mrow"><span id="MJXc-Node-2806" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-2807" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2808" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-253">C_i</script> l'évènement ``le candidat connait la réponse à la <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-254-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2809" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2810" class="mjx-mrow"><span id="MJXc-Node-2811" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-254">i</script>-ème question.''. On a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-255-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2812" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2813" class="mjx-mrow"><span id="MJXc-Node-2814" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2815" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2816" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2817" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2818" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span id="MJXc-Node-2819" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2820" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2821" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-2822" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-2823" class="mjx-texatom" style=""><span id="MJXc-Node-2824" class="mjx-mrow"><span id="MJXc-Node-2825" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-2826" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2827" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span></span></span><span id="MJXc-Node-2828" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2829" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2830" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2831" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span id="MJXc-Node-2832" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2833" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2834" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2835" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-2836" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2837" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span id="MJXc-Node-2838" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2839" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2840" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-2841" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.386em; padding-right: 0.071em;"><span id="MJXc-Node-2842" class="mjx-texatom" style=""><span id="MJXc-Node-2843" class="mjx-mrow"><span id="MJXc-Node-2844" class="mjx-texatom"><span id="MJXc-Node-2845" class="mjx-mrow"><span id="MJXc-Node-2846" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.247em;"><span id="MJXc-Node-2850" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-2847" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-2848" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2849" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span></span></span></span></span></span></span></span><span id="MJXc-Node-2851" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2852" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2853" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2854" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span id="MJXc-Node-2855" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2856" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2857" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2858" class="mjx-texatom"><span id="MJXc-Node-2859" class="mjx-mrow"><span id="MJXc-Node-2860" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.247em;"><span id="MJXc-Node-2864" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-2861" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-2862" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2863" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span></span></span></span><span id="MJXc-Node-2865" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2866" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2867" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2868" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-2869" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2870" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2871" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-2872" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.719em;"><span id="MJXc-Node-2873" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.087em; vertical-align: -0.719em;"></span></span><span id="MJXc-Node-2874" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-2875" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2876" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2877" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2878" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2879" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2880" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2881" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.925em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.925em; top: -1.402em;"><span id="MJXc-Node-2882" class="mjx-mrow"><span id="MJXc-Node-2883" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2884" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2885" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-2886" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span><span class="mjx-denominator" style="width: 2.925em; bottom: -0.719em;"><span id="MJXc-Node-2887" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.925em;"></span></span><span class="mjx-vsize" style="height: 2.122em; vertical-align: -0.719em;"></span></span><span id="MJXc-Node-2888" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.289em; padding-bottom: 0.353em;">&nbsp;</span></span><span id="MJXc-Node-2889" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>C</mi><mi>i</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>C</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mover><msub><mi>C</mi><mi>i</mi></msub><mo stretchy="false">¯</mo></mover></mrow></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><msub><mi>C</mi><mi>i</mi></msub><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>×</mo><mi>p</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>×</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mn>3</mn><mi>p</mi></mrow><mn>4</mn></mfrac><mtext>&nbsp;</mtext><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-255">P(A_i)=P_{C_i}(A_i)P(C_i)+P_{\bar{C_i}}(A_i)P(\bar{C_i})=1\times p+\frac14\times (1-p)=\frac{1+3p}4\ .</script>
</li><li>
Les questions étant toutes indépendantes, on est en présence d'un schéma de Bernoulli et la variable <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-256-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2890" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2891" class="mjx-mrow"><span id="MJXc-Node-2892" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></span></span><script type="math/tex" id="MathJax-Element-256">S</script> qui compte le nombre de bonnes réponses suit une loi binomiale de paramètres 40 et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-257-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2893" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2894" class="mjx-mrow"><span id="MJXc-Node-2895" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.754em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.481em; top: -1.512em;"><span id="MJXc-Node-2896" class="mjx-mrow" style=""><span id="MJXc-Node-2897" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2898" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2899" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-2900" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.481em; bottom: -0.622em;"><span id="MJXc-Node-2901" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.754em;"></span></span><span class="mjx-vsize" style="height: 1.509em; vertical-align: -0.44em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>1</mn><mo>+</mo><mn>3</mn><mi>p</mi></mrow><mn>4</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-257">\frac{1+3p}4</script>.
</li><li> L'espérance de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-258-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2902" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2903" class="mjx-mrow"><span id="MJXc-Node-2904" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></span></span><script type="math/tex" id="MathJax-Element-258">S</script> vaut <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-259-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;40&lt;/mn&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2905" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2906" class="mjx-mrow"><span id="MJXc-Node-2907" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-2908" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2909" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-2910" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2911" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2912" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">40</span></span><span id="MJXc-Node-2913" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.754em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.481em; top: -1.512em;"><span id="MJXc-Node-2914" class="mjx-mrow" style=""><span id="MJXc-Node-2915" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2916" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2917" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-2918" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.481em; bottom: -0.622em;"><span id="MJXc-Node-2919" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.754em;"></span></span><span class="mjx-vsize" style="height: 1.509em; vertical-align: -0.44em;"></span></span><span id="MJXc-Node-2920" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2921" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-2922" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2923" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2924" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2925" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-2926" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2927" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mn>40</mn><mfrac><mrow><mn>1</mn><mo>+</mo><mn>3</mn><mi>p</mi></mrow><mn>4</mn></mfrac><mo>=</mo><mn>10</mn><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>3</mn><mi>p</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-259">E(S)=40\frac{1+3p}4=10(1+3p)</script>. On a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-260-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2928" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2929" class="mjx-mrow"><span id="MJXc-Node-2930" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-2931" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2932" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-2933" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2934" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-2935" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>36</mn></math></span></span><script type="math/tex" id="MathJax-Element-260">E(S)\ge 36</script> si et seulement si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-261-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2936" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2937" class="mjx-mrow"><span id="MJXc-Node-2938" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-2939" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2940" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2941" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2942" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-2943" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2944" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2945" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-2946" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>3</mn><mi>p</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>36</mn></math></span></span><script type="math/tex" id="MathJax-Element-261">10(1+3p)\ge 36</script>, ce qui revient à dire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-262-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mn&gt;15&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-2947" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2948" class="mjx-mrow"><span id="MJXc-Node-2949" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2950" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-2951" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.34em;"><span id="MJXc-Node-2952" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.633em;"><span id="MJXc-Node-2953" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">15</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.395em; vertical-align: -0.448em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>≥</mo><mfrac><mn>13</mn><mn>15</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-262">p\ge\frac{13}{15}</script>.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>12</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Minimum et maximum de deux dés<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1258"> [Signaler une erreur]</a><a id="exo12" href="javascript:ajouteexo(1258,12,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1258&#39;,&#39;imageenonce1258&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce1258"></a><div class="inner" id="innerenonce1258">On lance deux dés équilibrés, on note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-263-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2954" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2955" class="mjx-mrow"><span id="MJXc-Node-2956" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-2957" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2958" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-263">U_1</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-264-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2959" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2960" class="mjx-mrow"><span id="MJXc-Node-2961" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-2962" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2963" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-264">U_2</script> les variables aléatoires correspondant aux résultats obtenus.
On appelle <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-265-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo movablelimits=&quot;true&quot; form=&quot;prefix&quot;&gt;min&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2964" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2965" class="mjx-mrow"><span id="MJXc-Node-2966" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-2967" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-2968" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">min</span></span><span id="MJXc-Node-2969" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-2970" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-2971" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2972" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span id="MJXc-Node-2973" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-2974" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-2975" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2976" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span><span id="MJXc-Node-2977" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mo movablelimits="true" form="prefix">min</mo><mo stretchy="false">(</mo><msub><mi>U</mi><mn>1</mn></msub><mo>,</mo><msub><mi>U</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-265">X=\min(U_1,U_2)</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-266-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo movablelimits=&quot;true&quot; form=&quot;prefix&quot;&gt;max&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2978" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2979" class="mjx-mrow"><span id="MJXc-Node-2980" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-2981" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-2982" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.133em; padding-bottom: 0.328em;">max</span></span><span id="MJXc-Node-2983" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-2984" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-2985" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2986" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span id="MJXc-Node-2987" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-2988" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-2989" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2990" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span><span id="MJXc-Node-2991" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mo movablelimits="true" form="prefix">max</mo><mo stretchy="false">(</mo><msub><mi>U</mi><mn>1</mn></msub><mo>,</mo><msub><mi>U</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-266">Y=\max(U_1,U_2)</script>.
<ol class="enumeratechiffre">
<li> Donner la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-267-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2992" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2993" class="mjx-mrow"><span id="MJXc-Node-2994" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-267">X</script>. En déduire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-268-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2995" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2996" class="mjx-mrow"><span id="MJXc-Node-2997" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-2998" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-2999" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3000" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-268">E(X)</script>.
</li><li> Exprimer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-269-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3001" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3002" class="mjx-mrow"><span id="MJXc-Node-3003" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3004" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-3005" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>+</mo><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-269">X+Y</script> en fonction de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-270-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3006" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3007" class="mjx-mrow"><span id="MJXc-Node-3008" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3009" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3010" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-270">U_1</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-271-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3011" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3012" class="mjx-mrow"><span id="MJXc-Node-3013" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3014" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3015" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-271">U_2</script>. En déduire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-272-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3016" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3017" class="mjx-mrow"><span id="MJXc-Node-3018" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3019" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3020" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-3021" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-272">E(Y)</script>.
</li><li> Exprimer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-273-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3022" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3023" class="mjx-mrow"><span id="MJXc-Node-3024" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3025" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-273">XY</script> en fonction de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-274-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3026" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3027" class="mjx-mrow"><span id="MJXc-Node-3028" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3029" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3030" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-274">U_1</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-275-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3031" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3032" class="mjx-mrow"><span id="MJXc-Node-3033" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3034" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3035" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-275">U_2</script>. En déduire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-276-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;Cov&lt;/mtext&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3036" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3037" class="mjx-mrow"><span id="MJXc-Node-3038" class="mjx-texatom"><span id="MJXc-Node-3039" class="mjx-mrow"><span id="MJXc-Node-3040" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">Cov</span></span></span></span><span id="MJXc-Node-3041" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3042" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3043" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-3044" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-3045" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext>Cov</mtext></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-276">\textrm{Cov}(X,Y)</script>. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-277-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3046" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3047" class="mjx-mrow"><span id="MJXc-Node-3048" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-277">X</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-278-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3049" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3050" class="mjx-mrow"><span id="MJXc-Node-3051" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-278">Y</script> sont-elles indépendantes?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1258&#39;,&#39;imageindication1258&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication1258"></a><div class="inner" id="innerindication1258" style="display:none;"><ol class="enumeratechiffre">
<li> Écrire l'événement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-279-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3052" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3053" class="mjx-mrow"><span id="MJXc-Node-3054" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3055" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3056" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-279">X=i</script> comme réunion de trois événements disjoints.
</li><li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-280-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3057" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3058" class="mjx-mrow"><span id="MJXc-Node-3059" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3060" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3061" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-3062" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-3063" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-280">(X,Y)</script> est une permutation de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-281-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3064" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3065" class="mjx-mrow"><span id="MJXc-Node-3066" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3067" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3068" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3069" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3070" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-3071" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3072" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3073" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3074" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>U</mi><mn>1</mn></msub><mo>,</mo><msub><mi>U</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-281">(U_1,U_2)</script>.
</li><li>
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1258&#39;,&#39;imagecorrige1258&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige1258"></a><div class="inner" id="innercorrige1258" style="display:none;"><ol class="enumeratechiffre">
<li> Pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-282-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3075" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3076" class="mjx-mrow"><span id="MJXc-Node-3077" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-3078" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3079" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-3080" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-3081" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-3082" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-3083" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-282">i=1,\dots,6</script>, l'événement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-283-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3084" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3085" class="mjx-mrow"><span id="MJXc-Node-3086" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3087" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3088" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-283">X=i</script> est réunion disjointe des trois événements suivants :
<ul class="rien">
<li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-284-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3089" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3090" class="mjx-mrow"><span id="MJXc-Node-3091" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-284">A</script> : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-285-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3092" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3093" class="mjx-mrow"><span id="MJXc-Node-3094" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3095" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3096" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3097" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3098" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>1</mn></msub><mo>=</mo><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-285">U_1=i</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-286-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3099" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3100" class="mjx-mrow"><span id="MJXc-Node-3101" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3102" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3103" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3104" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3105" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>2</mn></msub><mo>=</mo><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-286">U_2=i</script>;
</li><li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-287-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3106" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3107" class="mjx-mrow"><span id="MJXc-Node-3108" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-287">B</script> : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-288-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3109" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3110" class="mjx-mrow"><span id="MJXc-Node-3111" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3112" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3113" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3114" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3115" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>1</mn></msub><mo>=</mo><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-288">U_1=i</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-289-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3116" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3117" class="mjx-mrow"><span id="MJXc-Node-3118" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3119" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3120" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3121" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-3122" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>2</mn></msub><mo>&gt;</mo><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-289">U_2>i</script>;
</li><li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-290-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3123" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3124" class="mjx-mrow"><span id="MJXc-Node-3125" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></span></span><script type="math/tex" id="MathJax-Element-290">C</script> : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-291-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3126" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3127" class="mjx-mrow"><span id="MJXc-Node-3128" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3129" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3130" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3131" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-3132" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>1</mn></msub><mo>&gt;</mo><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-291">U_1>i</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-292-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3133" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3134" class="mjx-mrow"><span id="MJXc-Node-3135" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3136" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3137" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3138" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3139" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>2</mn></msub><mo>=</mo><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-292">U_2=i</script>.
</li></ul>
Par indépendance des variables aléatoires <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-293-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3140" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3141" class="mjx-mrow"><span id="MJXc-Node-3142" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3143" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3144" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-293">U_1</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-294-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3145" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3146" class="mjx-mrow"><span id="MJXc-Node-3147" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3148" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3149" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-294">U_2</script>, on en déduit que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-295-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;et&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3150" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3151" class="mjx-mrow"><span id="MJXc-Node-3152" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3153" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3154" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3155" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3156" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3157" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.368em;"><span id="MJXc-Node-3158" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.733em;"><span id="MJXc-Node-3159" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3160" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-3161" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.289em; padding-bottom: 0.353em;">&nbsp;</span></span><span id="MJXc-Node-3162" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3163" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3164" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-3165" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3166" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3167" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.267em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.267em; top: -1.368em;"><span id="MJXc-Node-3168" class="mjx-mrow"><span id="MJXc-Node-3169" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-3170" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3171" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="mjx-denominator" style="width: 2.267em; bottom: -0.733em;"><span id="MJXc-Node-3172" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.267em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3173" class="mjx-texatom"><span id="MJXc-Node-3174" class="mjx-mrow"><span id="MJXc-Node-3175" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">&nbsp;et&nbsp;</span></span></span></span><span id="MJXc-Node-3176" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3177" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3178" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-3179" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3180" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3181" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.267em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.267em; top: -1.368em;"><span id="MJXc-Node-3182" class="mjx-mrow"><span id="MJXc-Node-3183" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-3184" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3185" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="mjx-denominator" style="width: 2.267em; bottom: -0.733em;"><span id="MJXc-Node-3186" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.267em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3187" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>36</mn></mfrac><mo>,</mo><mtext>&nbsp;</mtext><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>6</mn><mo>−</mo><mi>i</mi></mrow><mn>36</mn></mfrac><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;et&nbsp;</mtext></mrow><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>6</mn><mo>−</mo><mi>i</mi></mrow><mn>36</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-295">P(A)=\frac1{36},\ P(B)=\frac{6-i}{36}\textrm{ et }P(C)=\frac{6-i}{36}.</script>
Il vient <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-296-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;11&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3188" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3189" class="mjx-mrow"><span id="MJXc-Node-3190" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3191" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3192" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3193" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3194" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-3195" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3196" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3197" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">11</span></span><span id="MJXc-Node-3198" class="mjx-texatom"><span id="MJXc-Node-3199" class="mjx-mrow"><span id="MJXc-Node-3200" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-3201" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>11</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>36</mn></math></span></span><script type="math/tex" id="MathJax-Element-296">P(X=1)=11/36</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-297-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3202" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3203" class="mjx-mrow"><span id="MJXc-Node-3204" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3205" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3206" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3207" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3208" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-3209" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3210" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3211" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-3212" class="mjx-texatom"><span id="MJXc-Node-3213" class="mjx-mrow"><span id="MJXc-Node-3214" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-3215" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mn>9</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>36</mn></math></span></span><script type="math/tex" id="MathJax-Element-297">P(X=2)=9/36</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-298-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3216" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3217" class="mjx-mrow"><span id="MJXc-Node-3218" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3219" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3220" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3221" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3222" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-3223" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3224" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3225" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-3226" class="mjx-texatom"><span id="MJXc-Node-3227" class="mjx-mrow"><span id="MJXc-Node-3228" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-3229" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mn>7</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>36</mn></math></span></span><script type="math/tex" id="MathJax-Element-298">P(X=3)=7/36</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-299-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3230" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3231" class="mjx-mrow"><span id="MJXc-Node-3232" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3233" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3234" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3235" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3236" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-3237" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3238" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3239" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-3240" class="mjx-texatom"><span id="MJXc-Node-3241" class="mjx-mrow"><span id="MJXc-Node-3242" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-3243" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>4</mn><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>36</mn></math></span></span><script type="math/tex" id="MathJax-Element-299">P(X=4)=5/36</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-300-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3244" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3245" class="mjx-mrow"><span id="MJXc-Node-3246" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3247" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3248" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3249" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3250" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-3251" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3252" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3253" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-3254" class="mjx-texatom"><span id="MJXc-Node-3255" class="mjx-mrow"><span id="MJXc-Node-3256" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-3257" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>5</mn><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>36</mn></math></span></span><script type="math/tex" id="MathJax-Element-300">P(X=5)=3/36</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-301-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3258" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3259" class="mjx-mrow"><span id="MJXc-Node-3260" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3261" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3262" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3263" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3264" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-3265" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3266" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3267" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-3268" class="mjx-texatom"><span id="MJXc-Node-3269" class="mjx-mrow"><span id="MJXc-Node-3270" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-3271" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>6</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>36</mn></math></span></span><script type="math/tex" id="MathJax-Element-301">P(X=6)=1/36</script>.
On en déduit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-302-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;91&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3272" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3273" class="mjx-mrow"><span id="MJXc-Node-3274" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3275" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3276" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3277" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3278" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3279" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">91</span></span><span id="MJXc-Node-3280" class="mjx-texatom"><span id="MJXc-Node-3281" class="mjx-mrow"><span id="MJXc-Node-3282" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-3283" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mn>91</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>36</mn></math></span></span><script type="math/tex" id="MathJax-Element-302">E(X)=91/36</script>.
</li><li> On a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-303-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3284" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3285" class="mjx-mrow"><span id="MJXc-Node-3286" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3287" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3288" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-3289" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3290" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3291" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3292" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3293" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3294" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3295" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3296" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>+</mo><mi>Y</mi><mo>=</mo><msub><mi>U</mi><mn>1</mn></msub><mo>+</mo><msub><mi>U</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-303">X+Y=U_1+U_2</script> car <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-304-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3297" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3298" class="mjx-mrow"><span id="MJXc-Node-3299" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3300" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3301" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3302" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3303" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-3304" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3305" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3306" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3307" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>U</mi><mn>1</mn></msub><mo>,</mo><msub><mi>U</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-304">(U_1,U_2)</script> est une permutation de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-305-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3308" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3309" class="mjx-mrow"><span id="MJXc-Node-3310" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3311" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3312" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-3313" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-3314" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-305">(X,Y)</script>. Il vient <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-306-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3315" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3316" class="mjx-mrow"><span id="MJXc-Node-3317" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3318" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3319" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3320" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3321" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-3322" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3323" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3324" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3325" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3326" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3327" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3328" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3329" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3330" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3331" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-3332" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3333" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3334" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3335" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3336" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3337" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3338" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3339" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3340" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3341" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3342" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3343" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3344" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3345" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3346" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3347" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3348" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mi>U</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mi>U</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>7</mn></math></span></span><script type="math/tex" id="MathJax-Element-306">E(X+Y)=E(X)+E(Y)=E(U_1)+E(U_2)=7</script>, puisque chaque <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-307-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3349" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3350" class="mjx-mrow"><span id="MJXc-Node-3351" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3352" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3353" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-307">U_i</script> suit une loi uniforme
sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-308-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3354" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3355" class="mjx-mrow"><span id="MJXc-Node-3356" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-3357" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-3358" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-3359" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-3360" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-3361" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-3362" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>6</mn><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-308">\{1,\dots,6\}</script>, son espérance vaut <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-309-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3363" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3364" class="mjx-mrow"><span id="MJXc-Node-3365" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3366" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-3367" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3368" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-3369" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3370" class="mjx-texatom"><span id="MJXc-Node-3371" class="mjx-mrow"><span id="MJXc-Node-3372" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-3373" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-3374" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3375" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-3376" class="mjx-texatom"><span id="MJXc-Node-3377" class="mjx-mrow"><span id="MJXc-Node-3378" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-3379" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>6</mn><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo>=</mo><mn>7</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-309">(6+1)/2=7/2</script>. On en déduit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-310-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;161&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3380" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3381" class="mjx-mrow"><span id="MJXc-Node-3382" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3383" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3384" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-3385" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3386" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3387" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">161</span></span><span id="MJXc-Node-3388" class="mjx-texatom"><span id="MJXc-Node-3389" class="mjx-mrow"><span id="MJXc-Node-3390" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-3391" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">36</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>161</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>36</mn></math></span></span><script type="math/tex" id="MathJax-Element-310">E(Y)=161/36</script>.
</li><li> On a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-311-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3392" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3393" class="mjx-mrow"><span id="MJXc-Node-3394" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3395" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-3396" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3397" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3398" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3399" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3400" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3401" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3402" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mi>Y</mi><mo>=</mo><msub><mi>U</mi><mn>1</mn></msub><msub><mi>U</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-311">XY=U_1U_2</script>. On en déduit
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-312-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3403" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3404" class="mjx-mrow"><span id="MJXc-Node-3405" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3406" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3407" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3408" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-3409" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3410" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3411" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3412" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3413" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3414" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3415" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3416" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3417" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3418" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3419" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3420" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3421" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3422" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3423" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3424" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3425" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3426" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3427" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3428" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3429" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-3430" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3431" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3432" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mi>U</mi><mn>1</mn></msub><msub><mi>U</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mi>U</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mi>U</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-312">E(XY)=E(U_1U_2)=E(U_1)E(U_2)</script>
par indépendance de ces deux variables aléatoires. D'où
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-313-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;Cov&lt;/mtext&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1225&lt;/mn&gt;&lt;mn&gt;1296&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3433" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3434" class="mjx-mrow"><span id="MJXc-Node-3435" class="mjx-texatom"><span id="MJXc-Node-3436" class="mjx-mrow"><span id="MJXc-Node-3437" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">Cov</span></span></span></span><span id="MJXc-Node-3438" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3439" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3440" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-3441" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-3442" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3443" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3444" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3445" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3446" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3447" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-3448" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3449" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3450" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3451" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3452" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3453" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3454" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3455" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3456" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-3457" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3458" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3459" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.2em; top: -1.368em;"><span id="MJXc-Node-3460" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1225</span></span></span><span class="mjx-denominator" style="width: 2.2em; bottom: -0.733em;"><span id="MJXc-Node-3461" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1296</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.2em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3462" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mtext>Cov</mtext></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>−</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1225</mn><mn>1296</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-313">\textrm{Cov}(X,Y)=E(XY)-E(X)E(Y)=\frac{1225}{1296}.</script>
Les deux variables ne sont pas indépendantes puisque leur covariance n'est pas nulle.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>13</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Simuler une variable aléatoire discrète<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2060"> [Signaler une erreur]</a><a id="exo13" href="javascript:ajouteexo(2060,13,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2060&#39;,&#39;imageenonce2060&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2060"></a><div class="inner" id="innerenonce2060">La plupart des langages de programmation dispose d'une fonction ALEA() qui renvoie un nombre aléatoire suivant la loi uniforme sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-314-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3463" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3464" class="mjx-mrow"><span id="MJXc-Node-3465" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-3466" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-3467" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-3468" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-3469" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-314">[0,1]</script>. Écrire un algorithme dont les données sont un entier <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-315-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3470" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3471" class="mjx-mrow"><span id="MJXc-Node-3472" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-3473" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.458em;">≥</span></span><span id="MJXc-Node-3474" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>≥</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-315">N\geq 1</script> et un tableau <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-316-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3475" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3476" class="mjx-mrow"><span id="MJXc-Node-3477" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span><span id="MJXc-Node-3478" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-3479" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-3480" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span><span id="MJXc-Node-3481" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-3482" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.328em;">…</span></span><span id="MJXc-Node-3483" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-3484" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span><span id="MJXc-Node-3485" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-3486" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-3487" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><mo>…</mo><mo>,</mo><mi>p</mi><mo stretchy="false">[</mo><mi>N</mi><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-316">p[1],\dots,p[N]</script> de réels positifs vérifiant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-317-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/munderover&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3488" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3489" class="mjx-mrow"><span id="MJXc-Node-3490" class="mjx-munderover"><span class="mjx-base"><span id="MJXc-Node-3491" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.522em; padding-bottom: 0.522em;">∑</span></span></span><span class="mjx-stack" style="vertical-align: -0.31em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.422em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-3497" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-3492" class="mjx-texatom" style=""><span id="MJXc-Node-3493" class="mjx-mrow"><span id="MJXc-Node-3494" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">i</span></span><span id="MJXc-Node-3495" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-3496" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span></span><span id="MJXc-Node-3498" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span><span id="MJXc-Node-3499" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-3500" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">i</span></span><span id="MJXc-Node-3501" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span><span id="MJXc-Node-3502" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-3503" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>p</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-317">\sum_{i=1}^n p[i]=1</script> et qui retourne un nombre aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-318-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3504" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3505" class="mjx-mrow"><span id="MJXc-Node-3506" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-318">X</script> tel que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-319-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3507" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3508" class="mjx-mrow"><span id="MJXc-Node-3509" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3510" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3511" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3512" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-3513" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">i</span></span><span id="MJXc-Node-3514" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-3515" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-3516" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span><span id="MJXc-Node-3517" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-3518" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">i</span></span><span id="MJXc-Node-3519" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-319">P(X=i)=p[i]</script>.</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2060&#39;,&#39;imageindication2060&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2060"></a><div class="inner" id="innerindication2060" style="display:none;"></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2060&#39;,&#39;imagecorrige2060&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2060"></a><div class="inner" id="innercorrige2060" style="display:none;">Une méthode possible est basée sur la fonction de répartition de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-320-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3520" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3521" class="mjx-mrow"><span id="MJXc-Node-3522" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-320">X</script>. Plus précisément, notons
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-321-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3523" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3524" class="mjx-mrow"><span id="MJXc-Node-3525" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3526" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">s</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3527" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-3528" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3529" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-321">s_0=0</script> et, pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-322-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3530" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3531" class="mjx-mrow"><span id="MJXc-Node-3532" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-3533" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3534" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-3535" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-3536" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-3537" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-3538" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-322">i=1,\dots,N</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-323-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3539" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3540" class="mjx-mrow"><span id="MJXc-Node-3541" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3542" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">s</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3543" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span id="MJXc-Node-3544" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3545" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-3546" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">s</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3547" class="mjx-texatom" style=""><span id="MJXc-Node-3548" class="mjx-mrow"><span id="MJXc-Node-3549" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-3550" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3551" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-3552" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3553" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-3554" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3555" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mi>i</mi></msub><mo>=</mo><msub><mi>s</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>p</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-323">s_i=s_{i-1}+p_i</script>. Alors, si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-324-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3556" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3557" class="mjx-mrow"><span id="MJXc-Node-3558" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-324">U</script> suit une loi uniforme sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-325-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3559" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3560" class="mjx-mrow"><span id="MJXc-Node-3561" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-3562" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-3563" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-3564" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-3565" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-325">[0,1]</script>,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-326-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3566" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3567" class="mjx-mrow"><span id="MJXc-Node-3568" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3569" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3570" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span><span id="MJXc-Node-3571" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">∈</span></span><span id="MJXc-Node-3572" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span><span id="MJXc-Node-3573" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3574" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">s</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3575" class="mjx-texatom" style=""><span id="MJXc-Node-3576" class="mjx-mrow"><span id="MJXc-Node-3577" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-3578" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3579" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-3580" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-3581" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-3582" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">s</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3583" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span id="MJXc-Node-3584" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span><span id="MJXc-Node-3585" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3586" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3587" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-3588" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">s</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3589" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span id="MJXc-Node-3590" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3591" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-3592" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">s</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3593" class="mjx-texatom" style=""><span id="MJXc-Node-3594" class="mjx-mrow"><span id="MJXc-Node-3595" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-3596" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3597" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-3598" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3599" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-3600" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3601" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>U</mi><mo>∈</mo><mo stretchy="false">]</mo><msub><mi>s</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>s</mi><mi>i</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>s</mi><mi>i</mi></msub><mo>−</mo><msub><mi>s</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>p</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-326">P(U\in ]s_{i-1},s_i])=s_i-s_{i-1}=p_i</script>. L'algorithme suivant reprend cette idée.
<p></p><div class="codebox"><div style="font-family: &#39;Courier New&#39;;font-size:10pt" dir="ltr">
<br>Données :
<br>&nbsp;&nbsp;N, p[1],...,p[N]
<br>Variables :
<br>
<br>Traitement :
<br>&nbsp;&nbsp;S[0]=0
<br>&nbsp;&nbsp;Pour i allant de 1 à N faire
<br>&nbsp;&nbsp;&nbsp;&nbsp;S[i]=S[i-1]+p[i]
<br>&nbsp;&nbsp;Fin Pour
<br>&nbsp;&nbsp;x=ALEA()
<br>&nbsp;&nbsp;Pour i allant de 1 à N faire
<br>&nbsp;&nbsp;&nbsp;&nbsp; Si S[i-1]&lt;x&lt;=S[i] alors retourner i.
<br>&nbsp;&nbsp;Fin Pour
<br>
<br></div></div><p>
Une erreur classique serait d'utiliser la fonction ALEA() à l'intérieur de la deuxième boucle Pour. On ne tirerait pas une seule fois un nombre au hasard, mais plusieurs fois et cela n'aurait plus du tout le même sens (d'ailleurs, on pourrait sortir de la boucle pour sans avoir retourné quelque chose).</p></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>14</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Deux fois pile<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1293"> [Signaler une erreur]</a><a id="exo14" href="javascript:ajouteexo(1293,14,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1293&#39;,&#39;imageenonce1293&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce1293"></a><div class="inner" id="innerenonce1293">On joue à pile ou face avec une pièce non équilibrée. A chaque lancer, la probabilité d'obtenir pile est 2/3, et donc celle d'obtenir face est 1/3.
Les lancers sont supposés indépendants, et on note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-327-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3602" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3603" class="mjx-mrow"><span id="MJXc-Node-3604" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-327">X</script> la variable aléatoire réelle égale au nombre de lancers nécessaires pour obtenir, pour la première fois, deux piles consécutifs.
Pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-328-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3605" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3606" class="mjx-mrow"><span id="MJXc-Node-3607" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-3608" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.458em;">≥</span></span><span id="MJXc-Node-3609" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-328">n\geq 1</script>, on note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-329-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3610" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3611" class="mjx-mrow"><span id="MJXc-Node-3612" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3613" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3614" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-329">p_n</script> la probabilité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-330-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3615" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3616" class="mjx-mrow"><span id="MJXc-Node-3617" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3618" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3619" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3620" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-3621" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-3622" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>n</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-330">P(X=n)</script>.
<ol class="enumeratechiffre">
<li> Expliciter les événements <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-331-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3623" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3624" class="mjx-mrow"><span id="MJXc-Node-3625" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3626" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3627" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-3628" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span><span id="MJXc-Node-3629" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-331">(X=2)</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-332-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3630" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3631" class="mjx-mrow"><span id="MJXc-Node-3632" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3633" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3634" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-3635" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span><span id="MJXc-Node-3636" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-332">(X=3)</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-333-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3637" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3638" class="mjx-mrow"><span id="MJXc-Node-3639" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3640" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3641" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-3642" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span><span id="MJXc-Node-3643" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>4</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-333">(X=4)</script>, et déterminer la valeur de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-334-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3644" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3645" class="mjx-mrow"><span id="MJXc-Node-3646" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3647" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3648" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-334">p_2</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-335-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3649" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3650" class="mjx-mrow"><span id="MJXc-Node-3651" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3652" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3653" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-335">p_3</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-336-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3654" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3655" class="mjx-mrow"><span id="MJXc-Node-3656" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3657" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3658" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>4</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-336">p_4</script>.
</li><li> Montrer que l'on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-337-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3659" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3660" class="mjx-mrow"><span id="MJXc-Node-3661" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3662" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3663" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-3664" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-3665" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.341em;"><span id="MJXc-Node-3666" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.656em;"><span id="MJXc-Node-3667" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">9</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.291em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.413em; vertical-align: -0.464em;"></span></span><span id="MJXc-Node-3668" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3669" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3670" class="mjx-texatom" style=""><span id="MJXc-Node-3671" class="mjx-mrow"><span id="MJXc-Node-3672" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-3673" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-3674" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span></span></span><span id="MJXc-Node-3675" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-3676" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.341em;"><span id="MJXc-Node-3677" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.655em;"><span id="MJXc-Node-3678" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.291em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.412em; vertical-align: -0.463em;"></span></span><span id="MJXc-Node-3679" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3680" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3681" class="mjx-texatom" style=""><span id="MJXc-Node-3682" class="mjx-mrow"><span id="MJXc-Node-3683" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-3684" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-3685" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>2</mn><mn>9</mn></mfrac><msub><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msub><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-337">p_n=\frac{2}{9}p_{n-2}+\frac{1}{3}p_{n-1}</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-338-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3686" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3687" class="mjx-mrow"><span id="MJXc-Node-3688" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-3689" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.458em;">≥</span></span><span id="MJXc-Node-3690" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-338">n\geq 4</script>.
</li><li> En déduire l'expression de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-339-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3691" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3692" class="mjx-mrow"><span id="MJXc-Node-3693" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3694" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3695" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-339">p_n</script> pour tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-340-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3696" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3697" class="mjx-mrow"><span id="MJXc-Node-3698" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-340">n</script>.
</li><li> Rappeler, pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-341-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3699" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3700" class="mjx-mrow"><span id="MJXc-Node-3701" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em; padding-right: 0.014em;">q</span></span><span id="MJXc-Node-3702" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-3703" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span><span id="MJXc-Node-3704" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-3705" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-3706" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-3707" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-3708" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>∈</mo><mo stretchy="false">]</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">[</mo></math></span></span><script type="math/tex" id="MathJax-Element-341">q\in]-1,1[</script>, l'expression de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-342-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3709" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3710" class="mjx-mrow"><span id="MJXc-Node-3711" class="mjx-munderover"><span class="mjx-base"><span id="MJXc-Node-3712" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.522em; padding-bottom: 0.522em;">∑</span></span></span><span class="mjx-stack" style="vertical-align: -0.31em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.351em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-3718" class="mjx-texatom" style=""><span id="MJXc-Node-3719" class="mjx-mrow"><span id="MJXc-Node-3720" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-3721" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.133em; padding-bottom: 0.328em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-3713" class="mjx-texatom" style=""><span id="MJXc-Node-3714" class="mjx-mrow"><span id="MJXc-Node-3715" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-3716" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-3717" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span></span></span></span></span></span><span id="MJXc-Node-3722" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-3723" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.014em;"><span id="MJXc-Node-3724" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em; padding-right: 0.014em;">q</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.096em; padding-right: 0.071em;"><span id="MJXc-Node-3725" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mi>n</mi><msup><mi>q</mi><mi>n</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-342">\sum_{n=0}^{+\infty}nq^n</script>, et calculer alors <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-343-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3726" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3727" class="mjx-mrow"><span id="MJXc-Node-3728" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-3729" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3730" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3731" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-343">E(X)</script>. Interpréter.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1293&#39;,&#39;imageindication1293&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication1293"></a><div class="inner" id="innerindication1293" style="display:none;"><ol class="enumeratechiffre">
<li> Introduire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-344-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3732" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3733" class="mjx-mrow"><span id="MJXc-Node-3734" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-3735" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-3736" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>k</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-344">P_k</script> (resp. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-345-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3737" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3738" class="mjx-mrow"><span id="MJXc-Node-3739" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3740" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-3741" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>k</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-345">F_k</script>) l'événement "on obtient pile" (resp. face) au <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-346-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3742" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3743" class="mjx-mrow"><span id="MJXc-Node-3744" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-3745" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>−</mo></math></span></span><script type="math/tex" id="MathJax-Element-346">k-</script>ième lancer.
</li><li> Raisonner par disjonction de cas en utilisant la formule des probabilités totales, suivant la valeur du premier lancer!
</li><li> Revoir les suites récurrentes d'ordre 2.
</li><li>
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1293&#39;,&#39;imagecorrige1293&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige1293"></a><div class="inner" id="innercorrige1293" style="display:none;"><ol class="enumeratechiffre">
<li> On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-347-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3746" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3747" class="mjx-mrow"><span id="MJXc-Node-3748" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-3749" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-3750" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>k</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-347">P_k</script> (resp. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-348-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3751" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3752" class="mjx-mrow"><span id="MJXc-Node-3753" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3754" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-3755" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>k</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-348">F_k</script>) l'événement on obtient pile (resp. face) au <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-349-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3756" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3757" class="mjx-mrow"><span id="MJXc-Node-3758" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-3759" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>−</mo></math></span></span><script type="math/tex" id="MathJax-Element-349">k-</script>ième lancer. L'événement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-350-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3760" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3761" class="mjx-mrow"><span id="MJXc-Node-3762" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3763" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3764" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3765" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-3766" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-350">(X=2)</script> correspond à :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-351-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27F9;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3767" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3768" class="mjx-mrow"><span id="MJXc-Node-3769" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3770" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3771" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3772" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-3773" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3774" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3775" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-3776" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3777" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3778" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-3779" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3780" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3781" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-3782" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟹</span></span><span id="MJXc-Node-3783" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-3784" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-3785" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3786" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3787" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3788" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-3789" class="mjx-mrow"><span id="MJXc-Node-3790" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-3791" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-3792" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-3793" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3794" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-3795" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3796" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3797" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.379em;"><span id="MJXc-Node-3798" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-3799" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.111em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3800" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>P</mi><mn>1</mn></msub><msub><mi>P</mi><mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo stretchy="false">⟹</mo><mspace width="thickmathspace"></mspace><msub><mi>p</mi><mn>2</mn></msub><mo>=</mo><msup><mrow><mo>(</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><mfrac><mn>4</mn><mn>9</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-351">(X=2)=P_1P_2\implies p_2=\left(\frac{2}{3}\right)^2=\frac4{9}.</script>
De même,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-352-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27F9;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;27&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3801" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3802" class="mjx-mrow"><span id="MJXc-Node-3803" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3804" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3805" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3806" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-3807" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3808" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3809" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3810" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3811" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3812" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-3813" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3814" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3815" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-3816" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3817" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3818" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-3819" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟹</span></span><span id="MJXc-Node-3820" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-3821" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-3822" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3823" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3824" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3825" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-3826" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-3827" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3828" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3829" class="mjx-mrow"><span id="MJXc-Node-3830" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-3831" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-3832" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-3833" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3834" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-3835" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3836" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3837" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.379em;"><span id="MJXc-Node-3838" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.74em;"><span id="MJXc-Node-3839" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">27</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.119em; vertical-align: -0.74em;"></span></span><span id="MJXc-Node-3840" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>F</mi><mn>1</mn></msub><msub><mi>P</mi><mn>2</mn></msub><msub><mi>P</mi><mn>3</mn></msub><mspace width="thickmathspace"></mspace><mo stretchy="false">⟹</mo><mspace width="thickmathspace"></mspace><msub><mi>p</mi><mn>3</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mrow><mo>(</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><mfrac><mn>4</mn><mn>27</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-352">(X=3)=F_1P_2P_3\implies p_3=\frac{1}{3}\left(\frac{2}{3}\right)^2=\frac4{27}.</script>
Pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-353-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3841" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3842" class="mjx-mrow"><span id="MJXc-Node-3843" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3844" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3845" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3846" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-3847" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>4</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-353">(X=4)</script>, cela se corse un peu !
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-354-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27F9;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;27&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3848" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3849" class="mjx-mrow"><span id="MJXc-Node-3850" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3851" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3852" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3853" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-3854" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3855" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3856" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3857" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3858" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3859" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3860" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3861" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3862" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-3863" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3864" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3865" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-3866" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3867" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span id="MJXc-Node-3868" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3869" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-3870" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3871" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3872" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3873" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3874" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3875" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-3876" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3877" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3878" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-3879" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3880" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span id="MJXc-Node-3881" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-3882" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟹</span></span><span id="MJXc-Node-3883" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-3884" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-3885" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3886" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span id="MJXc-Node-3887" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3888" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-3889" class="mjx-mrow"><span id="MJXc-Node-3890" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-3891" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-3892" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-3893" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3894" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-3895" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3896" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3897" class="mjx-mrow"><span id="MJXc-Node-3898" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-3899" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-3900" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-3901" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3902" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-3903" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3904" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3905" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-3906" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-3907" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-3908" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-3909" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3910" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span><span id="MJXc-Node-3911" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-3912" class="mjx-mrow"><span id="MJXc-Node-3913" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-3914" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-3915" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-3916" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3917" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-3918" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3919" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3920" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.379em;"><span id="MJXc-Node-3921" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.74em;"><span id="MJXc-Node-3922" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">27</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.119em; vertical-align: -0.74em;"></span></span><span id="MJXc-Node-3923" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>4</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>F</mi><mn>1</mn></msub><msub><mi>F</mi><mn>2</mn></msub><msub><mi>P</mi><mn>3</mn></msub><msub><mi>P</mi><mn>4</mn></msub><mo>∪</mo><msub><mi>P</mi><mn>1</mn></msub><msub><mi>F</mi><mn>2</mn></msub><msub><mi>P</mi><mn>3</mn></msub><msub><mi>P</mi><mn>4</mn></msub><mspace width="thickmathspace"></mspace><mo stretchy="false">⟹</mo><mspace width="thickmathspace"></mspace><msub><mi>p</mi><mn>4</mn></msub><mo>=</mo><msup><mrow><mo>(</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>)</mo></mrow><mn>2</mn></msup><msup><mrow><mo>(</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>)</mo></mrow><msup><mrow><mo>(</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>)</mo></mrow><mn>3</mn></msup><mo>=</mo><mfrac><mn>4</mn><mn>27</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-354">(X=4)=F_1F_2P_3P_4\cup P_1F_2P_3P_4\implies p_4=\left(\frac{1}{3}\right)^2\left(\frac 23\right)^2+\left(\frac 13\right)\left(\frac 23\right)^3=\frac 4{27}.</script>
</li><li> On s'inspire du calcul de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-355-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3924" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3925" class="mjx-mrow"><span id="MJXc-Node-3926" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3927" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3928" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>4</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-355">p_4</script> : pour obtenir <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-356-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3929" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3930" class="mjx-mrow"><span id="MJXc-Node-3931" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3932" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3933" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-356">X=n</script>, on peut :
<ul class="rien">
<li> ou bien avoir obtenu pile au 1er lancer. Dans ce cas, on a forcément obtenu face au second lancer (sinon <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-357-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3934" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3935" class="mjx-mrow"><span id="MJXc-Node-3936" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3937" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3938" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-357">X=2</script>), donc avec une probabilité de 1/3. Puis il reste <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-358-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3939" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3940" class="mjx-mrow"><span id="MJXc-Node-3941" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-3942" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3943" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>−</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-358">n-2</script> lancers, et le premier "double pile" doit arriver au bout du <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-359-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3944" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3945" class="mjx-mrow"><span id="MJXc-Node-3946" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-3947" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3948" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>−</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-359">n-2</script>-ème. Ceci se produit avec une probabilité valant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-360-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3949" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3950" class="mjx-mrow"><span id="MJXc-Node-3951" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3952" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3953" class="mjx-texatom" style=""><span id="MJXc-Node-3954" class="mjx-mrow"><span id="MJXc-Node-3955" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-3956" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3957" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-360">p_{n-2}</script>. On a donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-361-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;et&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3958" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3959" class="mjx-mrow"><span id="MJXc-Node-3960" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3961" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3962" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-3963" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3964" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-3965" class="mjx-texatom"><span id="MJXc-Node-3966" class="mjx-mrow"><span id="MJXc-Node-3967" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-3968" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-3969" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3970" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3971" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3972" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3973" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-3974" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-3975" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3976" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3977" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3978" class="mjx-texatom" style=""><span id="MJXc-Node-3979" class="mjx-mrow"><span id="MJXc-Node-3980" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-3981" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3982" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span id="MJXc-Node-3983" class="mjx-texatom"><span id="MJXc-Node-3984" class="mjx-mrow"><span id="MJXc-Node-3985" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">&nbsp;et&nbsp;</span></span></span></span><span id="MJXc-Node-3986" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3987" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3988" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-3989" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3990" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3991" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3992" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3993" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-3994" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-3995" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3996" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>P</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msub><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;et&nbsp;</mtext></mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>P</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-361">P(X=n|P_1)=\frac 13p_{n-2}\textrm{ et }P(P_1)=\frac 23.</script>
</li><li> ou bien avoir obtenu face au 1er lancer. Il reste <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-362-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-3997" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3998" class="mjx-mrow"><span id="MJXc-Node-3999" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4000" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4001" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-362">n-1</script> lancers où il faut obtenir le premier double pile au bout du <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-363-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4002" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4003" class="mjx-mrow"><span id="MJXc-Node-4004" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4005" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4006" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-363">n-1</script>-ème, ce qui se produit avec une probabilité valant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-364-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4007" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4008" class="mjx-mrow"><span id="MJXc-Node-4009" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4010" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4011" class="mjx-texatom" style=""><span id="MJXc-Node-4012" class="mjx-mrow"><span id="MJXc-Node-4013" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4014" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4015" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-364">p_{n-1}</script>. On a donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-365-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;et&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4016" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4017" class="mjx-mrow"><span id="MJXc-Node-4018" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4019" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4020" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4021" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4022" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4023" class="mjx-texatom"><span id="MJXc-Node-4024" class="mjx-mrow"><span id="MJXc-Node-4025" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-4026" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4027" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4028" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4029" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4030" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4031" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-4032" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4033" class="mjx-texatom" style=""><span id="MJXc-Node-4034" class="mjx-mrow"><span id="MJXc-Node-4035" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4036" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4037" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-4038" class="mjx-texatom"><span id="MJXc-Node-4039" class="mjx-mrow"><span id="MJXc-Node-4040" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">&nbsp;et&nbsp;</span></span></span></span><span id="MJXc-Node-4041" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4042" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4043" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4044" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4045" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4046" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4047" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4048" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-4049" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-4050" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-4051" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>F</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;et&nbsp;</mtext></mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-365">P(X=n|F_1)=p_{n-1}\textrm{ et }P(F_1)=\frac 13.</script>
</li></ul>
D'après la formule des probabilités totales, on trouve :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-366-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4052" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4053" class="mjx-mrow"><span id="MJXc-Node-4054" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4055" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4056" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4057" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4058" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-4059" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-4060" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-4061" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4062" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4063" class="mjx-texatom" style=""><span id="MJXc-Node-4064" class="mjx-mrow"><span id="MJXc-Node-4065" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4066" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4067" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span id="MJXc-Node-4068" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4069" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-4070" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-4071" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-4072" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4073" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4074" class="mjx-texatom" style=""><span id="MJXc-Node-4075" class="mjx-mrow"><span id="MJXc-Node-4076" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4077" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4078" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-4079" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>p</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>2</mn><mn>9</mn></mfrac><msub><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msub><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-366">p_n=\frac{2}{9}p_{n-2}+\frac{1}{3}p_{n-1}.</script>
</li><li> On a une classique formule de récurrence linéaire d'ordre 2. L'équation caractéristique <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-367-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4080" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4081" class="mjx-mrow"><span id="MJXc-Node-4082" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4083" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4084" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-4085" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4086" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-4087" class="mjx-texatom"><span id="MJXc-Node-4088" class="mjx-mrow"><span id="MJXc-Node-4089" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-4090" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-4091" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4092" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-4093" class="mjx-texatom"><span id="MJXc-Node-4094" class="mjx-mrow"><span id="MJXc-Node-4095" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-4096" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>r</mi><mn>2</mn></msup><mo>=</mo><mi>r</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn><mo>+</mo><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>9</mn></math></span></span><script type="math/tex" id="MathJax-Element-367">r^2=r/3+2/9</script> a pour solution <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-368-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4097" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4098" class="mjx-mrow"><span id="MJXc-Node-4099" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-4100" class="mjx-texatom"><span id="MJXc-Node-4101" class="mjx-mrow"><span id="MJXc-Node-4102" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-4103" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-368">2/3</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-369-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4104" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4105" class="mjx-mrow"><span id="MJXc-Node-4106" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4107" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4108" class="mjx-texatom"><span id="MJXc-Node-4109" class="mjx-mrow"><span id="MJXc-Node-4110" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-4111" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-369">-1/3</script>. On en déduit finalement que, pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-370-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4112" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4113" class="mjx-mrow"><span id="MJXc-Node-4114" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4115" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-4116" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-370">n\geq 2</script>, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-371-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;&amp;#x03B1;&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;&amp;#x03B2;&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4117" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4118" class="mjx-mrow"><span id="MJXc-Node-4119" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4120" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4121" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4122" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4123" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">α</span></span><span id="MJXc-Node-4124" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4125" class="mjx-mrow"><span id="MJXc-Node-4126" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-4127" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-4128" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-4129" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-4130" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4131" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4132" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4133" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.007em;">β</span></span><span id="MJXc-Node-4134" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4135" class="mjx-mrow"><span id="MJXc-Node-4136" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-4137" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.478em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.478em; top: -1.368em;"><span id="MJXc-Node-4138" class="mjx-mrow"><span id="MJXc-Node-4139" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4140" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="width: 1.478em; bottom: -0.733em;"><span id="MJXc-Node-4141" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.478em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-4142" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4143" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4144" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>p</mi><mi>n</mi></msub><mo>=</mo><mi>α</mi><msup><mrow><mo>(</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>)</mo></mrow><mi>n</mi></msup><mo>+</mo><mi>β</mi><msup><mrow><mo>(</mo><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><mn>3</mn></mfrac><mo>)</mo></mrow><mi>n</mi></msup><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-371">p_n=\alpha\left(\frac{2}{3}\right)^n+\beta\left(\frac{-1}{3}\right)^n.</script>
On détermine <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-372-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03B1;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4145" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4146" class="mjx-mrow"><span id="MJXc-Node-4147" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">α</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math></span></span><script type="math/tex" id="MathJax-Element-372">\alpha</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-373-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03B2;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4148" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4149" class="mjx-mrow"><span id="MJXc-Node-4150" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.007em;">β</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math></span></span><script type="math/tex" id="MathJax-Element-373">\beta</script> en testant sur les premiers termes (<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-374-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4151" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4152" class="mjx-mrow"><span id="MJXc-Node-4153" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4154" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4155" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-374">p_2</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-375-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4156" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4157" class="mjx-mrow"><span id="MJXc-Node-4158" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4159" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4160" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-375">p_3</script>). On obtient :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-376-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4161" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4162" class="mjx-mrow"><span id="MJXc-Node-4163" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4164" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4165" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4166" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4167" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-4168" class="mjx-mrow"><span id="MJXc-Node-4169" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-4170" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-4171" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-4172" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-4173" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4174" class="mjx-texatom" style=""><span id="MJXc-Node-4175" class="mjx-mrow"><span id="MJXc-Node-4176" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4177" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4178" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-4179" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4180" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.379em;"><span id="MJXc-Node-4181" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-4182" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.111em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-4183" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4184" class="mjx-mrow"><span id="MJXc-Node-4185" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-4186" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.478em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.478em; top: -1.368em;"><span id="MJXc-Node-4187" class="mjx-mrow"><span id="MJXc-Node-4188" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4189" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="width: 1.478em; bottom: -0.733em;"><span id="MJXc-Node-4190" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.478em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-4191" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4192" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4193" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>p</mi><mi>n</mi></msub><mo>=</mo><msup><mrow><mo>(</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><msup><mrow><mo>(</mo><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><mn>3</mn></mfrac><mo>)</mo></mrow><mi>n</mi></msup><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-376">p_n=\left(\frac{2}{3}\right)^{n+1}+\frac{4}{3}\left(\frac{-1}{3}\right)^n.</script>
Remarquons que cette formule reste valable pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-377-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4194" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4195" class="mjx-mrow"><span id="MJXc-Node-4196" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4197" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4198" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-377">n=1</script> puisqu'on a alors <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-378-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4199" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4200" class="mjx-mrow"><span id="MJXc-Node-4201" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4202" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4203" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4204" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4205" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-378">p_1=0</script>.
</li><li> Il est bien connu que pour tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-379-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4206" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4207" class="mjx-mrow"><span id="MJXc-Node-4208" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span><span id="MJXc-Node-4209" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">∈</span></span><span id="MJXc-Node-4210" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span><span id="MJXc-Node-4211" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4212" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4213" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4214" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4215" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>∈</mo><mo stretchy="false">]</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">[</mo></math></span></span><script type="math/tex" id="MathJax-Element-379">q\in]-1,1[</script>, on a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-380-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4216" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4217" class="mjx-mrow"><span id="MJXc-Node-4218" class="mjx-munderover"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.176em; padding-top: 0.141em; padding-left: 0.132em;"><span id="MJXc-Node-4225" class="mjx-texatom" style=""><span id="MJXc-Node-4226" class="mjx-mrow"><span id="MJXc-Node-4227" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4228" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-op"><span id="MJXc-Node-4219" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.082em;"><span id="MJXc-Node-4220" class="mjx-texatom" style=""><span id="MJXc-Node-4221" class="mjx-mrow"><span id="MJXc-Node-4222" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4223" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4224" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span></span></span></span><span id="MJXc-Node-4229" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4230" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.014em;"><span id="MJXc-Node-4231" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0.096em; padding-right: 0.071em;"><span id="MJXc-Node-4232" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4233" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4234" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.549em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.549em; top: -1.178em;"><span id="MJXc-Node-4235" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-denominator" style="width: 3.549em; bottom: -1.045em;"><span id="MJXc-Node-4236" class="mjx-mrow"><span id="MJXc-Node-4237" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4238" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4239" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4240" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span><span id="MJXc-Node-4241" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4242" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4243" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.549em;"></span></span><span class="mjx-vsize" style="height: 2.223em; vertical-align: -1.045em;"></span></span><span id="MJXc-Node-4244" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mi>n</mi><msup><mi>q</mi><mi>n</mi></msup><mo>=</mo><mfrac><mi>q</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>q</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-380">\sum_{n=0}^{+\infty}nq^n=\frac{q}{(1-q)^2}.</script>
On en déduit, après un peu de calculs :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-381-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;15&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4245" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4246" class="mjx-mrow"><span id="MJXc-Node-4247" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-4248" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4249" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4250" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4251" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4252" class="mjx-munderover MJXc-space3"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.176em; padding-top: 0.141em; padding-left: 0.132em;"><span id="MJXc-Node-4259" class="mjx-texatom" style=""><span id="MJXc-Node-4260" class="mjx-mrow"><span id="MJXc-Node-4261" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4262" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-op"><span id="MJXc-Node-4253" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.082em;"><span id="MJXc-Node-4254" class="mjx-texatom" style=""><span id="MJXc-Node-4255" class="mjx-mrow"><span id="MJXc-Node-4256" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4257" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4258" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span id="MJXc-Node-4263" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4264" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4265" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4266" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4267" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4268" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.368em;"><span id="MJXc-Node-4269" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">15</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.719em;"><span id="MJXc-Node-4270" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.087em; vertical-align: -0.719em;"></span></span><span id="MJXc-Node-4271" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mi>n</mi><msub><mi>p</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>15</mn><mn>4</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-381">E(X)=\sum_{n=1}^{+\infty}np_n=\frac{15}{4}.</script>
En moyenne, si on répète un grand nombre de fois l'expérience aléatoire, il faudra 15/4 lancers (non entier!) pour obtenir pour la première fois deux piles consécutifs.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>15</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Maximum d'une loi de Poisson<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1279"> [Signaler une erreur]</a><a id="exo15" href="javascript:ajouteexo(1279,15,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1279&#39;,&#39;imageenonce1279&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce1279"></a><div class="inner" id="innerenonce1279">Soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-382-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4272" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4273" class="mjx-mrow"><span id="MJXc-Node-4274" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-382">X</script> une variable aléatoire réelle suivant une loi de Poisson <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-383-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4275" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4276" class="mjx-mrow"><span id="MJXc-Node-4277" class="mjx-texatom"><span id="MJXc-Node-4278" class="mjx-mrow"><span id="MJXc-Node-4279" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.393em; padding-bottom: 0.393em; padding-right: 0.037em;">P</span></span></span></span><span id="MJXc-Node-4280" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-4281" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">λ</span></span><span id="MJXc-Node-4282" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">P</mi></mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-383">\mathcal P(\lambda)</script>.
Pour quelle(s) valeur(s) de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-384-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4283" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4284" class="mjx-mrow"><span id="MJXc-Node-4285" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">k</span></span><span id="MJXc-Node-4286" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-4287" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-4288" class="mjx-mrow"><span id="MJXc-Node-4289" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-384">k\in\mathbb N</script> la probabilité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-385-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4290" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4291" class="mjx-mrow"><span id="MJXc-Node-4292" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4293" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-4294" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4295" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-4296" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">k</span></span><span id="MJXc-Node-4297" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-385">P(X=k)</script> est maximale?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1279&#39;,&#39;imageindication1279&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication1279"></a><div class="inner" id="innerindication1279" style="display:none;">Calculer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-386-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4298" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4299" class="mjx-mrow"><span id="MJXc-Node-4300" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4301" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4302" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4303" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4304" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4305" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4306" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4307" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4308" class="mjx-texatom"><span id="MJXc-Node-4309" class="mjx-mrow"><span id="MJXc-Node-4310" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-4311" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4312" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4313" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4314" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4315" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4316" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-386">P(X=k+1)/P(X=k)</script>.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1279&#39;,&#39;imagecorrige1279&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige1279"></a><div class="inner" id="innercorrige1279" style="display:none;">On a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-387-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4317" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4318" class="mjx-mrow"><span id="MJXc-Node-4319" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 6.158em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 6.158em; top: -1.542em;"><span id="MJXc-Node-4320" class="mjx-mrow"><span id="MJXc-Node-4321" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4322" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4323" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4324" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4325" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4326" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4327" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4328" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 6.158em; bottom: -1.042em;"><span id="MJXc-Node-4329" class="mjx-mrow"><span id="MJXc-Node-4330" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4331" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4332" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4333" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4334" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4335" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 6.158em;"></span></span><span class="mjx-vsize" style="height: 2.585em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-4336" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4337" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.443em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.443em; top: -1.396em;"><span id="MJXc-Node-4338" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span><span class="mjx-denominator" style="width: 2.443em; bottom: -0.818em;"><span id="MJXc-Node-4339" class="mjx-mrow"><span id="MJXc-Node-4340" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4341" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4342" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.443em;"></span></span><span class="mjx-vsize" style="height: 2.214em; vertical-align: -0.818em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mi>λ</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-387">\frac{P(X=k+1)}{P(X=k)}=\frac{\lambda}{k+1}</script>
d'où
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-388-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27FA;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4343" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4344" class="mjx-mrow"><span id="MJXc-Node-4345" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 6.158em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 6.158em; top: -1.542em;"><span id="MJXc-Node-4346" class="mjx-mrow"><span id="MJXc-Node-4347" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4348" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4349" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4350" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4351" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4352" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4353" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4354" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 6.158em; bottom: -1.042em;"><span id="MJXc-Node-4355" class="mjx-mrow"><span id="MJXc-Node-4356" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4357" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4358" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4359" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4360" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4361" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 6.158em;"></span></span><span class="mjx-vsize" style="height: 2.585em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-4362" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-4363" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4364" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-4365" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟺</span></span><span id="MJXc-Node-4366" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-4367" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4368" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-4369" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span><span id="MJXc-Node-4370" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4371" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>≥</mo><mn>1</mn><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mi>k</mi><mo>≤</mo><mi>λ</mi><mo>−</mo><mn>1.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-388">\frac{P(X=k+1)}{P(X=k)}\geq 1\iff k\leq \lambda-1.</script>
On distingue alors trois cas :
<ol class="enumeratechiffre">
<li> Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-389-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4372" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4373" class="mjx-mrow"><span id="MJXc-Node-4374" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span><span id="MJXc-Node-4375" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4376" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4377" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&lt;</span></span><span id="MJXc-Node-4378" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>−</mo><mn>1</mn><mo>&lt;</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-389">\lambda-1<0</script>, c'est-à-dire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-390-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4379" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4380" class="mjx-mrow"><span id="MJXc-Node-4381" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span><span id="MJXc-Node-4382" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">∈</span></span><span id="MJXc-Node-4383" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span><span id="MJXc-Node-4384" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-4385" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4386" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4387" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>∈</mo><mo stretchy="false">]</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">[</mo></math></span></span><script type="math/tex" id="MathJax-Element-390">\lambda\in ]0,1[</script>, la suite <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-391-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4388" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4389" class="mjx-mrow"><span id="MJXc-Node-4390" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4391" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4392" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4393" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4394" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4395" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4396" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4397" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-391">(P(X=k))</script> est strictement décroissante, et donc son maximum est atteint en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-392-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4398" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4399" class="mjx-mrow"><span id="MJXc-Node-4400" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4401" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4402" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4403" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4404" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-4405" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-392">P(X=0)</script>.
</li><li> si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-393-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4406" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4407" class="mjx-mrow"><span id="MJXc-Node-4408" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-393">\lambda</script> est un entier, la suite est strictement croissante jusque <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-394-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4409" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4410" class="mjx-mrow"><span id="MJXc-Node-4411" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span><span id="MJXc-Node-4412" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4413" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-394">\lambda-1</script>, strictement décroissante à partir de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-395-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4414" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4415" class="mjx-mrow"><span id="MJXc-Node-4416" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-395">\lambda</script> et le maximum est atteint en deux points : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-396-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4417" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4418" class="mjx-mrow"><span id="MJXc-Node-4419" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4420" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4421" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4422" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4423" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span><span id="MJXc-Node-4424" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4425" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4426" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>λ</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-396">P(X=\lambda-1)</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-397-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4427" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4428" class="mjx-mrow"><span id="MJXc-Node-4429" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4430" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4431" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4432" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4433" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span><span id="MJXc-Node-4434" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>λ</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-397">P(X=\lambda)</script>.
</li><li> Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-398-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4435" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4436" class="mjx-mrow"><span id="MJXc-Node-4437" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-398">\lambda</script> n'est pas un entier, la suite est strictement croissante jusque <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-399-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;&amp;#x230A;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;&amp;#x230B;&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4438" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4439" class="mjx-mrow"><span id="MJXc-Node-4440" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">⌊</span></span><span id="MJXc-Node-4441" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span><span id="MJXc-Node-4442" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4443" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4444" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">⌋</span></span><span id="MJXc-Node-4445" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4446" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">⌊</mo><mi>λ</mi><mo>−</mo><mn>1</mn><mo fence="false" stretchy="false">⌋</mo><mo>+</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-399">\lfloor \lambda-1\rfloor+1</script>, puis strictement décroissante ensuite. La valeur maximale est donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-400-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;&amp;#x230A;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;&amp;#x230B;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4447" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4448" class="mjx-mrow"><span id="MJXc-Node-4449" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4450" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4451" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4452" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4453" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">⌊</span></span><span id="MJXc-Node-4454" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span><span id="MJXc-Node-4455" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">⌋</span></span><span id="MJXc-Node-4456" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mo fence="false" stretchy="false">⌊</mo><mi>λ</mi><mo fence="false" stretchy="false">⌋</mo><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-400">P(X=\lfloor \lambda\rfloor)</script>.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>16</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Retrouver une loi connaissant son conditionnement<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1280"> [Signaler une erreur]</a><a id="exo16" href="javascript:ajouteexo(1280,16,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1280&#39;,&#39;imageenonce1280&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce1280"></a><div class="inner" id="innerenonce1280">Soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-401-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4457" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4458" class="mjx-mrow"><span id="MJXc-Node-4459" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-401">X</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-402-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4460" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4461" class="mjx-mrow"><span id="MJXc-Node-4462" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-402">Y</script> deux variables aléatoires. On suppose que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-403-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4463" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4464" class="mjx-mrow"><span id="MJXc-Node-4465" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-403">X</script> suit une loi de Poisson <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-404-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4466" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4467" class="mjx-mrow"><span id="MJXc-Node-4468" class="mjx-texatom"><span id="MJXc-Node-4469" class="mjx-mrow"><span id="MJXc-Node-4470" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.393em; padding-bottom: 0.393em; padding-right: 0.037em;">P</span></span></span></span><span id="MJXc-Node-4471" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-4472" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">λ</span></span><span id="MJXc-Node-4473" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">P</mi></mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-404">\mathcal P(\lambda)</script> et que la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-405-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4474" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4475" class="mjx-mrow"><span id="MJXc-Node-4476" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-405">Y</script> conditionnée par <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-406-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4477" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4478" class="mjx-mrow"><span id="MJXc-Node-4479" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-4480" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4481" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-4482" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4483" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>n</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-406">(X=n)</script> est la loi binomiale <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-407-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4484" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4485" class="mjx-mrow"><span id="MJXc-Node-4486" class="mjx-texatom"><span id="MJXc-Node-4487" class="mjx-mrow"><span id="MJXc-Node-4488" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.007em;">B</span></span></span></span><span id="MJXc-Node-4489" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-4490" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4491" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-4492" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span><span id="MJXc-Node-4493" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">B</mi></mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-407">\mathcal B(n,p)</script>, pour tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-408-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4494" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4495" class="mjx-mrow"><span id="MJXc-Node-4496" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4497" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-4498" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-4499" class="mjx-mrow"><span id="MJXc-Node-4500" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-408">n\in\mathbb N</script>. Quelle est la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-409-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4501" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4502" class="mjx-mrow"><span id="MJXc-Node-4503" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-409">Y</script>?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1280&#39;,&#39;imageindication1280&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication1280"></a><div class="inner" id="innerindication1280" style="display:none;">Utiliser la formule des probabilités totales. Il faut aussi triturer la somme un petit peu (ne sommer que sur les termes non nuls, puis faire un changement d'indices pour retrouver la fonction exponentielle...).</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1280&#39;,&#39;imagecorrige1280&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige1280"></a><div class="inner" id="innercorrige1280" style="display:none;">On remarque d'abord que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-410-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4504" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4505" class="mjx-mrow"><span id="MJXc-Node-4506" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-410">Y</script> est à valeurs entières. D'après la formule des probabilités totales,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-411-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0 thickmathspace&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4507" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4508" class="mjx-mrow"><span id="MJXc-Node-4509" class="mjx-mtable" style="vertical-align: -7.195em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-4510" class="mjx-mtr" style="height: 3.118em;"><span id="MJXc-Node-4511" class="mjx-mtd" style="padding: 0px; text-align: right; width: 4.147em;"><span id="MJXc-Node-4512" class="mjx-mrow" style="margin-top: 0.705em;"><span id="MJXc-Node-4513" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4514" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4515" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-4516" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4517" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4518" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4519" class="mjx-mtd" style="padding: 0px 0.139em 0px 0px; width: 1.056em;"><span id="MJXc-Node-4520" class="mjx-mrow" style="margin-top: 0.705em;"><span id="MJXc-Node-4521" class="mjx-mi"></span><span id="MJXc-Node-4522" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4523" class="mjx-mtd" style="padding: 0px 0px 0px 0.139em; text-align: left; width: 13.135em;"><span id="MJXc-Node-4524" class="mjx-mrow"><span id="MJXc-Node-4525" class="mjx-mi"></span><span id="MJXc-Node-4526" class="mjx-munderover MJXc-space1"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.176em; padding-top: 0.141em; padding-left: 0.132em;"><span id="MJXc-Node-4533" class="mjx-texatom" style=""><span id="MJXc-Node-4534" class="mjx-mrow"><span id="MJXc-Node-4535" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4536" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-op"><span id="MJXc-Node-4527" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.082em;"><span id="MJXc-Node-4528" class="mjx-texatom" style=""><span id="MJXc-Node-4529" class="mjx-mrow"><span id="MJXc-Node-4530" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4531" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4532" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span></span></span></span><span id="MJXc-Node-4537" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4538" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4539" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-4540" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4541" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4542" class="mjx-texatom"><span id="MJXc-Node-4543" class="mjx-mrow"><span id="MJXc-Node-4544" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-4545" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4546" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4547" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4548" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4549" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4550" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4551" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4552" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4553" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4554" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-4555" class="mjx-mtr" style="height: 3.28em;"><span id="MJXc-Node-4556" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-4557" class="mjx-mrow" style="margin-top: 0.705em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4558" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-4559" class="mjx-mrow" style="margin-top: 0.705em;"><span id="MJXc-Node-4560" class="mjx-mi"></span><span id="MJXc-Node-4561" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4562" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-4563" class="mjx-mrow"><span id="MJXc-Node-4564" class="mjx-mi"></span><span id="MJXc-Node-4565" class="mjx-munderover MJXc-space1"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.176em; padding-top: 0.141em; padding-left: 0.132em;"><span id="MJXc-Node-4572" class="mjx-texatom" style=""><span id="MJXc-Node-4573" class="mjx-mrow"><span id="MJXc-Node-4574" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4575" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-op"><span id="MJXc-Node-4566" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.072em;"><span id="MJXc-Node-4567" class="mjx-texatom" style=""><span id="MJXc-Node-4568" class="mjx-mrow"><span id="MJXc-Node-4569" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4570" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4571" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span></span></span><span id="MJXc-Node-4576" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-4577" class="mjx-TeXmathchoice"><span id="MJXc-Node-4578" class="mjx-texatom"><span id="MJXc-Node-4579" class="mjx-mrow"><span id="MJXc-Node-4580" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-4581" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.6em;"><span class="mjx-numerator" style="width: 0.6em; top: -1.144em;"><span id="MJXc-Node-4582" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-denominator" style="width: 0.6em; bottom: -0.722em;"><span id="MJXc-Node-4583" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="mjx-vsize" style="height: 1.865em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-4584" class="mjx-TeXmathchoice"><span id="MJXc-Node-4585" class="mjx-texatom"><span id="MJXc-Node-4586" class="mjx-mrow"><span id="MJXc-Node-4587" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-4588" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4589" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4590" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-4591" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4592" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4593" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4594" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-4595" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4596" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4597" class="mjx-texatom" style=""><span id="MJXc-Node-4598" class="mjx-mrow"><span id="MJXc-Node-4599" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4600" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4601" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span id="MJXc-Node-4602" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4603" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4604" class="mjx-texatom" style=""><span id="MJXc-Node-4605" class="mjx-mrow"><span id="MJXc-Node-4606" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4607" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span></span></span></span><span id="MJXc-Node-4608" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.243em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.243em; top: -1.396em;"><span id="MJXc-Node-4609" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4610" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4611" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span><span class="mjx-denominator" style="width: 1.243em; bottom: -0.769em;"><span id="MJXc-Node-4612" class="mjx-mrow"><span id="MJXc-Node-4613" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4614" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.243em;"></span></span><span class="mjx-vsize" style="height: 2.165em; vertical-align: -0.769em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-4615" class="mjx-mtr" style="height: 3.28em;"><span id="MJXc-Node-4616" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-4617" class="mjx-mrow" style="margin-top: 0.705em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4618" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-4619" class="mjx-mrow" style="margin-top: 0.705em;"><span id="MJXc-Node-4620" class="mjx-mi"></span><span id="MJXc-Node-4621" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4622" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-4623" class="mjx-mrow"><span id="MJXc-Node-4624" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.557em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.557em; top: -1.583em;"><span id="MJXc-Node-4625" class="mjx-mrow"><span id="MJXc-Node-4626" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4627" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4628" class="mjx-texatom" style=""><span id="MJXc-Node-4629" class="mjx-mrow"><span id="MJXc-Node-4630" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4631" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span></span></span></span><span id="MJXc-Node-4632" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4633" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4634" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-4635" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4636" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4637" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span class="mjx-denominator" style="width: 3.557em; bottom: -0.769em;"><span id="MJXc-Node-4638" class="mjx-mrow"><span id="MJXc-Node-4639" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4640" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.557em;"></span></span><span class="mjx-vsize" style="height: 2.352em; vertical-align: -0.769em;"></span></span><span id="MJXc-Node-4641" class="mjx-munderover MJXc-space1"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.176em; padding-top: 0.141em; padding-left: 0.132em;"><span id="MJXc-Node-4648" class="mjx-texatom" style=""><span id="MJXc-Node-4649" class="mjx-mrow"><span id="MJXc-Node-4650" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4651" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-op"><span id="MJXc-Node-4642" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.072em;"><span id="MJXc-Node-4643" class="mjx-texatom" style=""><span id="MJXc-Node-4644" class="mjx-mrow"><span id="MJXc-Node-4645" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4646" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4647" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span></span></span><span id="MJXc-Node-4652" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 6.543em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 6.543em; top: -1.639em;"><span id="MJXc-Node-4653" class="mjx-mrow"><span id="MJXc-Node-4654" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4655" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4656" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4657" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-4658" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4659" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4660" class="mjx-texatom" style=""><span id="MJXc-Node-4661" class="mjx-mrow"><span id="MJXc-Node-4662" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4663" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4664" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span id="MJXc-Node-4665" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4666" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4667" class="mjx-texatom" style=""><span id="MJXc-Node-4668" class="mjx-mrow"><span id="MJXc-Node-4669" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4670" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4671" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 6.543em; bottom: -1.042em;"><span id="MJXc-Node-4672" class="mjx-mrow"><span id="MJXc-Node-4673" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4674" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4675" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4676" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4677" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4678" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 6.543em;"></span></span><span class="mjx-vsize" style="height: 2.681em; vertical-align: -1.042em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-4679" class="mjx-mtr" style="height: 2.652em;"><span id="MJXc-Node-4680" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-4681" class="mjx-mrow" style="margin-top: 0.583em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4682" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-4683" class="mjx-mrow" style="margin-top: 0.583em;"><span id="MJXc-Node-4684" class="mjx-mi"></span><span id="MJXc-Node-4685" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4686" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-4687" class="mjx-mrow"><span id="MJXc-Node-4688" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.557em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.557em; top: -1.583em;"><span id="MJXc-Node-4689" class="mjx-mrow"><span id="MJXc-Node-4690" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4691" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4692" class="mjx-texatom" style=""><span id="MJXc-Node-4693" class="mjx-mrow"><span id="MJXc-Node-4694" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4695" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span></span></span></span><span id="MJXc-Node-4696" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4697" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4698" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-4699" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4700" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4701" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span class="mjx-denominator" style="width: 3.557em; bottom: -0.769em;"><span id="MJXc-Node-4702" class="mjx-mrow"><span id="MJXc-Node-4703" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4704" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.557em;"></span></span><span class="mjx-vsize" style="height: 2.352em; vertical-align: -0.769em;"></span></span><span id="MJXc-Node-4705" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4706" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4707" class="mjx-texatom" style=""><span id="MJXc-Node-4708" class="mjx-mrow"><span id="MJXc-Node-4709" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4710" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4711" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4712" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-4713" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4714" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-4715" class="mjx-mtr" style="height: 2.558em;"><span id="MJXc-Node-4716" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-4717" class="mjx-mrow" style="margin-top: 0.639em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4718" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-4719" class="mjx-mrow" style="margin-top: 0.639em;"><span id="MJXc-Node-4720" class="mjx-mi"></span><span id="MJXc-Node-4721" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4722" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-4723" class="mjx-mrow"><span id="MJXc-Node-4724" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 4.287em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.287em; top: -1.639em;"><span id="MJXc-Node-4725" class="mjx-mrow"><span id="MJXc-Node-4726" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4727" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4728" class="mjx-texatom" style=""><span id="MJXc-Node-4729" class="mjx-mrow"><span id="MJXc-Node-4730" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4731" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-4732" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span></span></span></span><span id="MJXc-Node-4733" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4734" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span><span id="MJXc-Node-4735" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-4736" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4737" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4738" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span class="mjx-denominator" style="width: 4.287em; bottom: -0.769em;"><span id="MJXc-Node-4739" class="mjx-mrow"><span id="MJXc-Node-4740" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4741" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.287em;"></span></span><span class="mjx-vsize" style="height: 2.408em; vertical-align: -0.769em;"></span></span><span id="MJXc-Node-4742" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi></mi><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>X</mi><mo>=</mo><mi>n</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi></mi><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>=</mo><mi>k</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mi>n</mi><mi>k</mi></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><msup><mi>p</mi><mi>k</mi></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>λ</mi></mrow></msup><mfrac><msup><mi>λ</mi><mi>n</mi></msup><mrow><mi>n</mi><mo>!</mo></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mfrac><mrow><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>λ</mi></mrow></msup><msup><mi>p</mi><mi>k</mi></msup><msup><mi>λ</mi><mi>k</mi></msup></mrow><mrow><mi>k</mi><mo>!</mo></mrow></mfrac><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>=</mo><mi>k</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mfrac><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><msup><mi>λ</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mfrac><mrow><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>λ</mi></mrow></msup><msup><mi>p</mi><mi>k</mi></msup><msup><mi>λ</mi><mi>k</mi></msup></mrow><mrow><mi>k</mi><mo>!</mo></mrow></mfrac><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo><mi>λ</mi></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mfrac><mrow><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>p</mi><mi>λ</mi></mrow></msup><mo stretchy="false">(</mo><mi>λ</mi><mi>p</mi><msup><mo stretchy="false">)</mo><mi>k</mi></msup></mrow><mrow><mi>k</mi><mo>!</mo></mrow></mfrac><mo>.</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-411">\begin{eqnarray*}
P(Y=k)&=&\sum_{n=0}^{+\infty} P(Y=k|X=n)P(X=n)\\
&=&\sum_{n=k}^{+\infty} \binom nk p^k(1-p)^{n-k}e^{-\lambda}\frac{\lambda^n}{n!}\\
&=&\frac{e^{-\lambda}p^k \lambda^k}{k!}\sum_{n=k}^{+\infty}\frac{(1-p)^{n-k} \lambda^{n-k}}{(n-k)!}\\
&=&\frac{e^{-\lambda}p^k \lambda^k}{k!}e^{(1-p)\lambda}\\
&=&\frac{e^{-p\lambda}(\lambda p)^k}{k!}.
\end{eqnarray*}</script>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-412-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4743" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4744" class="mjx-mrow"><span id="MJXc-Node-4745" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-412">Y</script> suit donc une loi de Poisson <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-413-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4746" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4747" class="mjx-mrow"><span id="MJXc-Node-4748" class="mjx-texatom"><span id="MJXc-Node-4749" class="mjx-mrow"><span id="MJXc-Node-4750" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.411em; padding-bottom: 0.353em; padding-right: 0.037em;">P</span></span></span></span><span id="MJXc-Node-4751" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4752" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span><span id="MJXc-Node-4753" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-4754" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">P</mi></mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>p</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-413">\mathcal P(\lambda p)</script>.
</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>17</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Un problème chinois!<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1290"> [Signaler une erreur]</a><a id="exo17" href="javascript:ajouteexo(1290,17,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1290&#39;,&#39;imageenonce1290&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce1290"></a><div class="inner" id="innerenonce1290">On suppose qu'à la naissance, la probabilité qu'un nouveau-né soit un garçon est égale à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-414-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4755" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4756" class="mjx-mrow"><span id="MJXc-Node-4757" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-4758" class="mjx-texatom"><span id="MJXc-Node-4759" class="mjx-mrow"><span id="MJXc-Node-4760" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">/</span></span></span></span><span id="MJXc-Node-4761" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-414">1/2</script>. On suppose que
tous les couples ont des enfants jusqu'à obtenir un garçon. On souhaite évaluer la proportion de garçons dans une génération
de cette population. On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-415-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4762" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4763" class="mjx-mrow"><span id="MJXc-Node-4764" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-415">X</script> le nombre d'enfants d'un couple pris au hasard dans la population.
<ol class="enumeratechiffre">
<li> Donner la loi de la variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-416-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4765" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4766" class="mjx-mrow"><span id="MJXc-Node-4767" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-416">X</script>.
</li><li> On suppose qu'une génération en âge de procréer est constituée de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-417-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4768" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4769" class="mjx-mrow"><span id="MJXc-Node-4770" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-417">N</script> couples, et on note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-418-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x22EF;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4771" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4772" class="mjx-mrow"><span id="MJXc-Node-4773" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-4774" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4775" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span id="MJXc-Node-4776" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-4777" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.328em;">⋯</span></span><span id="MJXc-Node-4778" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-4779" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-4780" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4781" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>X</mi><mi>N</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-418">X_1,\cdots,X_N</script> le nombre d'enfants respectif de chaque couple. On note enfin <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-419-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4782" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4783" class="mjx-mrow"><span id="MJXc-Node-4784" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></span></span><script type="math/tex" id="MathJax-Element-419">P</script> la proportion de garçons issus de cette génération.
Exprimer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-420-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4785" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4786" class="mjx-mrow"><span id="MJXc-Node-4787" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></span></span><script type="math/tex" id="MathJax-Element-420">P</script> en fonction de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-421-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4788" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4789" class="mjx-mrow"><span id="MJXc-Node-4790" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-4791" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4792" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span id="MJXc-Node-4793" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-4794" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.328em;">…</span></span><span id="MJXc-Node-4795" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-4796" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-4797" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4798" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>X</mi><mi>N</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-421">X_1,\dots,X_N</script>.
</li><li> Quelle est la limite de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-422-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4799" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4800" class="mjx-mrow"><span id="MJXc-Node-4801" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></span></span><script type="math/tex" id="MathJax-Element-422">P</script> lorsque <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-423-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4802" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4803" class="mjx-mrow"><span id="MJXc-Node-4804" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-423">N</script> tend vers l'infini. Qu'en pensez-vous?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1290&#39;,&#39;imageindication1290&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication1290"></a><div class="inner" id="innerindication1290" style="display:none;"><ol class="enumeratechiffre">
<li> Loi géométrique.
</li><li>
</li><li> Appliquer la loi forte des grands nombres.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1290&#39;,&#39;imagecorrige1290&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige1290"></a><div class="inner" id="innercorrige1290" style="display:none;"><ol class="enumeratechiffre">
<li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-424-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4805" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4806" class="mjx-mrow"><span id="MJXc-Node-4807" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-424">X</script> est le temps d'attente du premier "succès" (avoir un garçon), avec probabilité de succès égale à 1/2.
On en déduit que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-425-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4808" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4809" class="mjx-mrow"><span id="MJXc-Node-4810" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-425">X</script> suit une loi géométrique de paramètre 1/2. En particulier,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-426-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4811" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4812" class="mjx-mrow"><span id="MJXc-Node-4813" class="mjx-texatom"><span id="MJXc-Node-4814" class="mjx-mrow"><span id="MJXc-Node-4815" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.47em; padding-bottom: 0.295em;">P</span></span></span></span><span id="MJXc-Node-4816" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4817" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4818" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4819" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4820" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4821" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4822" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-4823" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4824" class="mjx-texatom" style=""><span id="MJXc-Node-4825" class="mjx-mrow"><span id="MJXc-Node-4826" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4827" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">P</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>k</mi></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-426">\mathbb P(X=k)=2^{-k}</script>.
</li><li> Le nombre d'enfants total est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-427-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;&amp;#x22EF;&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4828" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4829" class="mjx-mrow"><span id="MJXc-Node-4830" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-4831" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4832" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4833" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4834" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋯</span></span><span id="MJXc-Node-4835" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4836" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-4837" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4838" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>X</mi><mi>N</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-427">X_1+\dots+X_N</script>, et le nombre de garçons est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-428-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4839" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4840" class="mjx-mrow"><span id="MJXc-Node-4841" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-428">N</script>. Ainsi, la proportion <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-429-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4842" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4843" class="mjx-mrow"><span id="MJXc-Node-4844" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></span></span><script type="math/tex" id="MathJax-Element-429">P</script> de garçons est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-430-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;&amp;#x22EF;&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4845" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4846" class="mjx-mrow"><span id="MJXc-Node-4847" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-4848" class="mjx-texatom"><span id="MJXc-Node-4849" class="mjx-mrow"><span id="MJXc-Node-4850" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-4851" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4852" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-4853" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4854" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4855" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4856" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋯</span></span><span id="MJXc-Node-4857" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4858" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-4859" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4860" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span id="MJXc-Node-4861" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>X</mi><mi>N</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-430">N/(X_1+\cdots+X_N)</script>.
</li><li> Les variables aléatoires <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-431-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4862" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4863" class="mjx-mrow"><span id="MJXc-Node-4864" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-4865" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4866" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-431">X_i</script> sont intégrables, identiquement distribuées, et deux à deux indépendantes. D'après la loi forte des grands nombres, presque sûrement,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-432-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mfrac&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;&amp;#x22EF;&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;&amp;#x22EF;&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mfrac&gt;&lt;/mfrac&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4867" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4868" class="mjx-mrow"><span id="MJXc-Node-4869" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 6.525em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 6.525em; top: -1.385em;"><span id="MJXc-Node-4870" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span><span class="mjx-denominator" style="width: 6.525em; bottom: -0.868em;"><span id="MJXc-Node-4871" class="mjx-mrow"><span id="MJXc-Node-4872" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-4873" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4874" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4875" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4876" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋯</span></span><span id="MJXc-Node-4877" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4878" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-4879" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4880" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 6.525em;"></span></span><span class="mjx-vsize" style="height: 2.253em; vertical-align: -0.868em;"></span></span><span id="MJXc-Node-4881" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4882" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.425em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.425em; top: -1.368em;"><span id="MJXc-Node-4883" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 4.425em; bottom: -1.482em;"><span id="MJXc-Node-4884" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.985em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 5.636em; top: -1.478em;"><span id="MJXc-Node-4885" class="mjx-mrow" style=""><span id="MJXc-Node-4886" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-4887" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4888" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4889" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4890" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋯</span></span><span id="MJXc-Node-4891" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4892" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-4893" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4894" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 5.636em; bottom: -0.628em;"><span id="MJXc-Node-4895" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.985em;"></span></span><span class="mjx-vsize" style="height: 1.489em; vertical-align: -0.444em;"></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.425em;"></span></span><span class="mjx-vsize" style="height: 2.849em; vertical-align: -1.482em;"></span></span><span id="MJXc-Node-4896" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">→</span></span><span id="MJXc-Node-4897" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.959em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.959em; top: -1.368em;"><span id="MJXc-Node-4898" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 2.959em; bottom: -1.042em;"><span id="MJXc-Node-4899" class="mjx-mrow"><span id="MJXc-Node-4900" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-4901" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4902" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-4903" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4904" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4905" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.959em;"></span></span><span class="mjx-vsize" style="height: 2.41em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-4906" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4907" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4908" class="mjx-texatom"><span id="MJXc-Node-4909" class="mjx-mrow"><span id="MJXc-Node-4910" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-4911" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>N</mi><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>X</mi><mi>N</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mfrac><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>X</mi><mi>N</mi></msub></mrow><mi>N</mi></mfrac></mfrac><mo stretchy="false">→</mo><mfrac><mn>1</mn><mrow><mi>E</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-432">\frac{N}{X_1+\dots+X_N}=\frac{1}{\frac{X_1+\cdots+X_N}N}\to \frac{1}{E(X_1)}=1/2.</script>
Ainsi, malgré cette politique nataliste, on n'observe pas de déséquilibre entre les filles et les garçons.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>18</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Le concierge<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1288"> [Signaler une erreur]</a><a id="exo18" href="javascript:ajouteexo(1288,18,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1288&#39;,&#39;imageenonce1288&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce1288"></a><div class="inner" id="innerenonce1288">Un concierge rentre d'une soirée. Il dispose de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-433-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4912" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4913" class="mjx-mrow"><span id="MJXc-Node-4914" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-433">n</script> clefs dont une seule ouvre la porte de son domicile,
mais il ne sait plus laquelle.
<ol class="enumeratechiffre">
<li> Il essaie les clefs les unes après les autres en éliminant après chaque essai la clef qui n'a pas convenu.
Trouver le nombre moyen d'essais nécessaires pour trouver la bonne clef.
</li><li> En réalité, la soirée était bien arrosée, et après chaque essai, le concierge remet la clef essayée dans le trousseau.
Trouver le nombre moyen d'essais nécessaires pour trouver la bonne clef.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1288&#39;,&#39;imageindication1288&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication1288"></a><div class="inner" id="innerindication1288" style="display:none;">Noter <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-434-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4915" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4916" class="mjx-mrow"><span id="MJXc-Node-4917" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-434">X</script> la variable aléatoire du nombre d'essais à effectuer avant d'ouvrir la porte. Il faut chercher la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-435-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4918" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4919" class="mjx-mrow"><span id="MJXc-Node-4920" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-435">X</script>. Il est peut-être plus facile de trouver <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-436-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4921" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4922" class="mjx-mrow"><span id="MJXc-Node-4923" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4924" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4925" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4926" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-4927" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4928" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-436">P(X>k)</script>.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1288&#39;,&#39;imagecorrige1288&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige1288"></a><div class="inner" id="innercorrige1288" style="display:none;"><ol class="enumeratechiffre">
<li> Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-437-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4929" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4930" class="mjx-mrow"><span id="MJXc-Node-4931" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-437">X</script> la variable aléatoire du nombre d'essais à effectuer avant d'ouvrir la porte.
Il est un peu difficile de calculer directement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-438-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4932" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4933" class="mjx-mrow"><span id="MJXc-Node-4934" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4935" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4936" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4937" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4938" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4939" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-438">P(X=k)</script>, nous allons plutôt calculer, pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-439-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4940" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4941" class="mjx-mrow"><span id="MJXc-Node-4942" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4943" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4944" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4945" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4946" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-4947" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4948" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4949" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4950" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-439">k=1,\dots,n-1</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-440-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4951" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4952" class="mjx-mrow"><span id="MJXc-Node-4953" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4954" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4955" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4956" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-4957" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4958" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-440">P(X>k)</script>. L'événement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-441-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4959" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4960" class="mjx-mrow"><span id="MJXc-Node-4961" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4962" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-4963" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-4964" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-4965" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-441">(X>k)</script> est réalisé si et seulement si, la première clé n'a pas convenu et la deuxième clé n'a pas convenu,et..., et la <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-442-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4966" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4967" class="mjx-mrow"><span id="MJXc-Node-4968" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-442">k</script>-ième clé n'a pas convenu. Pour la clé numéro <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-443-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4969" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4970" class="mjx-mrow"><span id="MJXc-Node-4971" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-443">i</script>, il y a avait une clé parmi <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-444-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4972" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4973" class="mjx-mrow"><span id="MJXc-Node-4974" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4975" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4976" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-4977" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4978" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>−</mo><mi>i</mi><mo>+</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-444">n-i+1</script> qui convenait. La probabilité que la clé numéro <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-445-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4979" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4980" class="mjx-mrow"><span id="MJXc-Node-4981" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-445">i</script> ne convenait pas est donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-446-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4982" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4983" class="mjx-mrow"><span id="MJXc-Node-4984" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4985" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4986" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 2.263em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 3.201em; top: -1.318em;"><span id="MJXc-Node-4987" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 3.201em; bottom: -0.693em;"><span id="MJXc-Node-4988" class="mjx-mrow" style=""><span id="MJXc-Node-4989" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4990" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4991" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-4992" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4993" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.263em;"></span></span><span class="mjx-vsize" style="height: 1.422em; vertical-align: -0.49em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>−</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-446">1-\frac{1}{n-i+1}</script>. Par la formule des probabilités composées, notant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-447-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4994" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4995" class="mjx-mrow"><span id="MJXc-Node-4996" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4997" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4998" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-447">A_i</script> l'événement "on fait au moins <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-448-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-4999" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5000" class="mjx-mrow"><span id="MJXc-Node-5001" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-448">i</script> essais et le <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-449-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5002" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5003" class="mjx-mrow"><span id="MJXc-Node-5004" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-449">i</script>-ème ne convient pas", on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-450-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0 thickmathspace&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x22EF;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x220F;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/munderover&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x220F;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/munderover&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5005" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5006" class="mjx-mrow"><span id="MJXc-Node-5007" class="mjx-mtable" style="vertical-align: -5.523em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-5008" class="mjx-mtr" style="height: 1.225em;"><span id="MJXc-Node-5009" class="mjx-mtd" style="padding: 0px; text-align: right; width: 4.236em;"><span id="MJXc-Node-5010" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5011" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5012" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5013" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-5014" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-5015" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-5016" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5017" class="mjx-mtd" style="padding: 0px 0.139em 0px 0px; width: 1.056em;"><span id="MJXc-Node-5018" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5019" class="mjx-mi"></span><span id="MJXc-Node-5020" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5021" class="mjx-mtd" style="padding: 0px 0px 0px 0.139em; text-align: left; width: 15.113em;"><span id="MJXc-Node-5022" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5023" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5024" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5025" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5026" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-5027" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-5028" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-5029" class="mjx-mtr" style="height: 1.375em;"><span id="MJXc-Node-5030" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-5031" class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5032" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-5033" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5034" class="mjx-mi"></span><span id="MJXc-Node-5035" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5036" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-5037" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5038" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5039" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5040" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5041" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-5042" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-5043" class="mjx-texatom"><span id="MJXc-Node-5044" class="mjx-mrow"><span id="MJXc-Node-5045" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-5046" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5047" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-5048" class="mjx-texatom" style=""><span id="MJXc-Node-5049" class="mjx-mrow"><span id="MJXc-Node-5050" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-5051" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5052" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5053" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5054" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5055" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5056" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5057" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-5058" class="mjx-texatom" style=""><span id="MJXc-Node-5059" class="mjx-mrow"><span id="MJXc-Node-5060" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-5061" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5062" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5063" class="mjx-texatom"><span id="MJXc-Node-5064" class="mjx-mrow"><span id="MJXc-Node-5065" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-5066" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5067" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-5068" class="mjx-texatom" style=""><span id="MJXc-Node-5069" class="mjx-mrow"><span id="MJXc-Node-5070" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-5071" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5072" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span id="MJXc-Node-5073" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5074" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋯</span></span><span id="MJXc-Node-5075" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5076" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5077" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5078" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5079" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-5080" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-5081" class="mjx-mtr" style="height: 3.339em;"><span id="MJXc-Node-5082" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-5083" class="mjx-mrow" style="margin-top: 0.783em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5084" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-5085" class="mjx-mrow" style="margin-top: 0.783em;"><span id="MJXc-Node-5086" class="mjx-mi"></span><span id="MJXc-Node-5087" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5088" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-5089" class="mjx-mrow"><span id="MJXc-Node-5090" class="mjx-mi"></span><span id="MJXc-Node-5091" class="mjx-munderover MJXc-space1"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.247em; padding-top: 0.141em; padding-left: 0.643em;"><span id="MJXc-Node-5098" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span><span class="mjx-op"><span id="MJXc-Node-5092" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∏</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.092em;"><span id="MJXc-Node-5093" class="mjx-texatom" style=""><span id="MJXc-Node-5094" class="mjx-mrow"><span id="MJXc-Node-5095" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-5096" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5097" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span id="MJXc-Node-5099" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-5100" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-5101" class="mjx-mrow"><span id="MJXc-Node-5102" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5103" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5104" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 4.09em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.09em; top: -1.368em;"><span id="MJXc-Node-5105" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 4.09em; bottom: -0.793em;"><span id="MJXc-Node-5106" class="mjx-mrow"><span id="MJXc-Node-5107" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5108" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5109" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-5110" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5111" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.09em;"></span></span><span class="mjx-vsize" style="height: 2.16em; vertical-align: -0.793em;"></span></span></span><span id="MJXc-Node-5112" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-5113" class="mjx-mtr" style="height: 3.339em;"><span id="MJXc-Node-5114" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-5115" class="mjx-mrow" style="margin-top: 0.783em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5116" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-5117" class="mjx-mrow" style="margin-top: 0.783em;"><span id="MJXc-Node-5118" class="mjx-mi"></span><span id="MJXc-Node-5119" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5120" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-5121" class="mjx-mrow"><span id="MJXc-Node-5122" class="mjx-mi"></span><span id="MJXc-Node-5123" class="mjx-munderover MJXc-space1"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.247em; padding-top: 0.141em; padding-left: 0.643em;"><span id="MJXc-Node-5130" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span><span class="mjx-op"><span id="MJXc-Node-5124" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∏</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.092em;"><span id="MJXc-Node-5125" class="mjx-texatom" style=""><span id="MJXc-Node-5126" class="mjx-mrow"><span id="MJXc-Node-5127" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-5128" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5129" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span id="MJXc-Node-5131" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 4.09em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.09em; top: -1.363em;"><span id="MJXc-Node-5132" class="mjx-mrow"><span id="MJXc-Node-5133" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5134" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5135" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="mjx-denominator" style="width: 4.09em; bottom: -0.793em;"><span id="MJXc-Node-5136" class="mjx-mrow"><span id="MJXc-Node-5137" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5138" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5139" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-5140" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5141" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.09em;"></span></span><span class="mjx-vsize" style="height: 2.155em; vertical-align: -0.793em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-5142" class="mjx-mtr" style="height: 2.267em;"><span id="MJXc-Node-5143" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-5144" class="mjx-mrow" style="margin-top: 0.396em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5145" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-5146" class="mjx-mrow" style="margin-top: 0.396em;"><span id="MJXc-Node-5147" class="mjx-mi"></span><span id="MJXc-Node-5148" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5149" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-5150" class="mjx-mrow"><span id="MJXc-Node-5151" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.543em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.543em; top: -1.396em;"><span id="MJXc-Node-5152" class="mjx-mrow"><span id="MJXc-Node-5153" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5154" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5155" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="mjx-denominator" style="width: 2.543em; bottom: -0.722em;"><span id="MJXc-Node-5156" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.543em;"></span></span><span class="mjx-vsize" style="height: 2.117em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-5157" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mi>k</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>k</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>−</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo>⋯</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi></mi><munderover><mo>∏</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>−</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi></mi><munderover><mo>∏</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><mfrac><mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mfrac><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow><mi>n</mi></mfrac><mo>.</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-450">\begin{eqnarray*}
P(X>k)&=&P(A_k)\\
&=&P(A_k|A_{k-1})P(A_{k-1}|A_{k-2})\cdots P(A_1)\\
&=&\prod_{i=1}^k \left(1-\frac{1}{n-i+1}\right)\\
&=&\prod_{i=1}^k \frac{n-i}{n-i+1}\\
&=&\frac{n-k}n.
\end{eqnarray*}</script>
On a ensuite
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-451-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5158" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5159" class="mjx-mrow"><span id="MJXc-Node-5160" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5161" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5162" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-5163" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5164" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-5165" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5166" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5167" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5168" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5169" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-5170" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-5171" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-5172" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5173" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5174" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5175" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5176" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5177" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5178" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-5179" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-5180" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-5181" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5182" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5183" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.8em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.8em; top: -1.368em;"><span id="MJXc-Node-5184" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.8em; bottom: -0.722em;"><span id="MJXc-Node-5185" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.8em;"></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-5186" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-451">P(X=k)=P(X>k-1)-P(X>k)=\frac 1n.</script>
Autrement dit, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-452-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5187" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5188" class="mjx-mrow"><span id="MJXc-Node-5189" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-452">X</script> suit une loi uniforme sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-453-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5190" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5191" class="mjx-mrow"><span id="MJXc-Node-5192" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-5193" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5194" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-5195" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-5196" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-5197" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5198" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-453">\{1,\dots,n\}</script>. Son espérance (le nombre moyen d'essais pour ouvrir la porte) vaut donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-454-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5199" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5200" class="mjx-mrow"><span id="MJXc-Node-5201" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-5202" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5203" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-5204" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5205" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5206" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.522em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.522em; top: -1.368em;"><span id="MJXc-Node-5207" class="mjx-mrow"><span id="MJXc-Node-5208" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5209" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5210" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="width: 2.522em; bottom: -0.711em;"><span id="MJXc-Node-5211" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.522em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-5212" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-454">E(X)=\frac{n+1}2.</script>
</li><li> On garde les mêmes notations. On a cette fois <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-455-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5213" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5214" class="mjx-mrow"><span id="MJXc-Node-5215" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5216" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5217" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5218" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5219" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span id="MJXc-Node-5220" class="mjx-texatom"><span id="MJXc-Node-5221" class="mjx-mrow"><span id="MJXc-Node-5222" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-5223" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5224" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5225" class="mjx-texatom" style=""><span id="MJXc-Node-5226" class="mjx-mrow"><span id="MJXc-Node-5227" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-5228" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5229" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5230" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5231" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5232" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.469em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.078em; top: -1.4em;"><span id="MJXc-Node-5233" class="mjx-mrow" style=""><span id="MJXc-Node-5234" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5235" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5236" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.078em; bottom: -0.524em;"><span id="MJXc-Node-5237" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.469em;"></span></span><span class="mjx-vsize" style="height: 1.36em; vertical-align: -0.37em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mi>n</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-455">P(A_i|A_{i-1})=\frac {n-1}n</script>
et donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-456-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5238" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5239" class="mjx-mrow"><span id="MJXc-Node-5240" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5241" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5242" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-5243" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-5244" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-5245" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5246" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5247" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.704em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.704em; top: -1.639em;"><span id="MJXc-Node-5248" class="mjx-mrow"><span id="MJXc-Node-5249" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5250" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5251" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5252" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5253" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5254" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5255" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span class="mjx-denominator" style="width: 3.704em; bottom: -0.826em;"><span id="MJXc-Node-5256" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5257" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5258" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.704em;"></span></span><span class="mjx-vsize" style="height: 2.464em; vertical-align: -0.826em;"></span></span><span id="MJXc-Node-5259" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>k</mi></msup></mrow><msup><mi>n</mi><mi>k</mi></msup></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-456">P(X>k)=\frac{(n-1)^k}{n^k}.</script>
On en déduit que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-457-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5260" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5261" class="mjx-mrow"><span id="MJXc-Node-5262" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5263" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5264" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-5265" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5266" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-5267" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5268" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5269" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.608em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.608em; top: -1.639em;"><span id="MJXc-Node-5270" class="mjx-mrow"><span id="MJXc-Node-5271" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5272" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5273" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5274" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5275" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5276" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5277" class="mjx-texatom" style=""><span id="MJXc-Node-5278" class="mjx-mrow"><span id="MJXc-Node-5279" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-5280" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5281" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 4.608em; bottom: -0.826em;"><span id="MJXc-Node-5282" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5283" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5284" class="mjx-texatom" style=""><span id="MJXc-Node-5285" class="mjx-mrow"><span id="MJXc-Node-5286" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-5287" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5288" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.608em;"></span></span><span class="mjx-vsize" style="height: 2.464em; vertical-align: -0.826em;"></span></span><span id="MJXc-Node-5289" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5290" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 3.704em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.704em; top: -1.639em;"><span id="MJXc-Node-5291" class="mjx-mrow"><span id="MJXc-Node-5292" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5293" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5294" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5295" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5296" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5297" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5298" class="mjx-texatom" style=""><span id="MJXc-Node-5299" class="mjx-mrow"><span id="MJXc-Node-5300" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 3.704em; bottom: -0.826em;"><span id="MJXc-Node-5301" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5302" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5303" class="mjx-texatom" style=""><span id="MJXc-Node-5304" class="mjx-mrow"><span id="MJXc-Node-5305" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.704em;"></span></span><span class="mjx-vsize" style="height: 2.464em; vertical-align: -0.826em;"></span></span><span id="MJXc-Node-5306" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5307" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.608em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.608em; top: -1.639em;"><span id="MJXc-Node-5308" class="mjx-mrow"><span id="MJXc-Node-5309" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5310" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5311" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5312" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5313" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5314" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5315" class="mjx-texatom" style=""><span id="MJXc-Node-5316" class="mjx-mrow"><span id="MJXc-Node-5317" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-5318" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5319" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 4.608em; bottom: -0.826em;"><span id="MJXc-Node-5320" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5321" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5322" class="mjx-texatom" style=""><span id="MJXc-Node-5323" class="mjx-mrow"><span id="MJXc-Node-5324" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-5325" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5326" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.608em;"></span></span><span class="mjx-vsize" style="height: 2.464em; vertical-align: -0.826em;"></span></span><span id="MJXc-Node-5327" class="mjx-mrow"><span id="MJXc-Node-5328" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-5329" class="mjx-mrow"><span id="MJXc-Node-5330" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5331" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5332" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 2.522em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.522em; top: -1.368em;"><span id="MJXc-Node-5333" class="mjx-mrow"><span id="MJXc-Node-5334" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5335" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5336" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="width: 2.522em; bottom: -0.722em;"><span id="MJXc-Node-5337" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.522em;"></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.722em;"></span></span></span><span id="MJXc-Node-5338" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span><span id="MJXc-Node-5339" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5340" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.8em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.8em; top: -1.368em;"><span id="MJXc-Node-5341" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.8em; bottom: -0.722em;"><span id="MJXc-Node-5342" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.8em;"></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-5343" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-5344" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 4.608em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.608em; top: -1.639em;"><span id="MJXc-Node-5345" class="mjx-mrow"><span id="MJXc-Node-5346" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5347" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5348" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5349" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5350" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5351" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5352" class="mjx-texatom" style=""><span id="MJXc-Node-5353" class="mjx-mrow"><span id="MJXc-Node-5354" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-5355" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5356" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 4.608em; bottom: -0.826em;"><span id="MJXc-Node-5357" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5358" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5359" class="mjx-texatom" style=""><span id="MJXc-Node-5360" class="mjx-mrow"><span id="MJXc-Node-5361" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-5362" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5363" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.608em;"></span></span><span class="mjx-vsize" style="height: 2.464em; vertical-align: -0.826em;"></span></span><span id="MJXc-Node-5364" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><msup><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></mfrac><mo>−</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi></mrow></msup></mrow><msup><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi></mrow></msup></mfrac><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><msup><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></mfrac><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mi>n</mi></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><mo>×</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><msup><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-457">P(X=k)=\frac{(n-1)^{k-1}}{n^{k-1}}-\frac{(n-1)^{k}}{n^{k}}=\frac{(n-1)^{k-1}}{n^{k-1}}\left(1-\frac{n-1}n\right)=\frac 1n\times \frac{(n-1)^{k-1}}{n^{k-1}}.</script>
En reconnaissant une loi géométrique de paramètre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-458-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5365" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5366" class="mjx-mrow"><span id="MJXc-Node-5367" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5368" class="mjx-texatom"><span id="MJXc-Node-5369" class="mjx-mrow"><span id="MJXc-Node-5370" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-5371" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-458">1/n</script>, on trouve que le nombre moyen d'essais nécessaires est
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-459-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5372" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5373" class="mjx-mrow"><span id="MJXc-Node-5374" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-5375" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5376" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-5377" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5378" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5379" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5380" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>n</mi><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-459">E(X)=n.</script>
Finalement, ce n'est pas si différent!
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>19</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Densité ou non?<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2333"> [Signaler une erreur]</a><a id="exo19" href="javascript:ajouteexo(2333,19,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2333&#39;,&#39;imageenonce2333&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2333"></a><div class="inner" id="innerenonce2333">Parmi les fonctions suivantes définies sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-460-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5381" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5382" class="mjx-mrow"><span id="MJXc-Node-5383" class="mjx-texatom"><span id="MJXc-Node-5384" class="mjx-mrow"><span id="MJXc-Node-5385" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">R</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-460">\mathbb R</script>, déterminer lesquelles sont la densité d'une variable aléatoire à densité. Calculer le cas échéant leur fonction de répartition et préciser si elles admettent une espérance.
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-461-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;left left left&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext mathvariant=&quot;bold&quot;&gt;1.&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable columnalign=&quot;left left&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;cos&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;si&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;sinon.&lt;/mtext&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; symmetric=&quot;true&quot;&gt;&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mspace width=&quot;1em&quot; /&gt;&lt;mspace width=&quot;1em&quot; /&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext mathvariant=&quot;bold&quot;&gt;2.&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext mathvariant=&quot;bold&quot;&gt;3.&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mspace width=&quot;1em&quot; /&gt;&lt;mspace width=&quot;1em&quot; /&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext mathvariant=&quot;bold&quot;&gt;4.&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable columnalign=&quot;left left&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;si&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;si&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;sinon&lt;/mtext&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; symmetric=&quot;true&quot;&gt;&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext mathvariant=&quot;bold&quot;&gt;5.&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable columnalign=&quot;left left&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;si&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;sinon&lt;/mtext&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; symmetric=&quot;true&quot;&gt;&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mspace width=&quot;1em&quot; /&gt;&lt;mspace width=&quot;1em&quot; /&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext mathvariant=&quot;bold&quot;&gt;6.&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;sin&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5386" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5387" class="mjx-mrow"><span id="MJXc-Node-5388" class="mjx-mtable" style="vertical-align: -4.958em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-5389" class="mjx-mtr" style="height: 2.675em;"><span id="MJXc-Node-5390" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: left; width: 14.907em;"><span id="MJXc-Node-5391" class="mjx-mrow"><span id="MJXc-Node-5392" class="mjx-texatom"><span id="MJXc-Node-5393" class="mjx-mrow"><span id="MJXc-Node-5394" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.328em; padding-bottom: 0.393em;">1.&nbsp;</span></span></span></span><span id="MJXc-Node-5395" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-5396" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5397" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span id="MJXc-Node-5398" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-5399" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-5400" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-5401" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-5402" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-5403" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.236em; padding-bottom: 1.236em;">{</span></span><span id="MJXc-Node-5404" class="mjx-mtable" style="vertical-align: -0.988em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-5405" class="mjx-mtr" style="height: 1.275em;"><span id="MJXc-Node-5406" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: left; width: 2.077em;"><span id="MJXc-Node-5407" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5408" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.133em; padding-bottom: 0.328em;">cos</span></span><span id="MJXc-Node-5409" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-5410" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5411" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 6.04em;"><span id="MJXc-Node-5412" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5413" class="mjx-texatom"><span id="MJXc-Node-5414" class="mjx-mrow"><span id="MJXc-Node-5415" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">&nbsp;si&nbsp;</span></span></span></span><span id="MJXc-Node-5416" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-5417" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-5418" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-5419" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-5420" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-5421" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-5422" class="mjx-texatom"><span id="MJXc-Node-5423" class="mjx-mrow"><span id="MJXc-Node-5424" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">/</span></span></span></span><span id="MJXc-Node-5425" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span><span id="MJXc-Node-5426" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-5427" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-5428" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: left;"><span id="MJXc-Node-5429" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5430" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5431" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-5432" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5433" class="mjx-texatom"><span id="MJXc-Node-5434" class="mjx-mrow"><span id="MJXc-Node-5435" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">&nbsp;sinon.</span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-5436" class="mjx-mo" style="width: 0.12em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5437" class="mjx-mtd" style="padding: 0px 0.5em; text-align: left; width: 2em;"><span id="MJXc-Node-5438" class="mjx-mrow" style="margin-top: 0.488em;"><span id="MJXc-Node-5439" class="mjx-mspace" style="width: 1em; height: 0px;"></span><span id="MJXc-Node-5440" class="mjx-mspace" style="width: 1em; height: 0px;"></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5441" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 14.968em;"><span id="MJXc-Node-5442" class="mjx-mrow" style="margin-top: 0.488em;"><span id="MJXc-Node-5443" class="mjx-texatom"><span id="MJXc-Node-5444" class="mjx-mrow"><span id="MJXc-Node-5445" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.328em; padding-bottom: 0.393em;">2.&nbsp;</span></span></span></span><span id="MJXc-Node-5446" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-5447" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5448" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span><span id="MJXc-Node-5449" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-5450" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-5451" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-5452" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-5453" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.735em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.454em; top: -1.341em;"><span id="MJXc-Node-5454" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.454em; bottom: -0.822em;"><span id="MJXc-Node-5455" class="mjx-mrow" style=""><span id="MJXc-Node-5456" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-5457" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-5458" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5459" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span><span class="mjx-sup" style="font-size: 73.4%; vertical-align: 0.393em; padding-left: 0px; padding-right: 0.068em;"><span id="MJXc-Node-5460" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.291em; width: 1.735em;"></span></span><span class="mjx-vsize" style="height: 1.53em; vertical-align: -0.581em;"></span></span><span id="MJXc-Node-5461" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-5462" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.256em; padding-bottom: 0.328em;">&nbsp;</span></span><span id="MJXc-Node-5463" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-5464" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-5465" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-5466" class="mjx-mrow"><span id="MJXc-Node-5467" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">R</span></span></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-5468" class="mjx-mtr" style="height: 4.35em;"><span id="MJXc-Node-5469" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: left;"><span id="MJXc-Node-5470" class="mjx-mrow" style="margin-top: 1.225em;"><span id="MJXc-Node-5471" class="mjx-texatom"><span id="MJXc-Node-5472" class="mjx-mrow"><span id="MJXc-Node-5473" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.328em; padding-bottom: 0.393em;">3.&nbsp;</span></span></span></span><span id="MJXc-Node-5474" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-5475" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5476" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span></span></span><span id="MJXc-Node-5477" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-5478" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-5479" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-5480" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-5481" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.533em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 3.583em; top: -1.367em;"><span id="MJXc-Node-5482" class="mjx-msubsup" style=""><span class="mjx-base"><span id="MJXc-Node-5483" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">e</span></span></span><span class="mjx-sup" style="font-size: 73.4%; vertical-align: 0.494em; padding-left: 0px; padding-right: 0.068em;"><span id="MJXc-Node-5484" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 3.583em; bottom: -0.99em;"><span id="MJXc-Node-5485" class="mjx-mrow" style=""><span id="MJXc-Node-5486" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-5487" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5488" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">e</span></span></span><span class="mjx-sup" style="font-size: 73.4%; vertical-align: 0.393em; padding-left: 0px; padding-right: 0.068em;"><span id="MJXc-Node-5489" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span></span><span id="MJXc-Node-5490" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-5491" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-5492" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5493" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span><span class="mjx-sup" style="font-size: 73.4%; vertical-align: 0.393em; padding-left: 0px; padding-right: 0.068em;"><span id="MJXc-Node-5494" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.291em; width: 2.533em;"></span></span><span class="mjx-vsize" style="height: 1.667em; vertical-align: -0.7em;"></span></span><span id="MJXc-Node-5495" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-5496" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.256em; padding-bottom: 0.328em;">&nbsp;</span></span><span id="MJXc-Node-5497" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-5498" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-5499" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-5500" class="mjx-mrow"><span id="MJXc-Node-5501" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">R</span></span></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5502" class="mjx-mtd" style="padding: 0.2em 0.5em 0px; text-align: left;"><span id="MJXc-Node-5503" class="mjx-mrow" style="margin-top: 1.225em;"><span id="MJXc-Node-5504" class="mjx-mspace" style="width: 1em; height: 0px;"></span><span id="MJXc-Node-5505" class="mjx-mspace" style="width: 1em; height: 0px;"></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5506" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-5507" class="mjx-mrow"><span id="MJXc-Node-5508" class="mjx-texatom"><span id="MJXc-Node-5509" class="mjx-mrow"><span id="MJXc-Node-5510" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.328em; padding-bottom: 0.393em;">4.&nbsp;</span></span></span></span><span id="MJXc-Node-5511" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-5512" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5513" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span></span><span id="MJXc-Node-5514" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-5515" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-5516" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-5517" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-5518" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-5519" class="mjx-mo" style="vertical-align: -0.801em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.652em; padding-bottom: 0.263em;">⎧</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.131em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎨</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.131em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="margin-top: -0.191em; padding-bottom: 1.171em;">⎩</span></span></span><span id="MJXc-Node-5520" class="mjx-mtable" style="vertical-align: -1.725em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-5521" class="mjx-mtr" style="height: 1.275em;"><span id="MJXc-Node-5522" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: left; width: 2.294em;"><span id="MJXc-Node-5523" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5524" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-5525" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-5526" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5527" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 5.745em;"><span id="MJXc-Node-5528" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5529" class="mjx-texatom"><span id="MJXc-Node-5530" class="mjx-mrow"><span id="MJXc-Node-5531" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">&nbsp;si&nbsp;</span></span></span></span><span id="MJXc-Node-5532" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-5533" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-5534" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-5535" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-5536" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-5537" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-5538" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-5539" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-5540" class="mjx-mtr" style="height: 1.475em;"><span id="MJXc-Node-5541" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: left;"><span id="MJXc-Node-5542" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5543" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-5544" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-5545" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5546" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-5547" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5548" class="mjx-texatom"><span id="MJXc-Node-5549" class="mjx-mrow"><span id="MJXc-Node-5550" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">&nbsp;si&nbsp;</span></span></span></span><span id="MJXc-Node-5551" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-5552" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-5553" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-5554" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-5555" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-5556" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-5557" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-5558" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-5559" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: left;"><span id="MJXc-Node-5560" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5561" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5562" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-5563" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5564" class="mjx-texatom"><span id="MJXc-Node-5565" class="mjx-mrow"><span id="MJXc-Node-5566" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">&nbsp;sinon</span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-5567" class="mjx-mo" style="width: 0.12em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-5568" class="mjx-mtr" style="height: 3.39em;"><span id="MJXc-Node-5569" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: left;"><span id="MJXc-Node-5570" class="mjx-mrow"><span id="MJXc-Node-5571" class="mjx-texatom"><span id="MJXc-Node-5572" class="mjx-mrow"><span id="MJXc-Node-5573" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.328em; padding-bottom: 0.393em;">5.&nbsp;</span></span></span></span><span id="MJXc-Node-5574" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-5575" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5576" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">5</span></span></span></span><span id="MJXc-Node-5577" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-5578" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-5579" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-5580" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-5581" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-5582" class="mjx-mo"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.496em; padding-bottom: 1.496em;">{</span></span><span id="MJXc-Node-5583" class="mjx-mtable" style="vertical-align: -1.345em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-5584" class="mjx-mtr" style="height: 1.99em;"><span id="MJXc-Node-5585" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: left; width: 1.465em;"><span id="MJXc-Node-5586" class="mjx-mrow" style="margin-top: -0.051em;"><span id="MJXc-Node-5587" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.225em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.732em; top: -1.341em;"><span id="MJXc-Node-5588" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.732em; bottom: -1.19em;"><span id="MJXc-Node-5589" class="mjx-mrow" style=""><span id="MJXc-Node-5590" class="mjx-texatom"><span id="MJXc-Node-5591" class="mjx-mrow"><span id="MJXc-Node-5592" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">|</span></span></span></span><span id="MJXc-Node-5593" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-5594" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5595" class="mjx-texatom"><span id="MJXc-Node-5596" class="mjx-mrow"><span id="MJXc-Node-5597" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">|</span></span></span></span></span><span class="mjx-sup" style="font-size: 73.4%; vertical-align: 0.669em; padding-left: 0px; padding-right: 0.068em;"><span id="MJXc-Node-5598" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.291em; width: 1.225em;"></span></span><span class="mjx-vsize" style="height: 1.79em; vertical-align: -0.842em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5599" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 4.134em;"><span id="MJXc-Node-5600" class="mjx-mrow" style="margin-top: -0.051em;"><span id="MJXc-Node-5601" class="mjx-texatom"><span id="MJXc-Node-5602" class="mjx-mrow"><span id="MJXc-Node-5603" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">&nbsp;si&nbsp;</span></span></span></span><span id="MJXc-Node-5604" class="mjx-texatom"><span id="MJXc-Node-5605" class="mjx-mrow"><span id="MJXc-Node-5606" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">|</span></span></span></span><span id="MJXc-Node-5607" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-5608" class="mjx-texatom"><span id="MJXc-Node-5609" class="mjx-mrow"><span id="MJXc-Node-5610" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">|</span></span></span></span><span id="MJXc-Node-5611" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">&gt;</span></span><span id="MJXc-Node-5612" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-5613" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-5614" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: left;"><span id="MJXc-Node-5615" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5616" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5617" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-5618" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5619" class="mjx-texatom"><span id="MJXc-Node-5620" class="mjx-mrow"><span id="MJXc-Node-5621" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">&nbsp;sinon</span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-5622" class="mjx-mo" style="width: 0.12em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5623" class="mjx-mtd" style="padding: 0.2em 0.5em 0px; text-align: left;"><span id="MJXc-Node-5624" class="mjx-mrow" style="margin-top: 0.845em;"><span id="MJXc-Node-5625" class="mjx-mspace" style="width: 1em; height: 0px;"></span><span id="MJXc-Node-5626" class="mjx-mspace" style="width: 1em; height: 0px;"></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5627" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-5628" class="mjx-mrow" style="margin-top: 0.845em;"><span id="MJXc-Node-5629" class="mjx-texatom"><span id="MJXc-Node-5630" class="mjx-mrow"><span id="MJXc-Node-5631" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.328em; padding-bottom: 0.393em;">6.&nbsp;</span></span></span></span><span id="MJXc-Node-5632" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-5633" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5634" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">6</span></span></span></span><span id="MJXc-Node-5635" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-5636" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-5637" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-5638" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-5639" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">sin</span></span><span id="MJXc-Node-5640" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-5641" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-5642" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-5643" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-5644" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-5645" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.256em; padding-bottom: 0.328em;">&nbsp;</span></span><span id="MJXc-Node-5646" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-5647" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-5648" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-5649" class="mjx-mrow"><span id="MJXc-Node-5650" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">R</span></span></span></span><span id="MJXc-Node-5651" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.328em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="left left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="bold">1.&nbsp;</mtext></mrow><msub><mi>f</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi>cos</mi><mo>⁡</mo><mi>x</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;si&nbsp;</mtext></mrow><mi>x</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;sinon.</mtext></mrow></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></mtd><mtd><mspace width="1em"></mspace><mspace width="1em"></mspace></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="bold">2.&nbsp;</mtext></mrow><msub><mi>f</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>,</mo><mtext>&nbsp;</mtext><mi>x</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></mtd></mtr><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="bold">3.&nbsp;</mtext></mrow><msub><mi>f</mi><mn>3</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><msup><mi>e</mi><mi>x</mi></msup><mrow><mo stretchy="false">(</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mo>,</mo><mtext>&nbsp;</mtext><mi>x</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></mtd><mtd><mspace width="1em"></mspace><mspace width="1em"></mspace></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="bold">4.&nbsp;</mtext></mrow><msub><mi>f</mi><mn>4</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>1</mn><mo>+</mo><mi>x</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;si&nbsp;</mtext></mrow><mi>x</mi><mo>∈</mo><mo stretchy="false">[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mn>1</mn><mo>−</mo><mi>x</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;si&nbsp;</mtext></mrow><mi>x</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;sinon</mtext></mrow></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></mtd></mtr><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="bold">5.&nbsp;</mtext></mrow><msub><mi>f</mi><mn>5</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mfrac><mn>1</mn><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><msup><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mn>3</mn></msup></mrow></mfrac></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;si&nbsp;</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>&gt;</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;sinon</mtext></mrow></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></mtd><mtd><mspace width="1em"></mspace><mspace width="1em"></mspace></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="bold">6.&nbsp;</mtext></mrow><msub><mi>f</mi><mn>6</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>,</mo><mtext>&nbsp;</mtext><mi>x</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow><mo>.</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-461">\begin{array}{lll}
\textbf{1. }f_1(x)=\left\{\begin{array}{ll}
\cos x&\textrm{ si }x\in [0,\pi/2]\\
0&\textrm{ sinon.}
\end{array}\right.
&\quad\quad&
\textbf{2. }f_2(x)=\frac{1}{1+x^2},\ x\in\mathbb R\\
\textbf{3. }f_3(x)=\frac{e^x}{(e^x+1)^2},\ x\in\mathbb R
&\quad\quad&
\textbf{4. }f_4(x)=\left\{
\begin{array}{ll}
1+x&\textrm{ si }x\in [-1,0]\\
1-x&\textrm{ si }x\in [0,1]\\
0&\textrm{ sinon}
\end{array}\right.\\
\textbf{5. }f_5(x)=\left\{
\begin{array}{ll}
\frac{1}{|x|^3}&\textrm{ si }|x|>1\\
0&\textrm{ sinon}
\end{array}\right.
&\quad\quad&
\textbf{6. }f_6(x)=\sin x+1,\ x\in\mathbb R.
\end{array}</script>
</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2333&#39;,&#39;imageindication2333&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2333"></a><div class="inner" id="innerindication2333" style="display:none;">Pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-462-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5652" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5653" class="mjx-mrow"><span id="MJXc-Node-5654" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-5655" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5656" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-462">f_3</script>, reconnaitre une forme du type <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-463-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;&amp;#x2032;&lt;/mo&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5657" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5658" class="mjx-mrow"><span id="MJXc-Node-5659" class="mjx-msup"><span class="mjx-base"><span id="MJXc-Node-5660" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5661" class="mjx-mo" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">′</span></span></span></span><span id="MJXc-Node-5662" class="mjx-texatom"><span id="MJXc-Node-5663" class="mjx-mrow"><span id="MJXc-Node-5664" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-5665" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5666" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5667" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>u</mi><mo>′</mo></msup><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msup><mi>u</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-463">u'/u^2</script>.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2333&#39;,&#39;imagecorrige2333&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2333"></a><div class="inner" id="innercorrige2333" style="display:none;">On vérifie d'abord que les fonctions données sont continues sauf en un nombre fini de points et positives sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-464-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5668" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5669" class="mjx-mrow"><span id="MJXc-Node-5670" class="mjx-texatom"><span id="MJXc-Node-5671" class="mjx-mrow"><span id="MJXc-Node-5672" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.47em; padding-bottom: 0.295em;">R</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-464">\mathbb R</script>. On doit donc vérifier que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-465-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5673" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5674" class="mjx-mrow"><span id="MJXc-Node-5675" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-5676" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.591em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-5681" class="mjx-texatom" style=""><span id="MJXc-Node-5682" class="mjx-mrow"><span id="MJXc-Node-5683" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5684" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-5677" class="mjx-texatom" style=""><span id="MJXc-Node-5678" class="mjx-mrow"><span id="MJXc-Node-5679" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5680" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span><span id="MJXc-Node-5685" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-5686" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5687" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-5688" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5689" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-5690" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi></math></span></span><script type="math/tex" id="MathJax-Element-465">\int_{-\infty}^{+\infty}f(t)dt</script> converge et est égal à 1 pour déterminer s'il s'agit d'une densité.
<ol class="enumeratechiffre">
<li> On a bien <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-466-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;cos&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;sin&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;sin&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5691" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5692" class="mjx-mrow"><span id="MJXc-Node-5693" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-5694" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.34em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-5696" class="mjx-texatom" style=""><span id="MJXc-Node-5697" class="mjx-mrow"><span id="MJXc-Node-5698" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-5699" class="mjx-texatom"><span id="MJXc-Node-5700" class="mjx-mrow"><span id="MJXc-Node-5701" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-5702" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-5695" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-5703" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">cos</span></span><span id="MJXc-Node-5704" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-5705" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5706" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5707" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5708" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5709" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">sin</span></span><span id="MJXc-Node-5710" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-5711" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5712" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-5713" class="mjx-texatom"><span id="MJXc-Node-5714" class="mjx-mrow"><span id="MJXc-Node-5715" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-5716" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-5717" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5718" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5719" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">sin</span></span><span id="MJXc-Node-5720" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-5721" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5722" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-5723" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5724" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5725" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mrow class="MJX-TeXAtom-ORD"><mi>π</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msubsup><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>π</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo stretchy="false">)</mo><mo>−</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-466">\int_0^{\pi/2}\cos(x)=\sin(\pi/2)-\sin(0)=1</script> : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-467-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5726" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5727" class="mjx-mrow"><span id="MJXc-Node-5728" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-5729" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5730" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-467">f_1</script> définit bien une densité de probabilité. Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-468-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5731" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5732" class="mjx-mrow"><span id="MJXc-Node-5733" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-5734" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5735" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-468">X_1</script> une variable aléatoire de densité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-469-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5736" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5737" class="mjx-mrow"><span id="MJXc-Node-5738" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-5739" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5740" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-469">f_1</script>. Sa fonction de répartition est alors donnée par <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-470-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5741" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5742" class="mjx-mrow"><span id="MJXc-Node-5743" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-5744" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5745" class="mjx-texatom" style=""><span id="MJXc-Node-5746" class="mjx-mrow"><span id="MJXc-Node-5747" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-5748" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5749" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span id="MJXc-Node-5750" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5751" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5752" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5753" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5754" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>X</mi><mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-470">F_{X_1}(x)=0</script> si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-471-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5755" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5756" class="mjx-mrow"><span id="MJXc-Node-5757" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5758" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&lt;</span></span><span id="MJXc-Node-5759" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&lt;</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-471">x<0</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-472-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;sin&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5760" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5761" class="mjx-mrow"><span id="MJXc-Node-5762" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-5763" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5764" class="mjx-texatom" style=""><span id="MJXc-Node-5765" class="mjx-mrow"><span id="MJXc-Node-5766" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-5767" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5768" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span id="MJXc-Node-5769" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5770" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5771" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5772" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5773" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">sin</span></span><span id="MJXc-Node-5774" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-5775" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5776" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5777" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>X</mi><mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-472">F_{X_1}(x)=\sin(x)</script> si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-473-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5778" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5779" class="mjx-mrow"><span id="MJXc-Node-5780" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5781" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">∈</span></span><span id="MJXc-Node-5782" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-5783" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-5784" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-5785" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-5786" class="mjx-texatom"><span id="MJXc-Node-5787" class="mjx-mrow"><span id="MJXc-Node-5788" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-5789" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-5790" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-473">x\in [0,\pi/2]</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-474-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5791" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5792" class="mjx-mrow"><span id="MJXc-Node-5793" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-5794" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5795" class="mjx-texatom" style=""><span id="MJXc-Node-5796" class="mjx-mrow"><span id="MJXc-Node-5797" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-5798" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5799" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span id="MJXc-Node-5800" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5801" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5802" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5803" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5804" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>X</mi><mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-474">F_{X_1}(x)=1</script> si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-475-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5805" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5806" class="mjx-mrow"><span id="MJXc-Node-5807" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5808" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-5809" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-5810" class="mjx-texatom"><span id="MJXc-Node-5811" class="mjx-mrow"><span id="MJXc-Node-5812" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-5813" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mi>π</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-475">x\geq\pi/2</script>. De plus, puisque <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-476-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5814" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5815" class="mjx-mrow"><span id="MJXc-Node-5816" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-5817" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5818" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-476">f_1</script> est non-nulle seulement sur l'intervalle <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-477-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5819" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5820" class="mjx-mrow"><span id="MJXc-Node-5821" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-5822" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-5823" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-5824" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-5825" class="mjx-texatom"><span id="MJXc-Node-5826" class="mjx-mrow"><span id="MJXc-Node-5827" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-5828" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-5829" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-477">[0,\pi/2]</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-478-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5830" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5831" class="mjx-mrow"><span id="MJXc-Node-5832" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-5833" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5834" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-478">X_1</script> admet une espérance donnée par
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-479-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;cos&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5835" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5836" class="mjx-mrow"><span id="MJXc-Node-5837" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-5838" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5839" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-5840" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5841" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-5842" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5843" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5844" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-5845" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 1.911em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-5847" class="mjx-texatom" style=""><span id="MJXc-Node-5848" class="mjx-mrow"><span id="MJXc-Node-5849" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-5850" class="mjx-texatom"><span id="MJXc-Node-5851" class="mjx-mrow"><span id="MJXc-Node-5852" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-5853" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-5846" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-5854" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5855" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">cos</span></span><span id="MJXc-Node-5856" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-5857" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5858" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5859" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5860" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-5861" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5862" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5863" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-5864" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-5865" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-5866" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5867" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-5868" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5869" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-5870" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mrow class="MJX-TeXAtom-ORD"><mi>π</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msubsup><mi>x</mi><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>π</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-479">E(X_1)=\int_0^{\pi/2}x\cos(x)dx=\frac 12(\pi -2)</script>
(intégrale qu'on peut calculer à l'aide d'une intégration par parties).
</li><li> On a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-480-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munder&gt;&lt;mo movablelimits=&quot;true&quot; form=&quot;prefix&quot;&gt;lim&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mi&gt;arctan&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;munder&gt;&lt;mo movablelimits=&quot;true&quot; form=&quot;prefix&quot;&gt;lim&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mi&gt;arctan&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mo&gt;&amp;#x2260;&lt;/mo&gt;&lt;mn&gt;1.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5871" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5872" class="mjx-mrow"><span id="MJXc-Node-5873" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-5874" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.162em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-5879" class="mjx-texatom" style=""><span id="MJXc-Node-5880" class="mjx-mrow"><span id="MJXc-Node-5881" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5882" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-5875" class="mjx-texatom" style=""><span id="MJXc-Node-5876" class="mjx-mrow"><span id="MJXc-Node-5877" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5878" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span><span id="MJXc-Node-5883" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 2.883em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.883em; top: -1.396em;"><span id="MJXc-Node-5884" class="mjx-mrow"><span id="MJXc-Node-5885" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-5886" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="mjx-denominator" style="width: 2.883em; bottom: -0.877em;"><span id="MJXc-Node-5887" class="mjx-mrow"><span id="MJXc-Node-5888" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5889" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5890" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-5891" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5892" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.883em;"></span></span><span class="mjx-vsize" style="height: 2.272em; vertical-align: -0.877em;"></span></span><span id="MJXc-Node-5893" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5894" class="mjx-munderover MJXc-space3"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-op" style="padding-left: 0.49em;"><span id="MJXc-Node-5895" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">lim</span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em;"><span id="MJXc-Node-5896" class="mjx-texatom" style=""><span id="MJXc-Node-5897" class="mjx-mrow"><span id="MJXc-Node-5898" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5899" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">→</span></span><span id="MJXc-Node-5900" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5901" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span></span><span id="MJXc-Node-5902" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">arctan</span></span><span id="MJXc-Node-5903" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-5904" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5905" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5906" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5907" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5908" class="mjx-munderover MJXc-space2"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-op" style="padding-left: 0.49em;"><span id="MJXc-Node-5909" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">lim</span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em;"><span id="MJXc-Node-5910" class="mjx-texatom" style=""><span id="MJXc-Node-5911" class="mjx-mrow"><span id="MJXc-Node-5912" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5913" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">→</span></span><span id="MJXc-Node-5914" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5915" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span></span><span id="MJXc-Node-5916" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">arctan</span></span><span id="MJXc-Node-5917" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-5918" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5919" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5920" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5921" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5922" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-5923" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.528em;">≠</span></span><span id="MJXc-Node-5924" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><munder><mo movablelimits="true" form="prefix">lim</mo><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo stretchy="false">→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>arctan</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><munder><mo movablelimits="true" form="prefix">lim</mo><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo stretchy="false">→</mo><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>arctan</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>π</mi><mo>≠</mo><mn>1.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-480">\int_{-\infty}^{+\infty}\frac{dx}{1+x^2}=\lim_{x\to+\infty}\arctan(x)-\lim_{x\to-\infty}\arctan(x)=\pi\neq 1.</script>
Ainsi, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-481-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5925" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5926" class="mjx-mrow"><span id="MJXc-Node-5927" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-5928" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5929" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-481">f_2</script> n'est pas la densité de probabilité d'une variable aléatoire.
</li><li> Remarquons que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-482-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5930" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5931" class="mjx-mrow"><span id="MJXc-Node-5932" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-5933" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5934" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-482">f_3</script> est de la forme <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-483-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;&amp;#x2032;&lt;/mo&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5935" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5936" class="mjx-mrow"><span id="MJXc-Node-5937" class="mjx-msup"><span class="mjx-base"><span id="MJXc-Node-5938" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5939" class="mjx-mo" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">′</span></span></span></span><span id="MJXc-Node-5940" class="mjx-texatom"><span id="MJXc-Node-5941" class="mjx-mrow"><span id="MJXc-Node-5942" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-5943" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5944" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5945" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>u</mi><mo>′</mo></msup><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msup><mi>u</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-483">u'/u^2</script> avec <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-484-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-5946" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5947" class="mjx-mrow"><span id="MJXc-Node-5948" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span><span id="MJXc-Node-5949" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5950" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5951" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5952" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5953" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-5954" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5955" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span id="MJXc-Node-5956" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5957" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-484">u(x)=e^x+1</script>. On a donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-485-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/msubsup&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/msubsup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5958" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5959" class="mjx-mrow"><span id="MJXc-Node-5960" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-5961" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.375em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-5963" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-5962" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span></span><span id="MJXc-Node-5964" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 3.995em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.995em; top: -1.37em;"><span id="MJXc-Node-5965" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5966" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5967" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span><span class="mjx-denominator" style="width: 3.995em; bottom: -1.045em;"><span id="MJXc-Node-5968" class="mjx-mrow"><span id="MJXc-Node-5969" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5970" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5971" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5972" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span id="MJXc-Node-5973" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5974" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5975" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5976" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5977" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.995em;"></span></span><span class="mjx-vsize" style="height: 2.414em; vertical-align: -1.045em;"></span></span><span id="MJXc-Node-5978" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-5979" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-5980" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5981" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-5982" class="mjx-mrow"><span id="MJXc-Node-5983" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">[</span></span><span id="MJXc-Node-5984" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.828em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.828em; top: -1.368em;"><span id="MJXc-Node-5985" class="mjx-mrow"><span id="MJXc-Node-5986" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5987" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="width: 2.828em; bottom: -0.793em;"><span id="MJXc-Node-5988" class="mjx-mrow"><span id="MJXc-Node-5989" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5990" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5991" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span id="MJXc-Node-5992" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5993" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.828em;"></span></span><span class="mjx-vsize" style="height: 2.16em; vertical-align: -0.793em;"></span></span><span id="MJXc-Node-5994" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">]</span></span></span></span><span class="mjx-stack" style="vertical-align: -1.009em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.625em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5996" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-5995" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span></span><span id="MJXc-Node-5997" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5998" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.727em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.727em; top: -1.368em;"><span id="MJXc-Node-5999" class="mjx-mrow"><span id="MJXc-Node-6000" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6001" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="width: 2.727em; bottom: -0.897em;"><span id="MJXc-Node-6002" class="mjx-mrow"><span id="MJXc-Node-6003" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6004" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6005" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span></span><span id="MJXc-Node-6006" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6007" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.727em;"></span></span><span class="mjx-vsize" style="height: 2.264em; vertical-align: -0.897em;"></span></span><span id="MJXc-Node-6008" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6009" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 2.798em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.798em; top: -1.368em;"><span id="MJXc-Node-6010" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 2.798em; bottom: -0.793em;"><span id="MJXc-Node-6011" class="mjx-mrow"><span id="MJXc-Node-6012" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6013" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6014" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span id="MJXc-Node-6015" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6016" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.798em;"></span></span><span class="mjx-vsize" style="height: 2.16em; vertical-align: -0.793em;"></span></span><span id="MJXc-Node-6017" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mfrac><msup><mi>e</mi><mi>x</mi></msup><mrow><mo stretchy="false">(</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mrow><mo>[</mo><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mo>]</mo></mrow><mi>a</mi><mi>b</mi></msubsup><mo>=</mo><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><mrow><msup><mi>e</mi><mi>b</mi></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msup><mi>e</mi><mi>a</mi></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-485">\int_a^b \frac{e^x}{(e^x+1)^2}dx=\left[\frac{-1}{e^x+1}\right]_a^b=\frac{-1}{e^b+1}+\frac{1}{e^a+1}.</script>
On fait tendre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-486-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6018" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6019" class="mjx-mrow"><span id="MJXc-Node-6020" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-486">a</script> vers <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-487-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6021" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6022" class="mjx-mrow"><span id="MJXc-Node-6023" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6024" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-487">-\infty</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-488-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6025" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6026" class="mjx-mrow"><span id="MJXc-Node-6027" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math></span></span><script type="math/tex" id="MathJax-Element-488">b</script> vers <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-489-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6028" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6029" class="mjx-mrow"><span id="MJXc-Node-6030" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6031" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-489">+\infty</script>. Utilisant les limites de l'exponentielle en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-490-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6032" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6033" class="mjx-mrow"><span id="MJXc-Node-6034" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6035" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-490">+\infty</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-491-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6036" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6037" class="mjx-mrow"><span id="MJXc-Node-6038" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6039" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-491">-\infty</script>, on en déduit que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-492-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6040" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6041" class="mjx-mrow"><span id="MJXc-Node-6042" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-6043" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.591em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-6048" class="mjx-texatom" style=""><span id="MJXc-Node-6049" class="mjx-mrow"><span id="MJXc-Node-6050" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6051" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6044" class="mjx-texatom" style=""><span id="MJXc-Node-6045" class="mjx-mrow"><span id="MJXc-Node-6046" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6047" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span><span id="MJXc-Node-6052" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 2.511em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 3.551em; top: -1.331em;"><span id="MJXc-Node-6053" class="mjx-msubsup" style=""><span class="mjx-base"><span id="MJXc-Node-6054" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6055" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 3.551em; bottom: -0.948em;"><span id="MJXc-Node-6056" class="mjx-mrow" style=""><span id="MJXc-Node-6057" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6058" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6059" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6060" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span id="MJXc-Node-6061" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6062" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6063" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6064" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6065" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.511em;"></span></span><span class="mjx-vsize" style="height: 1.612em; vertical-align: -0.67em;"></span></span><span id="MJXc-Node-6066" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-6067" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mfrac><msup><mi>e</mi><mi>x</mi></msup><mrow><mo stretchy="false">(</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mi>d</mi><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-492">\int_{-\infty}^{+\infty}\frac{e^x}{(e^x+1)^2}dx</script> converge et vaut 1. Ainsi, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-493-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6068" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6069" class="mjx-mrow"><span id="MJXc-Node-6070" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-6071" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6072" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-493">f_3</script> est bien la densité de probabilité d'une variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-494-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6073" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6074" class="mjx-mrow"><span id="MJXc-Node-6075" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6076" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6077" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-494">X_3</script>. La fonction de répartition <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-495-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6078" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6079" class="mjx-mrow"><span id="MJXc-Node-6080" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-6081" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6082" class="mjx-texatom" style=""><span id="MJXc-Node-6083" class="mjx-mrow"><span id="MJXc-Node-6084" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6085" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6086" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>X</mi><mn>3</mn></msub></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-495">F_{X_3}</script> de cette variable aléatoire est donnée par
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-496-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msubsup&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6087" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6088" class="mjx-mrow"><span id="MJXc-Node-6089" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-6090" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6091" class="mjx-texatom" style=""><span id="MJXc-Node-6092" class="mjx-mrow"><span id="MJXc-Node-6093" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6094" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6095" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span></span></span><span id="MJXc-Node-6096" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6097" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6098" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6099" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6100" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-6101" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.233em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-6106" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6102" class="mjx-texatom" style=""><span id="MJXc-Node-6103" class="mjx-mrow"><span id="MJXc-Node-6104" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6105" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span><span id="MJXc-Node-6107" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 3.995em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.995em; top: -1.37em;"><span id="MJXc-Node-6108" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6109" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6110" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span><span class="mjx-denominator" style="width: 3.995em; bottom: -1.045em;"><span id="MJXc-Node-6111" class="mjx-mrow"><span id="MJXc-Node-6112" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6113" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6114" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6115" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span id="MJXc-Node-6116" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6117" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6118" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6119" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6120" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.995em;"></span></span><span class="mjx-vsize" style="height: 2.414em; vertical-align: -1.045em;"></span></span><span id="MJXc-Node-6121" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-6122" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6123" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6124" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6125" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6126" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 2.679em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.679em; top: -1.368em;"><span id="MJXc-Node-6127" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 2.679em; bottom: -0.849em;"><span id="MJXc-Node-6128" class="mjx-mrow"><span id="MJXc-Node-6129" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6130" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6131" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span></span><span id="MJXc-Node-6132" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6133" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.679em;"></span></span><span class="mjx-vsize" style="height: 2.216em; vertical-align: -0.849em;"></span></span><span id="MJXc-Node-6134" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>X</mi><mn>3</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>t</mi></msubsup><mfrac><msup><mi>e</mi><mi>x</mi></msup><mrow><mo stretchy="false">(</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mrow><msup><mi>e</mi><mi>t</mi></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-496">F_{X_3}(t)=\int_{-\infty}^t \frac{e^x}{(e^x+1)^2}dx=1-\frac{1}{e^t+1}.</script>
De plus, l'intégrale <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-497-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6135" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6136" class="mjx-mrow"><span id="MJXc-Node-6137" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-6138" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.591em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-6143" class="mjx-texatom" style=""><span id="MJXc-Node-6144" class="mjx-mrow"><span id="MJXc-Node-6145" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6146" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6139" class="mjx-texatom" style=""><span id="MJXc-Node-6140" class="mjx-mrow"><span id="MJXc-Node-6141" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6142" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span><span id="MJXc-Node-6147" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 2.511em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 3.551em; top: -1.651em;"><span id="MJXc-Node-6148" class="mjx-mrow" style=""><span id="MJXc-Node-6149" class="mjx-texatom"><span id="MJXc-Node-6150" class="mjx-mrow"><span id="MJXc-Node-6151" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6152" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6153" class="mjx-texatom"><span id="MJXc-Node-6154" class="mjx-mrow"><span id="MJXc-Node-6155" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6156" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6157" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6158" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 3.551em; bottom: -0.948em;"><span id="MJXc-Node-6159" class="mjx-mrow" style=""><span id="MJXc-Node-6160" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6161" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6162" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6163" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span id="MJXc-Node-6164" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6165" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6166" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6167" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6168" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.511em;"></span></span><span class="mjx-vsize" style="height: 1.838em; vertical-align: -0.67em;"></span></span><span id="MJXc-Node-6169" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-6170" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msup><mi>e</mi><mi>x</mi></msup></mrow><mrow><mo stretchy="false">(</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mi>d</mi><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-497">\int_{-\infty}^{+\infty}\frac{|x|e^x}{(e^x+1)^2}dx</script> est convergente.
En effet, au voisinage de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-498-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6171" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6172" class="mjx-mrow"><span id="MJXc-Node-6173" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6174" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-498">-\infty</script>, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-499-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mo&gt;&amp;#x223C;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6175" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6176" class="mjx-mrow"><span id="MJXc-Node-6177" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.995em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.995em; top: -1.541em;"><span id="MJXc-Node-6178" class="mjx-mrow"><span id="MJXc-Node-6179" class="mjx-texatom"><span id="MJXc-Node-6180" class="mjx-mrow"><span id="MJXc-Node-6181" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6182" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6183" class="mjx-texatom"><span id="MJXc-Node-6184" class="mjx-mrow"><span id="MJXc-Node-6185" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6186" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6187" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6188" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span><span class="mjx-denominator" style="width: 3.995em; bottom: -1.045em;"><span id="MJXc-Node-6189" class="mjx-mrow"><span id="MJXc-Node-6190" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6191" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6192" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6193" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span id="MJXc-Node-6194" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6195" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6196" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6197" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6198" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.995em;"></span></span><span class="mjx-vsize" style="height: 2.586em; vertical-align: -1.045em;"></span></span><span id="MJXc-Node-6199" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-6200" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">∼</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6201" class="mjx-texatom" style=""><span id="MJXc-Node-6202" class="mjx-mrow"><span id="MJXc-Node-6203" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6204" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span><span id="MJXc-Node-6205" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-6206" class="mjx-mrow"><span id="MJXc-Node-6207" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6208" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6209" class="mjx-texatom"><span id="MJXc-Node-6210" class="mjx-mrow"><span id="MJXc-Node-6211" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6212" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6213" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6214" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msup><mi>e</mi><mi>x</mi></msup></mrow><mrow><mo stretchy="false">(</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><msub><mo>∼</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msup><mi>e</mi><mi>x</mi></msup></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-499">\frac{|x|e^x}{(e^x+1)^2}\sim_{-\infty}|x|e^x</script>
et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-500-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msubsup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6215" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6216" class="mjx-mrow"><span id="MJXc-Node-6217" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-6218" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.651em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-6223" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6219" class="mjx-texatom" style=""><span id="MJXc-Node-6220" class="mjx-mrow"><span id="MJXc-Node-6221" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6222" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span><span id="MJXc-Node-6224" class="mjx-texatom MJXc-space1"><span id="MJXc-Node-6225" class="mjx-mrow"><span id="MJXc-Node-6226" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6227" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6228" class="mjx-texatom"><span id="MJXc-Node-6229" class="mjx-mrow"><span id="MJXc-Node-6230" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6231" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6232" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6233" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span id="MJXc-Node-6234" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-6235" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mn>0</mn></msubsup><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msup><mi>e</mi><mi>x</mi></msup><mi>d</mi><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-500">\int_{-\infty}^0 |x|e^x dx</script> converge (par comparaison à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-501-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6236" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6237" class="mjx-mrow"><span id="MJXc-Node-6238" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6239" class="mjx-texatom"><span id="MJXc-Node-6240" class="mjx-mrow"><span id="MJXc-Node-6241" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-6242" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6243" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6244" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msup><mi>x</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-501">1/x^2</script> par exemple, ou par calcul en effectuant une intégration par parties). De même, au voisinage de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-502-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6245" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6246" class="mjx-mrow"><span id="MJXc-Node-6247" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6248" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-502">+\infty</script>,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-503-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mo&gt;&amp;#x223C;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6249" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6250" class="mjx-mrow"><span id="MJXc-Node-6251" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.995em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.995em; top: -1.541em;"><span id="MJXc-Node-6252" class="mjx-mrow"><span id="MJXc-Node-6253" class="mjx-texatom"><span id="MJXc-Node-6254" class="mjx-mrow"><span id="MJXc-Node-6255" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6256" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6257" class="mjx-texatom"><span id="MJXc-Node-6258" class="mjx-mrow"><span id="MJXc-Node-6259" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6260" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6261" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6262" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span><span class="mjx-denominator" style="width: 3.995em; bottom: -1.045em;"><span id="MJXc-Node-6263" class="mjx-mrow"><span id="MJXc-Node-6264" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6265" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6266" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6267" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span id="MJXc-Node-6268" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6269" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6270" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6271" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6272" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.995em;"></span></span><span class="mjx-vsize" style="height: 2.586em; vertical-align: -1.045em;"></span></span><span id="MJXc-Node-6273" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-6274" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">∼</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6275" class="mjx-texatom" style=""><span id="MJXc-Node-6276" class="mjx-mrow"><span id="MJXc-Node-6277" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6278" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span><span id="MJXc-Node-6279" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.234em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.234em; top: -1.541em;"><span id="MJXc-Node-6280" class="mjx-mrow"><span id="MJXc-Node-6281" class="mjx-texatom"><span id="MJXc-Node-6282" class="mjx-mrow"><span id="MJXc-Node-6283" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6284" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6285" class="mjx-texatom"><span id="MJXc-Node-6286" class="mjx-mrow"><span id="MJXc-Node-6287" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6288" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6289" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6290" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span><span class="mjx-denominator" style="width: 2.234em; bottom: -0.806em;"><span id="MJXc-Node-6291" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6292" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6293" class="mjx-texatom" style=""><span id="MJXc-Node-6294" class="mjx-mrow"><span id="MJXc-Node-6295" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-6296" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.234em;"></span></span><span class="mjx-vsize" style="height: 2.347em; vertical-align: -0.806em;"></span></span><span id="MJXc-Node-6297" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6298" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-6299" class="mjx-mrow"><span id="MJXc-Node-6300" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6301" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6302" class="mjx-texatom"><span id="MJXc-Node-6303" class="mjx-mrow"><span id="MJXc-Node-6304" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6305" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6306" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6307" class="mjx-texatom" style=""><span id="MJXc-Node-6308" class="mjx-mrow"><span id="MJXc-Node-6309" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6310" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msup><mi>e</mi><mi>x</mi></msup></mrow><mrow><mo stretchy="false">(</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><msub><mo>∼</mo><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msub><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msup><mi>e</mi><mi>x</mi></msup></mrow><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn><mi>x</mi></mrow></msup></mfrac><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>x</mi></mrow></msup></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-503">\frac{|x|e^x}{(e^x+1)^2}\sim_{+\infty}\frac{|x|e^x}{e^{2x}}=|x|e^{-x}</script>
et pour les mêmes raisons, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-504-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6311" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6312" class="mjx-mrow"><span id="MJXc-Node-6313" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-6314" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.508em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-6316" class="mjx-texatom" style=""><span id="MJXc-Node-6317" class="mjx-mrow"><span id="MJXc-Node-6318" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6319" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6315" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-6320" class="mjx-texatom MJXc-space1"><span id="MJXc-Node-6321" class="mjx-mrow"><span id="MJXc-Node-6322" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6323" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6324" class="mjx-texatom"><span id="MJXc-Node-6325" class="mjx-mrow"><span id="MJXc-Node-6326" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6327" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6328" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6329" class="mjx-texatom" style=""><span id="MJXc-Node-6330" class="mjx-mrow"><span id="MJXc-Node-6331" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6332" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span><span id="MJXc-Node-6333" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-6334" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>x</mi></mrow></msup><mi>d</mi><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-504">\int_0^{+\infty}|x|e^{-x}dx</script> converge. Ainsi, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-505-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6335" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6336" class="mjx-mrow"><span id="MJXc-Node-6337" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6338" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6339" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-505">X_3</script> admet une espérance. Par ailleurs, la fonction <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-506-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6340" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6341" class="mjx-mrow"><span id="MJXc-Node-6342" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-6343" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6344" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-506">f_3</script> est paire. En effet, pour tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-507-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6345" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6346" class="mjx-mrow"><span id="MJXc-Node-6347" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6348" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">∈</span></span><span id="MJXc-Node-6349" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-6350" class="mjx-mrow"><span id="MJXc-Node-6351" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.47em; padding-bottom: 0.295em;">R</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-507">x\in\mathbb R</script>, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-508-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;exp&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;exp&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;exp&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;exp&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;exp&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;exp&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;exp&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6352" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6353" class="mjx-mrow"><span id="MJXc-Node-6354" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-6355" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6356" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-6357" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6358" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6359" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6360" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6361" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6362" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 7.085em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 7.085em; top: -1.542em;"><span id="MJXc-Node-6363" class="mjx-mrow"><span id="MJXc-Node-6364" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.528em;">exp</span></span><span id="MJXc-Node-6365" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-6366" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6367" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6368" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6369" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 7.085em; bottom: -1.457em;"><span id="MJXc-Node-6370" class="mjx-mrow"><span id="MJXc-Node-6371" class="mjx-texatom"><span id="MJXc-Node-6372" class="mjx-mrow"><span id="MJXc-Node-6373" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-6374" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.528em;">exp</span></span><span id="MJXc-Node-6375" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-6376" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6377" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6378" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6379" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6380" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6381" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6382" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6383" class="mjx-texatom"><span id="MJXc-Node-6384" class="mjx-mrow"><span id="MJXc-Node-6385" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.851em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6386" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 7.085em;"></span></span><span class="mjx-vsize" style="height: 2.999em; vertical-align: -1.457em;"></span></span><span id="MJXc-Node-6387" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6388" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 10.297em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 10.297em; top: -1.542em;"><span id="MJXc-Node-6389" class="mjx-mrow"><span id="MJXc-Node-6390" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.528em;">exp</span></span><span id="MJXc-Node-6391" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-6392" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6393" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6394" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6395" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 10.297em; bottom: -1.457em;"><span id="MJXc-Node-6396" class="mjx-mrow"><span id="MJXc-Node-6397" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.528em;">exp</span></span><span id="MJXc-Node-6398" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-6399" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6400" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6401" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-6402" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6403" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6404" class="mjx-texatom"><span id="MJXc-Node-6405" class="mjx-mrow"><span id="MJXc-Node-6406" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-6407" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6408" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6409" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.528em;">exp</span></span><span id="MJXc-Node-6410" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-6411" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6412" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6413" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6414" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6415" class="mjx-texatom"><span id="MJXc-Node-6416" class="mjx-mrow"><span id="MJXc-Node-6417" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.851em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6418" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 10.297em;"></span></span><span class="mjx-vsize" style="height: 2.999em; vertical-align: -1.457em;"></span></span><span id="MJXc-Node-6419" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6420" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 6.307em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 6.307em; top: -1.542em;"><span id="MJXc-Node-6421" class="mjx-mrow"><span id="MJXc-Node-6422" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.528em;">exp</span></span><span id="MJXc-Node-6423" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-6424" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6425" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6426" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 6.307em; bottom: -1.457em;"><span id="MJXc-Node-6427" class="mjx-mrow"><span id="MJXc-Node-6428" class="mjx-texatom"><span id="MJXc-Node-6429" class="mjx-mrow"><span id="MJXc-Node-6430" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-6431" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.528em;">exp</span></span><span id="MJXc-Node-6432" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-6433" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6434" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6435" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6436" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6437" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6438" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6439" class="mjx-texatom"><span id="MJXc-Node-6440" class="mjx-mrow"><span id="MJXc-Node-6441" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.851em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6442" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 6.307em;"></span></span><span class="mjx-vsize" style="height: 2.999em; vertical-align: -1.457em;"></span></span><span id="MJXc-Node-6443" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>f</mi><mn>3</mn></msub><mo stretchy="false">(</mo><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn><msup><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mi>x</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mn>1</mn><mo>+</mo><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn><msup><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-508">f_3(-x)=\frac{\exp(-x)}{\big(\exp(-x)+1\big)^2}=\frac{\exp(-x)}{\exp(-2x)\big(1+\exp(x)\big)^2}=\frac{\exp(x)}{\big(\exp(x)+1\big)^2}.</script>
Ceci entraîne (par un résultat du cours, ou tout simplement en effectuant le changement de variables <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-509-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6444" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6445" class="mjx-mrow"><span id="MJXc-Node-6446" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span><span id="MJXc-Node-6447" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6448" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6449" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mo>−</mo><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-509">u=-x</script> dans l'intégrale) que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-510-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6450" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6451" class="mjx-mrow"><span id="MJXc-Node-6452" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-6453" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6454" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6455" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6456" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-6457" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6458" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6459" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-510">E(X_3)=0</script>.
</li><li> On a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-511-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msubsup&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6460" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6461" class="mjx-mrow"><span id="MJXc-Node-6462" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-6463" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.59em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-6465" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6464" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-6466" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6467" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6468" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6469" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6470" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6471" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-6472" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6473" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6474" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-6475" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-6476" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-511">\int_0^1 (1-x)dx=\frac 12</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-512-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msubsup&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6477" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6478" class="mjx-mrow"><span id="MJXc-Node-6479" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-6480" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.568em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-6485" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6481" class="mjx-texatom" style=""><span id="MJXc-Node-6482" class="mjx-mrow"><span id="MJXc-Node-6483" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6484" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span><span id="MJXc-Node-6486" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6487" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6488" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6489" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6490" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6491" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-6492" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6493" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6494" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-6495" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-6496" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow><mn>0</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-512">\int_{-1}^0 (1+x)dx=\frac 12</script>. Ainsi, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-513-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msubsup&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6497" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6498" class="mjx-mrow"><span id="MJXc-Node-6499" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-6500" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.59em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-6505" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6501" class="mjx-texatom" style=""><span id="MJXc-Node-6502" class="mjx-mrow"><span id="MJXc-Node-6503" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6504" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span><span id="MJXc-Node-6506" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-6507" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6508" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span id="MJXc-Node-6509" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6510" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6511" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6512" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-6513" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6514" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6515" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow><mn>1</mn></msubsup><msub><mi>f</mi><mn>4</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-513">\int_{-1}^1 f_4(x)dx=1</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-514-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6516" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6517" class="mjx-mrow"><span id="MJXc-Node-6518" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-6519" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6520" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>4</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-514">f_4</script> est la densité de probabilité d'une variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-515-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6521" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6522" class="mjx-mrow"><span id="MJXc-Node-6523" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6524" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6525" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>4</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-515">X_4</script>. La fonction de répartition de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-516-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6526" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6527" class="mjx-mrow"><span id="MJXc-Node-6528" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6529" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6530" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>4</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-516">X_4</script> vaut <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-517-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6531" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6532" class="mjx-mrow"><span id="MJXc-Node-6533" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-6534" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6535" class="mjx-texatom" style=""><span id="MJXc-Node-6536" class="mjx-mrow"><span id="MJXc-Node-6537" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6538" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6539" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span></span></span></span><span id="MJXc-Node-6540" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6541" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6542" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6543" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6544" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>X</mi><mn>4</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-517">F_{X_4}(t)=0</script> si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-518-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6545" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6546" class="mjx-mrow"><span id="MJXc-Node-6547" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6548" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-6549" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6550" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≤</mo><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-518">t\leq -1</script>. Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-519-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6551" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6552" class="mjx-mrow"><span id="MJXc-Node-6553" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6554" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">∈</span></span><span id="MJXc-Node-6555" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-6556" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6557" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6558" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-6559" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-6560" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-519">t\in [-1,0]</script>, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-520-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msubsup&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6561" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6562" class="mjx-mrow"><span id="MJXc-Node-6563" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-6564" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6565" class="mjx-texatom" style=""><span id="MJXc-Node-6566" class="mjx-mrow"><span id="MJXc-Node-6567" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6568" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6569" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span></span></span></span><span id="MJXc-Node-6570" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6571" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6572" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6573" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6574" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-6575" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.15em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-6580" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6576" class="mjx-texatom" style=""><span id="MJXc-Node-6577" class="mjx-mrow"><span id="MJXc-Node-6578" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6579" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span><span id="MJXc-Node-6581" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6582" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6583" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6584" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6585" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6586" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-6587" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6588" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6589" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-6590" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-6591" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-6592" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6593" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6594" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6595" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.95em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.95em; top: -1.528em;"><span id="MJXc-Node-6596" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6597" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6598" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span><span class="mjx-denominator" style="width: 0.95em; bottom: -0.711em;"><span id="MJXc-Node-6599" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.95em;"></span></span><span class="mjx-vsize" style="height: 2.239em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-6600" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>X</mi><mn>4</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow><mi>t</mi></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>t</mi><mo>+</mo><mfrac><msup><mi>t</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-520">F_{X_4}(t)=\int_{-1}^t (1+x)dx=\frac 12+t+\frac{t^2}2.</script>
Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-521-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6601" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6602" class="mjx-mrow"><span id="MJXc-Node-6603" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6604" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">∈</span></span><span id="MJXc-Node-6605" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-6606" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-6607" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-6608" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6609" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-521">t\in [0,1]</script>, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-522-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msubsup&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msubsup&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6610" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6611" class="mjx-mrow"><span id="MJXc-Node-6612" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-6613" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6614" class="mjx-texatom" style=""><span id="MJXc-Node-6615" class="mjx-mrow"><span id="MJXc-Node-6616" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6617" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6618" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span></span></span></span><span id="MJXc-Node-6619" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6620" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6621" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6622" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6623" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-6624" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.139em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-6629" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6625" class="mjx-texatom" style=""><span id="MJXc-Node-6626" class="mjx-mrow"><span id="MJXc-Node-6627" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6628" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span><span id="MJXc-Node-6630" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-6631" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6632" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span id="MJXc-Node-6633" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6634" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6635" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6636" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-6637" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6638" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6639" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-6640" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.15em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-6642" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6641" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-6643" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6644" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6645" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6646" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6647" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6648" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-6649" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6650" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6651" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-6652" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-6653" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-6654" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6655" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6656" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6657" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.95em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.95em; top: -1.528em;"><span id="MJXc-Node-6658" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6659" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6660" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span><span class="mjx-denominator" style="width: 0.95em; bottom: -0.711em;"><span id="MJXc-Node-6661" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.95em;"></span></span><span class="mjx-vsize" style="height: 2.239em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-6662" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>X</mi><mn>4</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow><mn>0</mn></msubsup><msub><mi>f</mi><mn>4</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>+</mo><msubsup><mo>∫</mo><mn>0</mn><mi>t</mi></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>t</mi><mo>−</mo><mfrac><msup><mi>t</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-522">F_{X_4}(t)=\int_{-1}^0 f_4(x)dx+\int_0^t (1-x)dt=\frac 12+t-\frac{t^2}2.</script>
Finalement, si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-523-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6663" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6664" class="mjx-mrow"><span id="MJXc-Node-6665" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6666" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-6667" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-523">t\geq 1</script>, on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-524-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6668" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6669" class="mjx-mrow"><span id="MJXc-Node-6670" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-6671" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6672" class="mjx-texatom" style=""><span id="MJXc-Node-6673" class="mjx-mrow"><span id="MJXc-Node-6674" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6675" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6676" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span></span></span></span><span id="MJXc-Node-6677" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6678" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6679" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6680" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6681" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>X</mi><mn>4</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-524">F_{X_4}(t)=1</script>. D'autre part, puisque <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-525-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6682" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6683" class="mjx-mrow"><span id="MJXc-Node-6684" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-6685" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6686" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>4</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-525">f_4</script> est nulle en dehors de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-526-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6687" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6688" class="mjx-mrow"><span id="MJXc-Node-6689" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-6690" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6691" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6692" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-6693" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6694" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-526">[-1,1]</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-527-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6695" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6696" class="mjx-mrow"><span id="MJXc-Node-6697" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6698" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6699" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>4</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-527">X_4</script> admet une espérance (il n'y a pas de problème de convergence d'intégrale). Enfin, puisque <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-528-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6700" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6701" class="mjx-mrow"><span id="MJXc-Node-6702" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-6703" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6704" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>4</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-528">f_4</script> est paire, on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-529-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6705" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6706" class="mjx-mrow"><span id="MJXc-Node-6707" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-6708" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6709" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6710" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6711" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span id="MJXc-Node-6712" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6713" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6714" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>4</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-529">E(X_4)=0</script>.
</li><li> On a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-530-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6715" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6716" class="mjx-mrow"><span id="MJXc-Node-6717" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-6718" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.079em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-6720" class="mjx-texatom" style=""><span id="MJXc-Node-6721" class="mjx-mrow"><span id="MJXc-Node-6722" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6723" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6719" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span><span id="MJXc-Node-6724" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 1.161em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.161em; top: -1.368em;"><span id="MJXc-Node-6725" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.161em; bottom: -0.805em;"><span id="MJXc-Node-6726" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6727" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6728" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.161em;"></span></span><span class="mjx-vsize" style="height: 2.173em; vertical-align: -0.805em;"></span></span><span id="MJXc-Node-6729" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-6730" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6731" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6732" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-6733" class="mjx-mrow"><span id="MJXc-Node-6734" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">[</span></span><span id="MJXc-Node-6735" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.661em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.661em; top: -1.368em;"><span id="MJXc-Node-6736" class="mjx-mrow"><span id="MJXc-Node-6737" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6738" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="width: 1.661em; bottom: -0.806em;"><span id="MJXc-Node-6739" class="mjx-mrow"><span id="MJXc-Node-6740" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-6741" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6742" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6743" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.661em;"></span></span><span class="mjx-vsize" style="height: 2.173em; vertical-align: -0.806em;"></span></span><span id="MJXc-Node-6744" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">]</span></span></span></span><span class="mjx-stack" style="vertical-align: -1.009em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.329em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6746" class="mjx-texatom" style=""><span id="MJXc-Node-6747" class="mjx-mrow"><span id="MJXc-Node-6748" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6749" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6745" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span><span id="MJXc-Node-6750" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6751" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-6752" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-6753" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-6754" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>∫</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mfrac><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mrow><mo>[</mo><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>]</mo></mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-530">\int_1^{+\infty}\frac{1}{x^3}dx=\left[\frac{-1}{2x^2}\right]_1^{+\infty}=\frac{1}2.</script>
De même, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-531-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6755" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6756" class="mjx-mrow"><span id="MJXc-Node-6757" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-6758" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.162em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-6763" class="mjx-texatom" style=""><span id="MJXc-Node-6764" class="mjx-mrow"><span id="MJXc-Node-6765" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6766" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6759" class="mjx-texatom" style=""><span id="MJXc-Node-6760" class="mjx-mrow"><span id="MJXc-Node-6761" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6762" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span><span id="MJXc-Node-6767" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 1.478em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.478em; top: -1.368em;"><span id="MJXc-Node-6768" class="mjx-mrow"><span id="MJXc-Node-6769" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6770" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="width: 1.478em; bottom: -0.805em;"><span id="MJXc-Node-6771" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6772" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6773" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.478em;"></span></span><span class="mjx-vsize" style="height: 2.173em; vertical-align: -0.805em;"></span></span><span id="MJXc-Node-6774" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-6775" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6776" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6777" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-6778" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-6779" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-6780" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msubsup><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-531">\int_{-\infty}^ {-1}\frac{-1}{x^3}dx=\frac 12.</script>
Ainsi, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-532-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6781" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6782" class="mjx-mrow"><span id="MJXc-Node-6783" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-6784" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6785" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>5</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-532">f_5</script> est bien la densité de probabilité d'une variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-533-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6786" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6787" class="mjx-mrow"><span id="MJXc-Node-6788" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6789" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6790" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>5</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-533">X_5</script>. Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-534-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6791" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6792" class="mjx-mrow"><span id="MJXc-Node-6793" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6794" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-6795" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6796" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≤</mo><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-534">t\leq -1</script>, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-535-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msubsup&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msup&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6797" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6798" class="mjx-mrow"><span id="MJXc-Node-6799" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-6800" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6801" class="mjx-texatom" style=""><span id="MJXc-Node-6802" class="mjx-mrow"><span id="MJXc-Node-6803" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6804" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6805" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span></span></span></span></span><span id="MJXc-Node-6806" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6807" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6808" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6809" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6810" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-6811" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.233em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-6816" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6812" class="mjx-texatom" style=""><span id="MJXc-Node-6813" class="mjx-mrow"><span id="MJXc-Node-6814" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6815" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span><span id="MJXc-Node-6817" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 1.478em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.478em; top: -1.368em;"><span id="MJXc-Node-6818" class="mjx-mrow"><span id="MJXc-Node-6819" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6820" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="width: 1.478em; bottom: -0.805em;"><span id="MJXc-Node-6821" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6822" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6823" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.478em;"></span></span><span class="mjx-vsize" style="height: 2.173em; vertical-align: -0.805em;"></span></span><span id="MJXc-Node-6824" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6825" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.45em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.45em; top: -1.368em;"><span id="MJXc-Node-6826" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.45em; bottom: -0.806em;"><span id="MJXc-Node-6827" class="mjx-mrow"><span id="MJXc-Node-6828" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-6829" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6830" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6831" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.45em;"></span></span><span class="mjx-vsize" style="height: 2.173em; vertical-align: -0.806em;"></span></span><span id="MJXc-Node-6832" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>X</mi><mn>5</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>t</mi></msubsup><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-535">F_{X_5}(t)=\int_{-\infty}^t\frac{-1}{x^3}=\frac{1}{2t^2}.</script>
Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-536-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6833" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6834" class="mjx-mrow"><span id="MJXc-Node-6835" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6836" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">∈</span></span><span id="MJXc-Node-6837" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-6838" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6839" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6840" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-6841" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6842" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-536">t\in [-1,1]</script>, alors on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-537-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6843" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6844" class="mjx-mrow"><span id="MJXc-Node-6845" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-6846" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6847" class="mjx-texatom" style=""><span id="MJXc-Node-6848" class="mjx-mrow"><span id="MJXc-Node-6849" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6850" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6851" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span></span></span></span></span><span id="MJXc-Node-6852" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6853" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6854" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6855" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6856" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-6857" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-6858" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-6859" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>X</mi><mn>5</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-537">F_{X_5}(t)=\frac 12.</script>
Enfin, si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-538-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6860" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6861" class="mjx-mrow"><span id="MJXc-Node-6862" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6863" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-6864" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-538">t\geq 1</script>, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-539-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msubsup&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msup&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6865" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6866" class="mjx-mrow"><span id="MJXc-Node-6867" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-6868" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6869" class="mjx-texatom" style=""><span id="MJXc-Node-6870" class="mjx-mrow"><span id="MJXc-Node-6871" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6872" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6873" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span></span></span></span></span><span id="MJXc-Node-6874" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6875" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6876" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6877" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6878" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-6879" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-6880" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-6881" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6882" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-6883" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.15em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-6885" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6884" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span><span id="MJXc-Node-6886" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 1.161em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.161em; top: -1.368em;"><span id="MJXc-Node-6887" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.161em; bottom: -0.805em;"><span id="MJXc-Node-6888" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6889" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6890" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.161em;"></span></span><span class="mjx-vsize" style="height: 2.173em; vertical-align: -0.805em;"></span></span><span id="MJXc-Node-6891" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-6892" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6893" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6894" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6895" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6896" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.45em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.45em; top: -1.368em;"><span id="MJXc-Node-6897" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.45em; bottom: -0.806em;"><span id="MJXc-Node-6898" class="mjx-mrow"><span id="MJXc-Node-6899" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-6900" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6901" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6902" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.45em;"></span></span><span class="mjx-vsize" style="height: 2.173em; vertical-align: -0.806em;"></span></span><span id="MJXc-Node-6903" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>X</mi><mn>5</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><msubsup><mo>∫</mo><mn>1</mn><mi>t</mi></msubsup><mfrac><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-539">F_{X_5}(t)=\frac 12+\int_1^t \frac{1}{x^3}dx=1-\frac{1}{2t^2}.</script>
D'autre part, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-540-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6904" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6905" class="mjx-mrow"><span id="MJXc-Node-6906" class="mjx-texatom"><span id="MJXc-Node-6907" class="mjx-mrow"><span id="MJXc-Node-6908" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6909" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6910" class="mjx-texatom"><span id="MJXc-Node-6911" class="mjx-mrow"><span id="MJXc-Node-6912" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6913" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-6914" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6915" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span id="MJXc-Node-6916" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6917" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6918" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6919" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6920" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.214em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.717em; top: -1.318em;"><span id="MJXc-Node-6921" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.717em; bottom: -1.16em;"><span id="MJXc-Node-6922" class="mjx-mrow" style=""><span id="MJXc-Node-6923" class="mjx-texatom"><span id="MJXc-Node-6924" class="mjx-mrow"><span id="MJXc-Node-6925" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6926" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6927" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6928" class="mjx-texatom"><span id="MJXc-Node-6929" class="mjx-mrow"><span id="MJXc-Node-6930" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.71em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6931" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.214em;"></span></span><span class="mjx-vsize" style="height: 1.752em; vertical-align: -0.82em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>f</mi><mn>5</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><msup><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mn>2</mn></msup></mrow></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-540">|x|f_5(x)=\frac{1}{|x|^2}</script> si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-541-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6932" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6933" class="mjx-mrow"><span id="MJXc-Node-6934" class="mjx-texatom"><span id="MJXc-Node-6935" class="mjx-mrow"><span id="MJXc-Node-6936" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6937" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6938" class="mjx-texatom"><span id="MJXc-Node-6939" class="mjx-mrow"><span id="MJXc-Node-6940" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6941" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-6942" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>≥</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-541">|x|\geq 1</script>, fonction qui est bien intégrable au voisinage de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-542-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6943" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6944" class="mjx-mrow"><span id="MJXc-Node-6945" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6946" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-542">+\infty</script> et de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-543-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6947" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6948" class="mjx-mrow"><span id="MJXc-Node-6949" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6950" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-543">-\infty</script>. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-544-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6951" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6952" class="mjx-mrow"><span id="MJXc-Node-6953" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6954" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6955" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>5</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-544">X_5</script> admet une espérance. Puisque <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-545-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6956" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6957" class="mjx-mrow"><span id="MJXc-Node-6958" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-6959" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6960" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>5</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-545">f_5</script> est paire, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-546-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6961" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6962" class="mjx-mrow"><span id="MJXc-Node-6963" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-6964" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6965" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6966" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6967" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span id="MJXc-Node-6968" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6969" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6970" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>5</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-546">E(X_5)=0</script>.
</li><li> On a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-547-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msubsup&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;cos&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6971" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6972" class="mjx-mrow"><span id="MJXc-Node-6973" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-6974" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.579em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-6976" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-6975" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-6977" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-6978" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6979" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span><span id="MJXc-Node-6980" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6981" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6982" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6983" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-6984" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-6985" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6986" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6987" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6988" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">cos</span></span><span id="MJXc-Node-6989" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-6990" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6991" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-6992" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6993" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6994" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mi>x</mi></msubsup><msub><mi>f</mi><mn>6</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi><mo>=</mo><mi>x</mi><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-547">\int_0^x f_6(t)dt=x-\cos(x)+1</script>. Ceci tend vers <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-548-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6995" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6996" class="mjx-mrow"><span id="MJXc-Node-6997" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6998" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-548">+\infty</script> si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-549-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-6999" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7000" class="mjx-mrow"><span id="MJXc-Node-7001" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-549">x</script> tend vers <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-550-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7002" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7003" class="mjx-mrow"><span id="MJXc-Node-7004" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7005" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-550">+\infty</script>. La fonction <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-551-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7006" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7007" class="mjx-mrow"><span id="MJXc-Node-7008" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-7009" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7010" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>6</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-551">f_6</script> n'est pas intégrable sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-552-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7011" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7012" class="mjx-mrow"><span id="MJXc-Node-7013" class="mjx-texatom"><span id="MJXc-Node-7014" class="mjx-mrow"><span id="MJXc-Node-7015" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.47em; padding-bottom: 0.295em;">R</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-552">\mathbb R</script>. Ce n'est pas une densité de probabilité.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>20</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Loi de Cauchy<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2346"> [Signaler une erreur]</a><a id="exo20" href="javascript:ajouteexo(2346,20,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2346&#39;,&#39;imageenonce2346&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2346"></a><div class="inner" id="innerenonce2346">Soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-553-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7016" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7017" class="mjx-mrow"><span id="MJXc-Node-7018" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-553">f</script> la fonction définie sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-554-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7019" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7020" class="mjx-mrow"><span id="MJXc-Node-7021" class="mjx-texatom"><span id="MJXc-Node-7022" class="mjx-mrow"><span id="MJXc-Node-7023" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">R</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-554">\mathbb R</script> par <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-555-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7024" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7025" class="mjx-mrow"><span id="MJXc-Node-7026" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7027" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-7028" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-7029" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-7030" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-7031" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.735em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.454em; top: -1.126em;"><span id="MJXc-Node-7032" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.454em; bottom: -0.822em;"><span id="MJXc-Node-7033" class="mjx-mrow" style=""><span id="MJXc-Node-7034" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-7035" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-7036" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7037" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span><span class="mjx-sup" style="font-size: 73.4%; vertical-align: 0.393em; padding-left: 0px; padding-right: 0.068em;"><span id="MJXc-Node-7038" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.291em; width: 1.735em;"></span></span><span class="mjx-vsize" style="height: 1.377em; vertical-align: -0.581em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>a</mi><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-555">f(x)=\frac{a}{1+x^2}</script>. Déterminer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-556-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7039" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7040" class="mjx-mrow"><span id="MJXc-Node-7041" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-556">a</script> pour que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-557-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7042" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7043" class="mjx-mrow"><span id="MJXc-Node-7044" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-557">f</script> soit la densité de probabilité d'une variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-558-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7045" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7046" class="mjx-mrow"><span id="MJXc-Node-7047" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-558">X</script>. Déterminer la fonction de répartition de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-559-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7048" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7049" class="mjx-mrow"><span id="MJXc-Node-7050" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-559">X</script>. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-560-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7051" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7052" class="mjx-mrow"><span id="MJXc-Node-7053" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-560">X</script> admet-elle une espérance?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2346&#39;,&#39;imageindication2346&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2346"></a><div class="inner" id="innerindication2346" style="display:none;"></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2346&#39;,&#39;imagecorrige2346&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2346"></a><div class="inner" id="innercorrige2346" style="display:none;">La fonction <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-561-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7054" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7055" class="mjx-mrow"><span id="MJXc-Node-7056" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-561">f</script> est continue sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-562-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7057" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7058" class="mjx-mrow"><span id="MJXc-Node-7059" class="mjx-texatom"><span id="MJXc-Node-7060" class="mjx-mrow"><span id="MJXc-Node-7061" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.47em; padding-bottom: 0.295em;">R</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-562">\mathbb R</script> et positive. Elle va être la densité de probabilité d'une variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-563-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7062" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7063" class="mjx-mrow"><span id="MJXc-Node-7064" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-563">X</script> si elle est intégrable et si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-564-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7065" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7066" class="mjx-mrow"><span id="MJXc-Node-7067" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-7068" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.591em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-7073" class="mjx-texatom" style=""><span id="MJXc-Node-7074" class="mjx-mrow"><span id="MJXc-Node-7075" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7076" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7069" class="mjx-texatom" style=""><span id="MJXc-Node-7070" class="mjx-mrow"><span id="MJXc-Node-7071" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7072" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span><span id="MJXc-Node-7077" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7078" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7079" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7080" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7081" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-7082" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7083" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7084" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-564">\int_{-\infty}^{+\infty}f(x)dx=1</script>. Or, une primitive de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-565-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7085" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7086" class="mjx-mrow"><span id="MJXc-Node-7087" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.725em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.439em; top: -1.318em;"><span id="MJXc-Node-7088" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.439em; bottom: -0.78em;"><span id="MJXc-Node-7089" class="mjx-mrow" style=""><span id="MJXc-Node-7090" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7091" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7092" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7093" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7094" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.725em;"></span></span><span class="mjx-vsize" style="height: 1.484em; vertical-align: -0.551em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-565">\frac{1}{1+x^2}</script> est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-566-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;arctan&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7095" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7096" class="mjx-mrow"><span id="MJXc-Node-7097" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">arctan</span></span><span id="MJXc-Node-7098" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-7099" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7100" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7101" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>arctan</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-566">\arctan(x)</script>. On a donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-567-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;munder&gt;&lt;mo movablelimits=&quot;true&quot; form=&quot;prefix&quot;&gt;lim&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mi&gt;arctan&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;munder&gt;&lt;mo movablelimits=&quot;true&quot; form=&quot;prefix&quot;&gt;lim&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mi&gt;arctan&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7102" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7103" class="mjx-mrow"><span id="MJXc-Node-7104" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-7105" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.162em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-7110" class="mjx-texatom" style=""><span id="MJXc-Node-7111" class="mjx-mrow"><span id="MJXc-Node-7112" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7113" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7106" class="mjx-texatom" style=""><span id="MJXc-Node-7107" class="mjx-mrow"><span id="MJXc-Node-7108" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7109" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span><span id="MJXc-Node-7114" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7115" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7116" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7117" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7118" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-7119" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7120" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7121" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-7122" class="mjx-munderover MJXc-space1"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-op" style="padding-left: 0.49em;"><span id="MJXc-Node-7123" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">lim</span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em;"><span id="MJXc-Node-7124" class="mjx-texatom" style=""><span id="MJXc-Node-7125" class="mjx-mrow"><span id="MJXc-Node-7126" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7127" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">→</span></span><span id="MJXc-Node-7128" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7129" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span></span><span id="MJXc-Node-7130" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">arctan</span></span><span id="MJXc-Node-7131" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-7132" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7133" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7134" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7135" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7136" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-7137" class="mjx-munderover MJXc-space1"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-op" style="padding-left: 0.49em;"><span id="MJXc-Node-7138" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">lim</span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em;"><span id="MJXc-Node-7139" class="mjx-texatom" style=""><span id="MJXc-Node-7140" class="mjx-mrow"><span id="MJXc-Node-7141" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7142" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">→</span></span><span id="MJXc-Node-7143" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7144" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span></span><span id="MJXc-Node-7145" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">arctan</span></span><span id="MJXc-Node-7146" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-7147" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7148" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7149" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7150" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7151" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-7152" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-7153" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mi>a</mi><munder><mo movablelimits="true" form="prefix">lim</mo><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo stretchy="false">→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>arctan</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>a</mi><munder><mo movablelimits="true" form="prefix">lim</mo><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo stretchy="false">→</mo><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>arctan</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>π</mi><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-567">\int_{-\infty}^{+\infty}f(x)dx=a\lim_{x\to+\infty}\arctan(x)-a\lim_{x\to-\infty}\arctan(x)=a\pi.</script>
Ainsi, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-568-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7154" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7155" class="mjx-mrow"><span id="MJXc-Node-7156" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-568">f</script> est la densité de probabilité d'une variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-569-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7157" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7158" class="mjx-mrow"><span id="MJXc-Node-7159" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-569">X</script> si et seulement si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-570-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7160" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7161" class="mjx-mrow"><span id="MJXc-Node-7162" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-7163" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7164" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.547em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.773em; top: -1.318em;"><span id="MJXc-Node-7165" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.773em; bottom: -0.524em;"><span id="MJXc-Node-7166" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.547em;"></span></span><span class="mjx-vsize" style="height: 1.302em; vertical-align: -0.37em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>1</mn><mi>π</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-570">a=\frac 1{\pi}</script>. Dans ce cas, la fonction de répartition de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-571-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7167" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7168" class="mjx-mrow"><span id="MJXc-Node-7169" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-571">X</script> est donnée par
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-572-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msubsup&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;/mfrac&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;arctan&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msubsup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;/mfrac&gt;&lt;mi&gt;arctan&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7170" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7171" class="mjx-mrow"><span id="MJXc-Node-7172" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-7173" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7174" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span id="MJXc-Node-7175" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7176" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-7177" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7178" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7179" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-7180" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.233em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-7185" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7181" class="mjx-texatom" style=""><span id="MJXc-Node-7182" class="mjx-mrow"><span id="MJXc-Node-7183" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7184" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span><span id="MJXc-Node-7186" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 4.234em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.234em; top: -1.368em;"><span id="MJXc-Node-7187" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 4.234em; bottom: -1.045em;"><span id="MJXc-Node-7188" class="mjx-mrow"><span id="MJXc-Node-7189" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-7190" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7191" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7192" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7193" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-7194" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7195" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-7196" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.234em;"></span></span><span class="mjx-vsize" style="height: 2.412em; vertical-align: -1.045em;"></span></span><span id="MJXc-Node-7197" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-7198" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7199" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7200" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.773em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.773em; top: -1.368em;"><span id="MJXc-Node-7201" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.773em; bottom: -0.722em;"><span id="MJXc-Node-7202" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.773em;"></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-7203" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7204" class="mjx-mrow"><span id="MJXc-Node-7205" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-7206" class="mjx-mrow"><span id="MJXc-Node-7207" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">arctan</span></span><span id="MJXc-Node-7208" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-7209" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7210" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7211" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span id="MJXc-Node-7212" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span><span class="mjx-stack" style="vertical-align: -0.31em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.505em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7217" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7213" class="mjx-texatom" style=""><span id="MJXc-Node-7214" class="mjx-mrow"><span id="MJXc-Node-7215" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7216" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span><span id="MJXc-Node-7218" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7219" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.773em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.773em; top: -1.368em;"><span id="MJXc-Node-7220" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.773em; bottom: -0.722em;"><span id="MJXc-Node-7221" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.773em;"></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-7222" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">arctan</span></span><span id="MJXc-Node-7223" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-7224" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7225" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-7226" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7227" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7228" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-7229" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-7230" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-7231" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>t</mi></msubsup><mfrac><mn>1</mn><mrow><mi>π</mi><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mi>π</mi></mfrac><msubsup><mrow><mo>[</mo><mrow><mi>arctan</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>t</mi></msubsup><mo>=</mo><mfrac><mn>1</mn><mi>π</mi></mfrac><mi>arctan</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-572">F_X(t)=\int_{-\infty}^t \frac{1}{\pi(1+x^2)}dx=\frac 1\pi\left[\arctan(x)\right]_{-\infty}^t=\frac1\pi\arctan(t)+\frac 12.</script>
Enfin, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-573-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7232" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7233" class="mjx-mrow"><span id="MJXc-Node-7234" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-573">X</script> n'admet pas d'espérance car la fonction <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-574-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7235" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7236" class="mjx-mrow"><span id="MJXc-Node-7237" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7238" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7239" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7240" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7241" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-574">xf(x)</script> n'est pas intégrable sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-575-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7242" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7243" class="mjx-mrow"><span id="MJXc-Node-7244" class="mjx-texatom"><span id="MJXc-Node-7245" class="mjx-mrow"><span id="MJXc-Node-7246" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.47em; padding-bottom: 0.295em;">R</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-575">\mathbb R</script>. En effet, au voisinage de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-576-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7247" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7248" class="mjx-mrow"><span id="MJXc-Node-7249" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7250" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-576">+\infty</script>, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-577-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mo&gt;&amp;#x223C;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7251" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7252" class="mjx-mrow"><span id="MJXc-Node-7253" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7254" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7255" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7256" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7257" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7258" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-7259" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">∼</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7260" class="mjx-texatom" style=""><span id="MJXc-Node-7261" class="mjx-mrow"><span id="MJXc-Node-7262" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7263" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span><span id="MJXc-Node-7264" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.345em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.345em; top: -1.368em;"><span id="MJXc-Node-7265" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.345em; bottom: -0.722em;"><span id="MJXc-Node-7266" class="mjx-mrow"><span id="MJXc-Node-7267" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-7268" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.345em;"></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.722em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>x</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>∼</mo><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msub><mfrac><mn>1</mn><mrow><mi>π</mi><mi>x</mi></mrow></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-577">xf(x)\sim_{+\infty}\frac{1}{\pi x}</script>
et on conclut par comparaison à une intégrale de Riemann divergente.</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>21</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Une densité<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2347"> [Signaler une erreur]</a><a id="exo21" href="javascript:ajouteexo(2347,21,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2347&#39;,&#39;imageenonce2347&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2347"></a><div class="inner" id="innerenonce2347">Soit la fonction <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-578-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7269" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7270" class="mjx-mrow"><span id="MJXc-Node-7271" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-578">f</script> définie sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-579-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7272" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7273" class="mjx-mrow"><span id="MJXc-Node-7274" class="mjx-texatom"><span id="MJXc-Node-7275" class="mjx-mrow"><span id="MJXc-Node-7276" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">R</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-579">\mathbb R</script> par <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-580-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;msqrt&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msqrt&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7277" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7278" class="mjx-mrow"><span id="MJXc-Node-7279" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7280" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-7281" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-7282" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-7283" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-7284" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.539em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.177em; top: -1.126em;"><span id="MJXc-Node-7285" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.177em; bottom: -1.028em;"><span id="MJXc-Node-7286" class="mjx-mrow" style=""><span id="MJXc-Node-7287" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-7288" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.522em; padding-bottom: 0.522em;">√</span></span><span class="mjx-box" style="padding-top: 0.236em; border-top: 1px solid;"><span id="MJXc-Node-7289" class="mjx-mrow"><span id="MJXc-Node-7290" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.291em; width: 1.539em;"></span></span><span class="mjx-vsize" style="height: 1.524em; vertical-align: -0.727em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>a</mi><mrow><mi>x</mi><msqrt><mi>x</mi></msqrt></mrow></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-580">f(x)=\frac{a}{x\sqrt x}</script> si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-581-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7291" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7292" class="mjx-mrow"><span id="MJXc-Node-7293" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-7294" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.458em;">≥</span></span><span id="MJXc-Node-7295" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-581">x\geq 1</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-582-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7296" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7297" class="mjx-mrow"><span id="MJXc-Node-7298" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7299" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-7300" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-7301" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-7302" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-7303" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-582">f(x)=0</script> sinon.
<ol class="enumeratechiffre">
<li> Déterminer le réel <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-583-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7304" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7305" class="mjx-mrow"><span id="MJXc-Node-7306" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-583">a</script> pour que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-584-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7307" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7308" class="mjx-mrow"><span id="MJXc-Node-7309" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-584">f</script> soit une densité de probabilité d'une certaine variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-585-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7310" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7311" class="mjx-mrow"><span id="MJXc-Node-7312" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-585">X</script>.
</li><li> Déterminer la fonction de répartition associée à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-586-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7313" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7314" class="mjx-mrow"><span id="MJXc-Node-7315" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-586">X</script>.
</li><li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-587-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7316" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7317" class="mjx-mrow"><span id="MJXc-Node-7318" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-587">X</script> admet-elle une espérance? Si oui, la déterminer.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2347&#39;,&#39;imageindication2347&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2347"></a><div class="inner" id="innerindication2347" style="display:none;"><ol class="enumeratechiffre">
<li> L'intégrale doit être égale à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-588-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7319" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7320" class="mjx-mrow"><span id="MJXc-Node-7321" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-588">1</script>.
</li><li> Calculer l'intégrale. Attention à la position par rapport à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-589-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7322" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7323" class="mjx-mrow"><span id="MJXc-Node-7324" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-589">1</script>.
</li><li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-590-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7325" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7326" class="mjx-mrow"><span id="MJXc-Node-7327" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7328" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7329" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7330" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7331" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-590">xf(x)</script> est-elle intégrable?
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2347&#39;,&#39;imagecorrige2347&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2347"></a><div class="inner" id="innercorrige2347" style="display:none;"><ol class="enumeratechiffre">
<li> La fonction <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-591-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7332" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7333" class="mjx-mrow"><span id="MJXc-Node-7334" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-591">f</script> est continue par morceaux et positive. De plus, la fonction <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-592-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7335" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7336" class="mjx-mrow"><span id="MJXc-Node-7337" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-592">f</script> est intégrable. Le seul problème est en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-593-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7338" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7339" class="mjx-mrow"><span id="MJXc-Node-7340" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7341" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-593">+\infty</script> et on sait qu'on a une intégrale de Riemann convergente. La fonction <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-594-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7342" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7343" class="mjx-mrow"><span id="MJXc-Node-7344" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-594">f</script> est une densité de probabilité d'une variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-595-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7345" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7346" class="mjx-mrow"><span id="MJXc-Node-7347" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-595">X</script> si et seulement si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-596-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7348" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7349" class="mjx-mrow"><span id="MJXc-Node-7350" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-7351" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.508em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-7355" class="mjx-texatom" style=""><span id="MJXc-Node-7356" class="mjx-mrow"><span id="MJXc-Node-7357" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7358" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7352" class="mjx-texatom" style=""><span id="MJXc-Node-7353" class="mjx-mrow"><span id="MJXc-Node-7354" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span><span id="MJXc-Node-7359" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7360" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7361" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7362" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7363" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-7364" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7365" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7366" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-596">\int_{1}^{+\infty}f(x)dx=1</script>. Mais
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-597-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;msqrt&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msqrt&gt;&lt;/mfrac&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7367" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7368" class="mjx-mrow"><span id="MJXc-Node-7369" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-7370" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.079em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-7372" class="mjx-texatom" style=""><span id="MJXc-Node-7373" class="mjx-mrow"><span id="MJXc-Node-7374" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7375" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7371" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span><span id="MJXc-Node-7376" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7377" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7378" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7379" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7380" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-7381" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7382" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7383" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-7384" class="mjx-mrow"><span id="MJXc-Node-7385" class="mjx-mo"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.52em; padding-bottom: 1.52em;">[</span></span><span id="MJXc-Node-7386" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.007em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.007em; top: -1.368em;"><span id="MJXc-Node-7387" class="mjx-mrow"><span id="MJXc-Node-7388" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7389" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-7390" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="mjx-denominator" style="width: 2.007em; bottom: -1.102em;"><span id="MJXc-Node-7391" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.236em; border-top: 1.3px solid;"><span id="MJXc-Node-7392" class="mjx-mrow"><span id="MJXc-Node-7393" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.007em;"></span></span><span class="mjx-vsize" style="height: 2.47em; vertical-align: -1.102em;"></span></span><span id="MJXc-Node-7394" class="mjx-mo"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.52em; padding-bottom: 1.52em;">]</span></span></span></span><span class="mjx-stack" style="vertical-align: -1.309em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 3.178em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7396" class="mjx-texatom" style=""><span id="MJXc-Node-7397" class="mjx-mrow"><span id="MJXc-Node-7398" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7399" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7395" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span><span id="MJXc-Node-7400" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7401" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-7402" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-7403" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>∫</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mrow><mo>[</mo><mfrac><mrow><mo>−</mo><mn>2</mn><mi>a</mi></mrow><msqrt><mi>x</mi></msqrt></mfrac><mo>]</mo></mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mo>=</mo><mn>2</mn><mi>a</mi><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-597">\int_1^{+\infty}f(x)dx=\left[\frac{-2a}{\sqrt x}\right]_1^{+\infty}=2a.</script>
Donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-598-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7404" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7405" class="mjx-mrow"><span id="MJXc-Node-7406" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-598">f</script> est une densité de probabilité si et seulement si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-599-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7407" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7408" class="mjx-mrow"><span id="MJXc-Node-7409" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-7410" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7411" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7412" class="mjx-texatom"><span id="MJXc-Node-7413" class="mjx-mrow"><span id="MJXc-Node-7414" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-7415" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-599">a=1/2</script>.
</li><li> Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-600-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7416" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7417" class="mjx-mrow"><span id="MJXc-Node-7418" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-7419" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7420" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>X</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-600">F_X</script> la fonction de répartition. Puisque <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-601-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7421" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7422" class="mjx-mrow"><span id="MJXc-Node-7423" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-601">f</script> est nulle sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-602-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7424" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7425" class="mjx-mrow"><span id="MJXc-Node-7426" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span><span id="MJXc-Node-7427" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7428" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span><span id="MJXc-Node-7429" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-7430" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7431" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">]</mo><mo>−</mo><mi mathvariant="normal">∞</mi><mo>,</mo><mn>1</mn><mo stretchy="false">[</mo></math></span></span><script type="math/tex" id="MathJax-Element-602">]-\infty,1[</script>, on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-603-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7432" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7433" class="mjx-mrow"><span id="MJXc-Node-7434" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-7435" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7436" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span id="MJXc-Node-7437" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7438" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-7439" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7440" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7441" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-603">F_X(t)=0</script> si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-604-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7442" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7443" class="mjx-mrow"><span id="MJXc-Node-7444" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-7445" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-7446" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≤</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-604">t\leq 1</script>. Pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-605-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7447" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7448" class="mjx-mrow"><span id="MJXc-Node-7449" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-7450" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-7451" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>&gt;</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-605">t>1</script>, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-606-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msubsup&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;msqrt&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msqrt&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;msqrt&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msqrt&gt;&lt;/mfrac&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msubsup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msqrt&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msqrt&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7452" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7453" class="mjx-mrow"><span id="MJXc-Node-7454" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-7455" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7456" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span id="MJXc-Node-7457" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7458" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-7459" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7460" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7461" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-7462" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.15em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-7464" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7463" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span><span id="MJXc-Node-7465" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 2.677em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.677em; top: -1.368em;"><span id="MJXc-Node-7466" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 2.677em; bottom: -1.102em;"><span id="MJXc-Node-7467" class="mjx-mrow"><span id="MJXc-Node-7468" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-7469" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7470" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.236em; border-top: 1.3px solid;"><span id="MJXc-Node-7471" class="mjx-mrow"><span id="MJXc-Node-7472" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.677em;"></span></span><span class="mjx-vsize" style="height: 2.47em; vertical-align: -1.102em;"></span></span><span id="MJXc-Node-7473" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-7474" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7475" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7476" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-7477" class="mjx-mrow"><span id="MJXc-Node-7478" class="mjx-mo"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.52em; padding-bottom: 1.52em;">[</span></span><span id="MJXc-Node-7479" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.605em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.605em; top: -1.368em;"><span id="MJXc-Node-7480" class="mjx-mrow"><span id="MJXc-Node-7481" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7482" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="width: 1.605em; bottom: -1.102em;"><span id="MJXc-Node-7483" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.236em; border-top: 1.3px solid;"><span id="MJXc-Node-7484" class="mjx-mrow"><span id="MJXc-Node-7485" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.605em;"></span></span><span class="mjx-vsize" style="height: 2.47em; vertical-align: -1.102em;"></span></span><span id="MJXc-Node-7486" class="mjx-mo"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.52em; padding-bottom: 1.52em;">]</span></span></span></span><span class="mjx-stack" style="vertical-align: -1.309em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 3.249em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7488" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7487" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span><span id="MJXc-Node-7489" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7490" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7491" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7492" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.394em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.394em; top: -1.368em;"><span id="MJXc-Node-7493" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.394em; bottom: -1.102em;"><span id="MJXc-Node-7494" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.144em; border-top: 1.3px solid;"><span id="MJXc-Node-7495" class="mjx-mrow"><span id="MJXc-Node-7496" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.394em;"></span></span><span class="mjx-vsize" style="height: 2.47em; vertical-align: -1.102em;"></span></span><span id="MJXc-Node-7497" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mn>1</mn><mi>t</mi></msubsup><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi><msqrt><mi>x</mi></msqrt></mrow></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mrow><mo>[</mo><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac><mo>]</mo></mrow><mn>1</mn><mi>t</mi></msubsup><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><msqrt><mi>t</mi></msqrt></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-606">F_X(t)=\int_1^t\frac{1}{2x\sqrt x}dx=\left[\frac{-1}{\sqrt x}\right]_1^t=1-\frac1{\sqrt t}.</script>
</li><li> La variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-607-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7498" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7499" class="mjx-mrow"><span id="MJXc-Node-7500" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-607">X</script> admet une espérance si la fonction <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-608-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x21A6;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7501" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7502" class="mjx-mrow"><span id="MJXc-Node-7503" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7504" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">↦</span></span><span id="MJXc-Node-7505" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7506" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7507" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7508" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7509" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo stretchy="false">↦</mo><mi>x</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-608">x\mapsto xf(x)</script> est intégrable. Mais on a affaire à une intégrale de Riemann divergence en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-609-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7510" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7511" class="mjx-mrow"><span id="MJXc-Node-7512" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7513" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-609">+\infty</script>. Donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-610-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7514" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7515" class="mjx-mrow"><span id="MJXc-Node-7516" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-610">X</script> n'admet pas d'espérance.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>22</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - La station-service<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2086"> [Signaler une erreur]</a><a id="exo22" href="javascript:ajouteexo(2086,22,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2086&#39;,&#39;imageenonce2086&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2086"></a><div class="inner" id="innerenonce2086">Dans une station-service, la demande hebdomadaire en essence, en milliers de litres,
est une variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-611-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7517" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7518" class="mjx-mrow"><span id="MJXc-Node-7519" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-611">X</script> de densité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-612-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn mathvariant=&quot;bold&quot;&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7520" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7521" class="mjx-mrow"><span id="MJXc-Node-7522" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7523" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-7524" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-7525" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-7526" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-7527" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">c</span></span><span id="MJXc-Node-7528" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-7529" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-7530" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-7531" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-7532" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7533" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7534" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span></span><span id="MJXc-Node-7535" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7536" class="mjx-texatom"><span id="MJXc-Node-7537" class="mjx-mrow"><span id="MJXc-Node-7538" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.328em; padding-bottom: 0.393em;">1</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.275em; padding-right: 0.071em;"><span id="MJXc-Node-7539" class="mjx-texatom" style=""><span id="MJXc-Node-7540" class="mjx-mrow"><span id="MJXc-Node-7541" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-7542" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-7543" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-7544" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-7545" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>4</mn></msup><msub><mrow class="MJX-TeXAtom-ORD"><mn mathvariant="bold">1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-612">f(x)=c(1-x)^4\mathbf 1_{[0,1]}</script>.
<ol class="enumeratechiffre">
<li> Déterminer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-613-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7546" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7547" class="mjx-mrow"><span id="MJXc-Node-7548" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">c</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math></span></span><script type="math/tex" id="MathJax-Element-613">c</script>.
</li><li> La station est réapprovisionnée chaque lundi à 20h. Quelle doit être la capacité du réservoir d'essence
pour que la probabilité d'épuiser ce réservoir soit inférieure à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-614-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7549" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7550" class="mjx-mrow"><span id="MJXc-Node-7551" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7552" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">10</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7553" class="mjx-texatom" style=""><span id="MJXc-Node-7554" class="mjx-mrow"><span id="MJXc-Node-7555" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-7556" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">5</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>5</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-614">10^{-5}</script>?
</li></ol>
</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2086&#39;,&#39;imageindication2086&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2086"></a><div class="inner" id="innerindication2086" style="display:none;"><ol class="enumeratechiffre">
<li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-615-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7557" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7558" class="mjx-mrow"><span id="MJXc-Node-7559" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-615">f</script> doit être une densité.
</li><li> Calculer la fonction de répartition de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-616-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7560" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7561" class="mjx-mrow"><span id="MJXc-Node-7562" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-616">X</script>.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2086&#39;,&#39;imagecorrige2086&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2086"></a><div class="inner" id="innercorrige2086" style="display:none;"><ol class="enumeratechiffre">
<li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-617-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7563" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7564" class="mjx-mrow"><span id="MJXc-Node-7565" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-617">f</script> doit être une densité de probabilité, et donc on doit avoir
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-618-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msubsup&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7566" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7567" class="mjx-mrow"><span id="MJXc-Node-7568" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-7569" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.161em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-7571" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7570" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-7572" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7573" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7574" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7575" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7576" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-7577" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7578" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7579" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mn>1.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-618">\int_0^1 f(x)dx=1.</script>
Or, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-619-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msubsup&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msubsup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7580" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7581" class="mjx-mrow"><span id="MJXc-Node-7582" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-7583" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.161em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-7585" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7584" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-7586" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7587" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7588" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7589" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7590" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-7591" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7592" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7593" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span><span id="MJXc-Node-7594" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7595" class="mjx-mrow"><span id="MJXc-Node-7596" class="mjx-mo"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.52em; padding-bottom: 1.52em;">[</span></span><span id="MJXc-Node-7597" class="mjx-mrow"><span id="MJXc-Node-7598" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7599" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.661em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.661em; top: -1.619em;"><span id="MJXc-Node-7600" class="mjx-mrow"><span id="MJXc-Node-7601" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7602" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7603" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7604" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7605" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7606" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7607" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span></span></span><span class="mjx-denominator" style="width: 3.661em; bottom: -0.733em;"><span id="MJXc-Node-7608" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.661em;"></span></span><span class="mjx-vsize" style="height: 2.352em; vertical-align: -0.733em;"></span></span></span><span id="MJXc-Node-7609" class="mjx-mo"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.52em; padding-bottom: 1.52em;">]</span></span></span></span><span class="mjx-stack" style="vertical-align: -1.309em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 3.26em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7611" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7610" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-7612" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7613" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.144em;"><span id="MJXc-Node-7614" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-7615" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 1.876em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-7616" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mi>c</mi><msubsup><mrow><mo>[</mo><mrow><mo>−</mo><mfrac><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>5</mn></msup></mrow><mn>5</mn></mfrac></mrow><mo>]</mo></mrow><mn>0</mn><mn>1</mn></msubsup><mo>=</mo><mfrac><mi>c</mi><mn>5</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-619">\int_0^1 f(x)dx=c\left[-\frac{(1-x)^5}{5}\right]_0^1=\frac c5.</script>
On a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-620-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7617" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7618" class="mjx-mrow"><span id="MJXc-Node-7619" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span><span id="MJXc-Node-7620" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7621" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>5</mn></math></span></span><script type="math/tex" id="MathJax-Element-620">c=5</script>.
</li><li> On va commencer par chercher la fonction de répartition <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-621-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7622" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7623" class="mjx-mrow"><span id="MJXc-Node-7624" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-7625" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7626" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span id="MJXc-Node-7627" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7628" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7629" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-621">F_X(x)</script> de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-622-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7630" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7631" class="mjx-mrow"><span id="MJXc-Node-7632" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-622">X</script>.
Elle est nulle à gauche de 0, égale à 1 à droite de 1, et si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-623-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7633" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7634" class="mjx-mrow"><span id="MJXc-Node-7635" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7636" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">∈</span></span><span id="MJXc-Node-7637" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-7638" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-7639" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-7640" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7641" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-623">x\in[0,1]</script>, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-624-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msubsup&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7642" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7643" class="mjx-mrow"><span id="MJXc-Node-7644" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-7645" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7646" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span id="MJXc-Node-7647" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7648" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7649" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7650" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7651" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-7652" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.15em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-7654" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7653" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-7655" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7656" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7657" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-7658" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7659" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-7660" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-7661" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7662" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7663" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7664" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7665" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7666" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7667" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7668" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7669" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7670" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span id="MJXc-Node-7671" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>x</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi><mo>=</mo><mn>1</mn><mo>−</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>5</mn></msup><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-624">F_X(x)=\int_0^xf(t)dt=1-(1-x)^5.</script>
On cherche alors <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-625-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7672" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7673" class="mjx-mrow"><span id="MJXc-Node-7674" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-625">x</script> tel que la probabilité de consommer plus de x milliers de litre dans la semaine
soit inférieure à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-626-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7675" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7676" class="mjx-mrow"><span id="MJXc-Node-7677" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7678" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7679" class="mjx-texatom" style=""><span id="MJXc-Node-7680" class="mjx-mrow"><span id="MJXc-Node-7681" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7682" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>5</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-626">10^{-5}</script>. Autrement dit, on cherche <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-627-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7683" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7684" class="mjx-mrow"><span id="MJXc-Node-7685" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-627">x</script> le plus petit possible tel que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-628-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7686" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7687" class="mjx-mrow"><span id="MJXc-Node-7688" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-7689" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7690" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span id="MJXc-Node-7691" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7692" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7693" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7694" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-7695" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7696" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7697" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-7698" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7699" class="mjx-texatom" style=""><span id="MJXc-Node-7700" class="mjx-mrow"><span id="MJXc-Node-7701" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7702" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>1</mn><mo>−</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>5</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-628">F_X(x)>1-10^{-5}</script>. Ceci est équivalent à
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-629-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27FA;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27FA;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;9.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7703" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7704" class="mjx-mrow"><span id="MJXc-Node-7705" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7706" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7707" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7708" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7709" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7710" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7711" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span id="MJXc-Node-7712" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-7713" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-7714" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7715" class="mjx-texatom" style=""><span id="MJXc-Node-7716" class="mjx-mrow"><span id="MJXc-Node-7717" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7718" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span></span></span><span id="MJXc-Node-7719" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-7720" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟺</span></span><span id="MJXc-Node-7721" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-7722" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7723" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7724" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7725" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-7726" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-7727" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7728" class="mjx-texatom" style=""><span id="MJXc-Node-7729" class="mjx-mrow"><span id="MJXc-Node-7730" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7731" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-7732" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-7733" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟺</span></span><span id="MJXc-Node-7734" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-7735" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7736" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-7737" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-7738" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-7739" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>5</mn></msup><mo>≤</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>5</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mn>1</mn><mo>−</mo><mi>x</mi><mo>≤</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mi>x</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mn>9.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-629">(1-x)^5\leq 10^{-5}\iff 1-x\leq 10^{-1}\iff x\geq 0,9.</script>
Le réservoir doit contenir au moins 900 litres.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>23</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Loi de Laplace<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2079"> [Signaler une erreur]</a><a id="exo23" href="javascript:ajouteexo(2079,23,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2079&#39;,&#39;imageenonce2079&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2079"></a><div class="inner" id="innerenonce2079">On considère une variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-630-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7740" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7741" class="mjx-mrow"><span id="MJXc-Node-7742" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-630">X</script> dont la densité est donnée par
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-631-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7743" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7744" class="mjx-mrow"><span id="MJXc-Node-7745" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7746" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-7747" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-7748" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-7749" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-7750" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">c</span></span><span id="MJXc-Node-7751" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7752" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7753" class="mjx-texatom" style=""><span id="MJXc-Node-7754" class="mjx-mrow"><span id="MJXc-Node-7755" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-7756" class="mjx-texatom"><span id="MJXc-Node-7757" class="mjx-mrow"><span id="MJXc-Node-7758" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">|</span></span></span></span><span id="MJXc-Node-7759" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-7760" class="mjx-texatom"><span id="MJXc-Node-7761" class="mjx-mrow"><span id="MJXc-Node-7762" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">|</span></span></span></span></span></span></span></span><span id="MJXc-Node-7763" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.328em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></msup><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-631">f(x)=ce^{-|x|}.</script>
<ol class="enumeratechiffre">
<li> Calculer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-632-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7764" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7765" class="mjx-mrow"><span id="MJXc-Node-7766" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">c</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math></span></span><script type="math/tex" id="MathJax-Element-632">c</script>.
</li><li> Démontrer que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-633-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7767" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7768" class="mjx-mrow"><span id="MJXc-Node-7769" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-633">X</script> admet des moments de tout ordre. Les calculer.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2079&#39;,&#39;imageindication2079&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2079"></a><div class="inner" id="innerindication2079" style="display:none;"><ol class="enumeratechiffre">
<li> Il faut que l'intégrale fasse 1.
</li><li> Moments impairs sont nuls. Moments pair à déterminer par récurrence.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2079&#39;,&#39;imagecorrige2079&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2079"></a><div class="inner" id="innercorrige2079" style="display:none;"><ol class="enumeratechiffre">
<li> Puisque <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-634-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7770" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7771" class="mjx-mrow"><span id="MJXc-Node-7772" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-634">f</script> est continue et positive, pour que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-635-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7773" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7774" class="mjx-mrow"><span id="MJXc-Node-7775" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-635">f</script> soit une densité, il faut et il suffit que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-636-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7776" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7777" class="mjx-mrow"><span id="MJXc-Node-7778" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-7779" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.517em; padding-right: 0.071em;"><span id="MJXc-Node-7780" class="mjx-texatom" style=""><span id="MJXc-Node-7781" class="mjx-mrow"><span id="MJXc-Node-7782" class="mjx-texatom"><span id="MJXc-Node-7783" class="mjx-mrow"><span id="MJXc-Node-7784" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.47em; padding-bottom: 0.295em;">R</span></span></span></span></span></span></span></span><span id="MJXc-Node-7785" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-7786" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7787" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7788" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7789" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-7790" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7791" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7792" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-636">\int_{\mathbb R}f(x)dx=1</script>. On calcule l'intégrale en séparant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-637-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7793" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7794" class="mjx-mrow"><span id="MJXc-Node-7795" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7796" class="mjx-texatom"><span id="MJXc-Node-7797" class="mjx-mrow"><span id="MJXc-Node-7798" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.47em; padding-bottom: 0.295em;">R</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7799" class="mjx-mo" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow><mo>+</mo></msub></math></span></span><script type="math/tex" id="MathJax-Element-637">\mathbb R_+</script> et
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-638-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7800" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7801" class="mjx-mrow"><span id="MJXc-Node-7802" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7803" class="mjx-texatom"><span id="MJXc-Node-7804" class="mjx-mrow"><span id="MJXc-Node-7805" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.47em; padding-bottom: 0.295em;">R</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7806" class="mjx-mo" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow><mo>−</mo></msub></math></span></span><script type="math/tex" id="MathJax-Element-638">\mathbb R_-</script> et on trouve <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-639-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7807" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7808" class="mjx-mrow"><span id="MJXc-Node-7809" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span><span id="MJXc-Node-7810" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7811" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7812" class="mjx-texatom"><span id="MJXc-Node-7813" class="mjx-mrow"><span id="MJXc-Node-7814" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-7815" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-639">c=1/2</script>.
</li><li> On a, pour tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-640-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7816" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7817" class="mjx-mrow"><span id="MJXc-Node-7818" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-7819" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-7820" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-640">n\geq 1</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-641-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7821" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7822" class="mjx-mrow"><span id="MJXc-Node-7823" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7824" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7825" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-7826" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7827" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7828" class="mjx-texatom" style=""><span id="MJXc-Node-7829" class="mjx-mrow"><span id="MJXc-Node-7830" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7831" class="mjx-texatom"><span id="MJXc-Node-7832" class="mjx-mrow"><span id="MJXc-Node-7833" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7834" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7835" class="mjx-texatom"><span id="MJXc-Node-7836" class="mjx-mrow"><span id="MJXc-Node-7837" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span></span></span></span></span><span id="MJXc-Node-7838" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7839" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">o</span></span><span id="MJXc-Node-7840" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7841" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7842" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7843" class="mjx-texatom" style=""><span id="MJXc-Node-7844" class="mjx-mrow"><span id="MJXc-Node-7845" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7846" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span id="MJXc-Node-7847" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mi>n</mi></msup><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></msup><mo>=</mo><mi>o</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-641">x^ne^{-|x|}=o(x^{-2})</script> en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-642-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;&amp;#x00B1;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7848" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7849" class="mjx-mrow"><span id="MJXc-Node-7850" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">±</span></span><span id="MJXc-Node-7851" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-642">\pm \infty</script>, ce qui prouve la convergence de l'intégrale.
Par imparité de la fonction <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-643-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x21A6;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7852" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7853" class="mjx-mrow"><span id="MJXc-Node-7854" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7855" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">↦</span></span><span id="MJXc-Node-7856" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-7857" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7858" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-7859" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7860" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7861" class="mjx-texatom" style=""><span id="MJXc-Node-7862" class="mjx-mrow"><span id="MJXc-Node-7863" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7864" class="mjx-texatom"><span id="MJXc-Node-7865" class="mjx-mrow"><span id="MJXc-Node-7866" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7867" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7868" class="mjx-texatom"><span id="MJXc-Node-7869" class="mjx-mrow"><span id="MJXc-Node-7870" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo stretchy="false">↦</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-643">x\mapsto x^n e^{-|x|}</script> si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-644-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7871" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7872" class="mjx-mrow"><span id="MJXc-Node-7873" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-644">n</script> est impair, les moments d'ordre impair sont nuls. Les moments
d'ordre pair sont calculés par récurrence. En effet, pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-645-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7874" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7875" class="mjx-mrow"><span id="MJXc-Node-7876" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-7877" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7878" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-7879" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-645">n=2p</script>, posons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-646-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7880" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7881" class="mjx-mrow"><span id="MJXc-Node-7882" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-7883" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7884" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span><span id="MJXc-Node-7885" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7886" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-7887" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.508em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-7889" class="mjx-texatom" style=""><span id="MJXc-Node-7890" class="mjx-mrow"><span id="MJXc-Node-7891" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7892" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7888" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-7893" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-7894" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7895" class="mjx-texatom" style=""><span id="MJXc-Node-7896" class="mjx-mrow"><span id="MJXc-Node-7897" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-7898" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span></span></span><span id="MJXc-Node-7899" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7900" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7901" class="mjx-texatom" style=""><span id="MJXc-Node-7902" class="mjx-mrow"><span id="MJXc-Node-7903" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7904" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span><span id="MJXc-Node-7905" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-7906" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>p</mi></msub><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msup><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn><mi>p</mi></mrow></msup><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>x</mi></mrow></msup><mi>d</mi><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-646">I_p=\int_0^{+\infty}x^{2p}e^{-x}dx</script>, de sorte
que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-647-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-7907" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7908" class="mjx-mrow"><span id="MJXc-Node-7909" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-7910" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7911" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-7912" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.115em; padding-right: 0.071em;"><span id="MJXc-Node-7913" class="mjx-texatom" style=""><span id="MJXc-Node-7914" class="mjx-mrow"><span id="MJXc-Node-7915" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-7916" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span></span></span><span id="MJXc-Node-7917" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7918" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7919" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-7920" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7921" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><msup><mi>X</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn><mi>p</mi></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><msub><mi>I</mi><mi>p</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-647">E(X^{2p})=I_p</script>. Alors, en intégrant par parties (deux fois), on trouve
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-648-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0 thickmathspace&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7922" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7923" class="mjx-mrow"><span id="MJXc-Node-7924" class="mjx-mtable" style="vertical-align: -0.999em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-7925" class="mjx-mtr" style="height: 2.497em;"><span id="MJXc-Node-7926" class="mjx-mtd" style="padding: 0px; text-align: right; width: 0.831em;"><span id="MJXc-Node-7927" class="mjx-mrow" style="margin-top: 0.543em;"><span id="MJXc-Node-7928" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-7929" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7930" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-7931" class="mjx-mtd" style="padding: 0px 0.139em 0px 0px; width: 1.056em;"><span id="MJXc-Node-7932" class="mjx-mrow" style="margin-top: 0.543em;"><span id="MJXc-Node-7933" class="mjx-mi"></span><span id="MJXc-Node-7934" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-7935" class="mjx-mtd" style="padding: 0px 0px 0px 0.139em; text-align: left; width: 29.544em;"><span id="MJXc-Node-7936" class="mjx-mrow"><span id="MJXc-Node-7937" class="mjx-mi"></span><span id="MJXc-Node-7938" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-7939" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.079em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-7941" class="mjx-texatom" style=""><span id="MJXc-Node-7942" class="mjx-mrow"><span id="MJXc-Node-7943" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7944" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7940" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-7945" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-7946" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-7947" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7948" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7949" class="mjx-texatom" style=""><span id="MJXc-Node-7950" class="mjx-mrow"><span id="MJXc-Node-7951" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-7952" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-7953" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7954" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-7955" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7956" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7957" class="mjx-texatom" style=""><span id="MJXc-Node-7958" class="mjx-mrow"><span id="MJXc-Node-7959" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7960" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span><span id="MJXc-Node-7961" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-7962" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7963" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7964" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-7965" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.079em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-7967" class="mjx-texatom" style=""><span id="MJXc-Node-7968" class="mjx-mrow"><span id="MJXc-Node-7969" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7970" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7966" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-7971" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-7972" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-7973" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7974" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-7975" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-7976" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7977" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7978" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7979" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7980" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7981" class="mjx-texatom" style=""><span id="MJXc-Node-7982" class="mjx-mrow"><span id="MJXc-Node-7983" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-7984" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-7985" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7986" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span id="MJXc-Node-7987" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7988" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7989" class="mjx-texatom" style=""><span id="MJXc-Node-7990" class="mjx-mrow"><span id="MJXc-Node-7991" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7992" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span><span id="MJXc-Node-7993" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-7994" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-7995" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7996" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-7997" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-7998" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7999" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-8000" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-8001" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8002" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-8003" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8004" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-8005" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8006" class="mjx-texatom" style=""><span id="MJXc-Node-8007" class="mjx-mrow"><span id="MJXc-Node-8008" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-8009" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8010" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-8011" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mtr><mtd><msub><mi>I</mi><mi>p</mi></msub></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi></mi><msubsup><mo>∫</mo><mn>0</mn><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mn>2</mn><mi>p</mi><msup><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>x</mi></mrow></msup><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mn>2</mn><mi>p</mi><mo stretchy="false">(</mo><mn>2</mn><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>x</mi></mrow></msup><mi>d</mi><mi>x</mi><mo>=</mo><mn>2</mn><mi>p</mi><mo stretchy="false">(</mo><mn>2</mn><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>.</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-648">\begin{eqnarray*}
I_p&=&\int_0^{+\infty}2px^{2p-1}e^{-x}dx=\int_0^{+\infty}2p(2p-1)x^{2p-2}e^{-x}dx=2p(2p-1)I_{p-1}.
\end{eqnarray*}</script>
On en déduit immédiatement que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-649-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8012" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8013" class="mjx-mrow"><span id="MJXc-Node-8014" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-8015" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8016" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span><span id="MJXc-Node-8017" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8018" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8019" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-8020" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-8021" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8022" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-8023" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-8024" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8025" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>p</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mi>p</mi><mo stretchy="false">)</mo><mo>!</mo><msub><mi>I</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-649">I_p=(2p)!I_0</script>, et il est aisé de voir que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-650-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8026" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8027" class="mjx-mrow"><span id="MJXc-Node-8028" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-8029" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8030" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-8031" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8032" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-650">I_0=1</script>. En conclusion, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-651-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8033" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8034" class="mjx-mrow"><span id="MJXc-Node-8035" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-8036" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8037" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-8038" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.115em; padding-right: 0.071em;"><span id="MJXc-Node-8039" class="mjx-texatom" style=""><span id="MJXc-Node-8040" class="mjx-mrow"><span id="MJXc-Node-8041" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-8042" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span></span></span><span id="MJXc-Node-8043" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8044" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8045" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-8046" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8047" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span><span id="MJXc-Node-8048" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8049" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8050" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-8051" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-8052" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8053" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><msup><mi>X</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn><mi>p</mi></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><msub><mi>I</mi><mi>p</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mi>p</mi><mo stretchy="false">)</mo><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-651">E(X^{2p})=I_p=(2p)!</script>.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>24</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Exponentiel des deux côtés!<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2078"> [Signaler une erreur]</a><a id="exo24" href="javascript:ajouteexo(2078,24,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2078&#39;,&#39;imageenonce2078&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2078"></a><div class="inner" id="innerenonce2078">Soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-652-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8054" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8055" class="mjx-mrow"><span id="MJXc-Node-8056" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-652">f</script> la fonction définie sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-653-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8057" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8058" class="mjx-mrow"><span id="MJXc-Node-8059" class="mjx-texatom"><span id="MJXc-Node-8060" class="mjx-mrow"><span id="MJXc-Node-8061" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">R</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-653">\mtr</script> par :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-654-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable columnalign=&quot;left left&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;msup&gt;&lt;mn&gt;.3&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;si&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;msup&gt;&lt;mn&gt;.3&lt;/mn&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;si&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mn&gt;0.&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; symmetric=&quot;true&quot;&gt;&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8062" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8063" class="mjx-mrow"><span id="MJXc-Node-8064" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-8065" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-8066" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-8067" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-8068" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-8069" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-8070" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.236em; padding-bottom: 1.236em;">{</span></span><span id="MJXc-Node-8071" class="mjx-mtable" style="vertical-align: -0.973em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-8072" class="mjx-mtr" style="height: 1.247em;"><span id="MJXc-Node-8073" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: left; width: 2.297em;"><span id="MJXc-Node-8074" class="mjx-mrow" style="margin-top: -0.153em;"><span id="MJXc-Node-8075" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span><span id="MJXc-Node-8076" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8077" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">.3</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.59em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8078" class="mjx-texatom" style=""><span id="MJXc-Node-8079" class="mjx-mrow"><span id="MJXc-Node-8080" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-8081" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8082" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 3.606em;"><span id="MJXc-Node-8083" class="mjx-mrow" style="margin-top: -0.153em;"><span id="MJXc-Node-8084" class="mjx-texatom"><span id="MJXc-Node-8085" class="mjx-mrow"><span id="MJXc-Node-8086" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">si&nbsp;</span></span></span></span><span id="MJXc-Node-8087" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-8088" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">&gt;</span></span><span id="MJXc-Node-8089" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-8090" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-8091" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: left;"><span id="MJXc-Node-8092" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-8093" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span><span id="MJXc-Node-8094" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8095" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">.3</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.59em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8096" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8097" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-8098" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-8099" class="mjx-texatom"><span id="MJXc-Node-8100" class="mjx-mrow"><span id="MJXc-Node-8101" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">si&nbsp;</span></span></span></span><span id="MJXc-Node-8102" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span><span id="MJXc-Node-8103" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">&lt;</span></span><span id="MJXc-Node-8104" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0.</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-8105" class="mjx-mo" style="width: 0.12em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi>a</mi><msup><mn>.3</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>x</mi></mrow></msup></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>si&nbsp;</mtext></mrow><mi>x</mi><mo>&gt;</mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>a</mi><msup><mn>.3</mn><mi>x</mi></msup></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>si&nbsp;</mtext></mrow><mi>x</mi><mo>&lt;</mo><mn>0.</mn></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-654">f(x)=\left\{
\begin{array}{ll}
a.3^{-x}&\textrm{si }x>0\\
a.3^x&\textrm{si }x<0.
\end{array}\right.</script>
<ol class="enumeratechiffre">
<li> Déterminer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-655-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8106" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8107" class="mjx-mrow"><span id="MJXc-Node-8108" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-655">a</script> pour que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-656-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8109" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8110" class="mjx-mrow"><span id="MJXc-Node-8111" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-656">f</script> soit une densité de probabilité.
</li><li> Soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-657-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8112" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8113" class="mjx-mrow"><span id="MJXc-Node-8114" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-657">X</script> une variable aléatoire admettant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-658-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8115" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8116" class="mjx-mrow"><span id="MJXc-Node-8117" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-658">f</script> pour densité. Déterminer la fonction de répartition de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-659-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8118" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8119" class="mjx-mrow"><span id="MJXc-Node-8120" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-659">X</script>. Montrer que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-660-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8121" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8122" class="mjx-mrow"><span id="MJXc-Node-8123" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-660">X</script> admet une espérance <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-661-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8124" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8125" class="mjx-mrow"><span id="MJXc-Node-8126" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-8127" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-8128" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-8129" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-661">E(X)</script> et la calculer.
</li><li> On pose <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-662-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8130" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8131" class="mjx-mrow"><span id="MJXc-Node-8132" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-8133" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-8134" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-8135" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.59em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8136" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><msup><mn>3</mn><mi>X</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-662">Y=3^X</script>. Déterminer la fonction de répartition de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-663-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8137" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8138" class="mjx-mrow"><span id="MJXc-Node-8139" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-663">Y</script>. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-664-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8140" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8141" class="mjx-mrow"><span id="MJXc-Node-8142" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-664">Y</script> admet-elle une espérance?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2078&#39;,&#39;imageindication2078&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2078"></a><div class="inner" id="innerindication2078" style="display:none;"><ol class="enumeratechiffre">
<li> Il suffit que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-665-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8143" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8144" class="mjx-mrow"><span id="MJXc-Node-8145" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-8146" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.662em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-8151" class="mjx-texatom" style=""><span id="MJXc-Node-8152" class="mjx-mrow"><span id="MJXc-Node-8153" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-8147" class="mjx-texatom" style=""><span id="MJXc-Node-8148" class="mjx-mrow"><span id="MJXc-Node-8149" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8150" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span><span id="MJXc-Node-8154" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-8155" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8156" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8157" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8158" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-8159" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8160" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8161" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-665">\int_{-\infty}^{\infty}f(x)dx=1</script>.
</li><li> C'est des petits calculs d'intégrale. Pour la fonction de répartition, séparer les cas <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-666-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8162" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8163" class="mjx-mrow"><span id="MJXc-Node-8164" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8165" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&lt;</span></span><span id="MJXc-Node-8166" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&lt;</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-666">x<0</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-667-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8167" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8168" class="mjx-mrow"><span id="MJXc-Node-8169" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8170" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-8171" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-667">x\geq 0</script>. Pour l'espérance, on pourra étudier la parité.
</li><li> Ecrire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-668-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27FA;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8172" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8173" class="mjx-mrow"><span id="MJXc-Node-8174" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8175" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-8176" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-8177" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8178" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8179" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-8180" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟺</span></span><span id="MJXc-Node-8181" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-8182" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8183" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-8184" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-8185" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-8186" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mo>…</mo><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-668">(Y\leq x)\iff (X\leq \dots)</script>. Donner une expression de la densité pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-669-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8187" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8188" class="mjx-mrow"><span id="MJXc-Node-8189" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8190" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-8191" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-669">x>1</script>. Montrer que l'intégrale définissant l'espérance est divergente au voisinage de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-670-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8192" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8193" class="mjx-mrow"><span id="MJXc-Node-8194" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-8195" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-670">+\infty</script>.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2078&#39;,&#39;imagecorrige2078&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2078"></a><div class="inner" id="innercorrige2078" style="display:none;"><ol class="enumeratechiffre">
<li> La fonction <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-671-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8196" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8197" class="mjx-mrow"><span id="MJXc-Node-8198" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-671">f</script> est continue sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-672-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8199" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8200" class="mjx-mrow"><span id="MJXc-Node-8201" class="mjx-texatom"><span id="MJXc-Node-8202" class="mjx-mrow"><span id="MJXc-Node-8203" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.47em; padding-bottom: 0.295em;">R</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-672">\mtr</script>, positive si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-673-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8204" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8205" class="mjx-mrow"><span id="MJXc-Node-8206" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-8207" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-8208" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≥</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-673">a\geq 0</script>, et on a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-674-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;msup&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8209" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8210" class="mjx-mrow"><span id="MJXc-Node-8211" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-8212" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.079em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-8214" class="mjx-texatom" style=""><span id="MJXc-Node-8215" class="mjx-mrow"><span id="MJXc-Node-8216" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-8217" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-8213" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-8218" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-8219" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.59em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8220" class="mjx-texatom" style=""><span id="MJXc-Node-8221" class="mjx-mrow"><span id="MJXc-Node-8222" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8223" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span><span id="MJXc-Node-8224" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-8225" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8226" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8227" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-8228" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.15em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-8230" class="mjx-texatom" style=""><span id="MJXc-Node-8231" class="mjx-mrow"><span id="MJXc-Node-8232" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-8229" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-8233" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-8234" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8235" class="mjx-texatom" style=""><span id="MJXc-Node-8236" class="mjx-mrow"><span id="MJXc-Node-8237" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8238" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8239" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ln</span></span><span id="MJXc-Node-8240" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-8241" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span><span id="MJXc-Node-8242" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-8243" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8244" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8245" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.701em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.701em; top: -1.368em;"><span id="MJXc-Node-8246" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.701em; bottom: -0.758em;"><span id="MJXc-Node-8247" class="mjx-mrow"><span id="MJXc-Node-8248" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ln</span></span><span id="MJXc-Node-8249" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-8250" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.701em;"></span></span><span class="mjx-vsize" style="height: 2.126em; vertical-align: -0.758em;"></span></span><span id="MJXc-Node-8251" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>∫</mo><mn>0</mn><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>x</mi></mrow></msup><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>x</mi><mi>ln</mi><mo>⁡</mo><mn>3</mn></mrow></msup><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mi>ln</mi><mo>⁡</mo><mn>3</mn></mrow></mfrac><mo>,</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-674">\int_0^{+\infty}3^{-x}dx=\int_0^{\infty}e^{-x\ln 3}dx=\frac{1}{\ln 3},</script>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-675-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msubsup&gt;&lt;msup&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8252" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8253" class="mjx-mrow"><span id="MJXc-Node-8254" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-8255" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.222em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-8260" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-8256" class="mjx-texatom" style=""><span id="MJXc-Node-8257" class="mjx-mrow"><span id="MJXc-Node-8258" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8259" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span><span id="MJXc-Node-8261" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-8262" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.59em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8263" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span id="MJXc-Node-8264" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-8265" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8266" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8267" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.701em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.701em; top: -1.368em;"><span id="MJXc-Node-8268" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.701em; bottom: -0.758em;"><span id="MJXc-Node-8269" class="mjx-mrow"><span id="MJXc-Node-8270" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ln</span></span><span id="MJXc-Node-8271" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-8272" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.701em;"></span></span><span class="mjx-vsize" style="height: 2.126em; vertical-align: -0.758em;"></span></span><span id="MJXc-Node-8273" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mn>0</mn></msubsup><msup><mn>3</mn><mi>x</mi></msup><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mi>ln</mi><mo>⁡</mo><mn>3</mn></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-675">\int_{-\infty}^0 3^xdx=\frac{1}{\ln 3}.</script>
Ainsi, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-676-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8274" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8275" class="mjx-mrow"><span id="MJXc-Node-8276" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-676">f</script> est une densité de probabilité si et seulement si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-677-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8277" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8278" class="mjx-mrow"><span id="MJXc-Node-8279" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-8280" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8281" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.203em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.701em; top: -1.368em;"><span id="MJXc-Node-8282" class="mjx-mrow" style=""><span id="MJXc-Node-8283" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ln</span></span><span id="MJXc-Node-8284" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-8285" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.701em; bottom: -0.611em;"><span id="MJXc-Node-8286" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.203em;"></span></span><span class="mjx-vsize" style="height: 1.4em; vertical-align: -0.432em;"></span></span><span id="MJXc-Node-8287" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mo>⁡</mo><mn>3</mn></mrow><mn>2</mn></mfrac><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-677">a=\frac{\ln 3}{2}.</script>
</li><li> Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-678-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8288" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8289" class="mjx-mrow"><span id="MJXc-Node-8290" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8291" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-8292" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≤</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-678">x\leq 0</script>, on a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-679-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msubsup&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8293" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8294" class="mjx-mrow"><span id="MJXc-Node-8295" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span><span id="MJXc-Node-8296" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8297" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8298" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8299" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8300" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-8301" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-8302" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-8303" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ln</span></span><span id="MJXc-Node-8304" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-8305" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-8306" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-8307" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.233em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-8312" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-8308" class="mjx-texatom" style=""><span id="MJXc-Node-8309" class="mjx-mrow"><span id="MJXc-Node-8310" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8311" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span></span><span id="MJXc-Node-8313" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-8314" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8315" class="mjx-texatom" style=""><span id="MJXc-Node-8316" class="mjx-mrow"><span id="MJXc-Node-8317" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-8318" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ln</span></span><span id="MJXc-Node-8319" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-8320" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span><span id="MJXc-Node-8321" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-8322" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-8323" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8324" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.14em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.14em; top: -1.424em;"><span id="MJXc-Node-8325" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8326" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.59em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8327" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span><span class="mjx-denominator" style="width: 1.14em; bottom: -0.711em;"><span id="MJXc-Node-8328" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.14em;"></span></span><span class="mjx-vsize" style="height: 2.135em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-8329" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>ln</mi><mo>⁡</mo><mn>3</mn><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>x</mi></msubsup><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi><mi>ln</mi><mo>⁡</mo><mn>3</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>=</mo><mfrac><msup><mn>3</mn><mi>x</mi></msup><mn>2</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-679">F(x)=\frac{1}{2}\ln 3\int_{-\infty}^x e^{t\ln 3}dt=\frac{3^x}{2}.</script>
Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-680-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8330" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8331" class="mjx-mrow"><span id="MJXc-Node-8332" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8333" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-8334" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-680">x\geq 0</script>, on a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-681-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msubsup&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8335" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8336" class="mjx-mrow"><span id="MJXc-Node-8337" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span><span id="MJXc-Node-8338" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8339" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8340" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8341" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8342" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span><span id="MJXc-Node-8343" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8344" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8345" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8346" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-8347" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.701em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.701em; top: -1.396em;"><span id="MJXc-Node-8348" class="mjx-mrow"><span id="MJXc-Node-8349" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ln</span></span><span id="MJXc-Node-8350" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-8351" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-denominator" style="width: 1.701em; bottom: -0.711em;"><span id="MJXc-Node-8352" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.701em;"></span></span><span class="mjx-vsize" style="height: 2.106em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-8353" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-8354" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.15em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-8356" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-8355" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span><span id="MJXc-Node-8357" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-8358" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8359" class="mjx-texatom" style=""><span id="MJXc-Node-8360" class="mjx-mrow"><span id="MJXc-Node-8361" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8362" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-8363" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ln</span></span><span id="MJXc-Node-8364" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-8365" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span><span id="MJXc-Node-8366" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-8367" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-8368" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8369" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-8370" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-8371" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-8372" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8373" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-8374" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-8375" class="mjx-mrow"><span id="MJXc-Node-8376" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.69em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.69em; top: -1.523em;"><span id="MJXc-Node-8377" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8378" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.59em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8379" class="mjx-texatom" style=""><span id="MJXc-Node-8380" class="mjx-mrow"><span id="MJXc-Node-8381" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8382" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 1.69em; bottom: -0.711em;"><span id="MJXc-Node-8383" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.69em;"></span></span><span class="mjx-vsize" style="height: 2.234em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-8384" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8385" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-8386" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-8387" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span></span><span id="MJXc-Node-8388" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span><span id="MJXc-Node-8389" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8390" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-8391" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8392" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.69em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.69em; top: -1.523em;"><span id="MJXc-Node-8393" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8394" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.59em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8395" class="mjx-texatom" style=""><span id="MJXc-Node-8396" class="mjx-mrow"><span id="MJXc-Node-8397" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8398" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 1.69em; bottom: -0.711em;"><span id="MJXc-Node-8399" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.69em;"></span></span><span class="mjx-vsize" style="height: 2.234em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-8400" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mi>ln</mi><mo>⁡</mo><mn>3</mn></mrow><mn>2</mn></mfrac><msubsup><mo>∫</mo><mn>0</mn><mi>x</mi></msubsup><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>t</mi><mi>ln</mi><mo>⁡</mo><mn>3</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo><mrow><mo>(</mo><mrow><mfrac><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>x</mi></mrow></msup><mn>2</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>x</mi></mrow></msup><mn>2</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-681">F(x)=F(0)+\frac{\ln 3}2\int_0^xe^{-t\ln 3}dt=\frac 12-\left(\frac{3^{-x}}{2}-\frac 12\right)=1-\frac{3^{-x}}{2}.</script>
La fonction <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-682-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8401" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8402" class="mjx-mrow"><span id="MJXc-Node-8403" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8404" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-8405" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8406" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8407" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-682">xf(x)</script> est négligeable au voisinage de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-683-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8408" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8409" class="mjx-mrow"><span id="MJXc-Node-8410" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-8411" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-683">+\infty</script> devant la fonction <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-684-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8412" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8413" class="mjx-mrow"><span id="MJXc-Node-8414" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-8415" class="mjx-texatom"><span id="MJXc-Node-8416" class="mjx-mrow"><span id="MJXc-Node-8417" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-8418" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8419" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8420" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msup><mi>x</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-684">1/x^2</script>, et il en est de même au voisinage de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-685-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8421" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8422" class="mjx-mrow"><span id="MJXc-Node-8423" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8424" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-685">-\infty</script> car cette fonction est impaire.
Elle est donc intégrable, et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-686-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8425" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8426" class="mjx-mrow"><span id="MJXc-Node-8427" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-686">X</script> admet bien une espérance. En outre, toujours par imparité de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-687-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x21A6;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8428" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8429" class="mjx-mrow"><span id="MJXc-Node-8430" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8431" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">↦</span></span><span id="MJXc-Node-8432" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8433" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-8434" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8435" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8436" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo stretchy="false">↦</mo><mi>x</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-687">x\mapsto xf(x)</script>, l'espérance est nulle!
</li><li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-688-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8437" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8438" class="mjx-mrow"><span id="MJXc-Node-8439" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-688">Y</script> prend ses valeurs dans <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-689-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8440" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8441" class="mjx-mrow"><span id="MJXc-Node-8442" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8443" class="mjx-texatom"><span id="MJXc-Node-8444" class="mjx-mrow"><span id="MJXc-Node-8445" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.47em; padding-bottom: 0.295em;">R</span></span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.615em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8446" class="mjx-mo" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow><mo>+</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-689">\mathbb R^+</script>, et on a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-690-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8447" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8448" class="mjx-mrow"><span id="MJXc-Node-8449" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8450" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8451" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-8452" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-8453" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8454" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8455" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8456" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8457" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8458" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8459" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.59em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8460" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span id="MJXc-Node-8461" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-8462" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8463" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8464" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8465" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8466" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-8467" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-8468" class="mjx-mrow"><span id="MJXc-Node-8469" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-8470" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-8471" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.773em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.773em; top: -1.396em;"><span id="MJXc-Node-8472" class="mjx-mrow"><span id="MJXc-Node-8473" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ln</span></span><span id="MJXc-Node-8474" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-8475" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="mjx-denominator" style="width: 1.773em; bottom: -0.758em;"><span id="MJXc-Node-8476" class="mjx-mrow"><span id="MJXc-Node-8477" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ln</span></span><span id="MJXc-Node-8478" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-8479" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.773em;"></span></span><span class="mjx-vsize" style="height: 2.154em; vertical-align: -0.758em;"></span></span></span><span id="MJXc-Node-8480" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span><span id="MJXc-Node-8481" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msup><mn>3</mn><mi>X</mi></msup><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mrow><mo>(</mo><mrow><mi>X</mi><mo>≤</mo><mfrac><mrow><mi>ln</mi><mo>⁡</mo><mi>x</mi></mrow><mrow><mi>ln</mi><mo>⁡</mo><mn>3</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-690">P(Y\leq x)=P(3^X\leq x)=P\left(X\leq \frac{\ln x}{\ln 3}\right).</script>
On en déduit : si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-691-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8482" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8483" class="mjx-mrow"><span id="MJXc-Node-8484" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8485" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-8486" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8487" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-8488" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-691">0\leq x\leq 1</script>,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-692-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8489" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8490" class="mjx-mrow"><span id="MJXc-Node-8491" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-8492" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8493" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span id="MJXc-Node-8494" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8495" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8496" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8497" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8498" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-8499" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-8500" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-8501" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8502" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.642em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8503" class="mjx-texatom" style=""><span id="MJXc-Node-8504" class="mjx-mrow"><span id="MJXc-Node-8505" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.253em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.773em; top: -1.383em;"><span id="MJXc-Node-8506" class="mjx-mrow" style=""><span id="MJXc-Node-8507" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ln</span></span><span id="MJXc-Node-8508" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-8509" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.773em; bottom: -0.687em;"><span id="MJXc-Node-8510" class="mjx-mrow" style=""><span id="MJXc-Node-8511" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ln</span></span><span id="MJXc-Node-8512" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-8513" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-line" style="border-bottom: 1px solid; top: -0.292em; width: 1.253em;"></span></span><span class="mjx-vsize" style="height: 1.464em; vertical-align: -0.486em;"></span></span></span></span></span></span><span id="MJXc-Node-8514" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8515" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.772em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.772em; top: -1.144em;"><span id="MJXc-Node-8516" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-denominator" style="width: 0.772em; bottom: -0.711em;"><span id="MJXc-Node-8517" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.772em;"></span></span><span class="mjx-vsize" style="height: 1.854em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-8518" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mi>ln</mi><mo>⁡</mo><mi>x</mi></mrow><mrow><mi>ln</mi><mo>⁡</mo><mn>3</mn></mrow></mfrac></mrow></msup><mo>=</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-692">F_Y(x)=\frac{1}{2}3^{\frac{\ln x}{\ln 3}}=\frac{x}{2}.</script>
Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-693-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8519" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8520" class="mjx-mrow"><span id="MJXc-Node-8521" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8522" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-8523" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-693">x>1</script>, on a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-694-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8524" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8525" class="mjx-mrow"><span id="MJXc-Node-8526" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-8527" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8528" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span id="MJXc-Node-8529" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8530" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8531" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8532" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8533" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-8534" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8535" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-8536" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-8537" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-8538" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8539" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.642em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8540" class="mjx-texatom" style=""><span id="MJXc-Node-8541" class="mjx-mrow"><span id="MJXc-Node-8542" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8543" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.253em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.773em; top: -1.383em;"><span id="MJXc-Node-8544" class="mjx-mrow" style=""><span id="MJXc-Node-8545" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ln</span></span><span id="MJXc-Node-8546" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-8547" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.773em; bottom: -0.687em;"><span id="MJXc-Node-8548" class="mjx-mrow" style=""><span id="MJXc-Node-8549" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ln</span></span><span id="MJXc-Node-8550" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-8551" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-line" style="border-bottom: 1px solid; top: -0.292em; width: 1.253em;"></span></span><span class="mjx-vsize" style="height: 1.464em; vertical-align: -0.486em;"></span></span></span></span></span></span><span id="MJXc-Node-8552" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8553" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-8554" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8555" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.272em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.272em; top: -1.368em;"><span id="MJXc-Node-8556" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.272em; bottom: -0.722em;"><span id="MJXc-Node-8557" class="mjx-mrow"><span id="MJXc-Node-8558" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-8559" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.272em;"></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-8560" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mfrac><mrow><mi>ln</mi><mo>⁡</mo><mi>x</mi></mrow><mrow><mi>ln</mi><mo>⁡</mo><mn>3</mn></mrow></mfrac></mrow></msup><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-694">F_Y(x)=1-\frac{1}{2}3^{-\frac{\ln x}{\ln 3}}=1-\frac{1}{2x}.</script>
En particulier, pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-695-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8561" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8562" class="mjx-mrow"><span id="MJXc-Node-8563" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8564" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-8565" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-695">x>1</script>, la densité de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-696-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8566" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8567" class="mjx-mrow"><span id="MJXc-Node-8568" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-696">Y</script> est :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-697-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;#x2032;&lt;/mo&gt;&lt;/msubsup&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8569" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8570" class="mjx-mrow"><span id="MJXc-Node-8571" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-8572" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8573" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span id="MJXc-Node-8574" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8575" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8576" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8577" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8578" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-8579" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-stack" style="vertical-align: -0.327em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.266em; padding-right: 0.071em;"><span id="MJXc-Node-8581" class="mjx-mo" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">′</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-8580" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span></span><span id="MJXc-Node-8582" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8583" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8584" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8585" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8586" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.661em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.661em; top: -1.368em;"><span id="MJXc-Node-8587" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.661em; bottom: -0.806em;"><span id="MJXc-Node-8588" class="mjx-mrow"><span id="MJXc-Node-8589" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-8590" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8591" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8592" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.661em;"></span></span><span class="mjx-vsize" style="height: 2.173em; vertical-align: -0.806em;"></span></span><span id="MJXc-Node-8593" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>f</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>F</mi><mi>Y</mi><mo>′</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-697">f_Y(x)=F_Y'(x)=\frac{1}{2x^2}.</script>
Au voisinage de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-698-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8594" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8595" class="mjx-mrow"><span id="MJXc-Node-8596" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-8597" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi mathvariant="normal">∞</mi></math></span></span><script type="math/tex" id="MathJax-Element-698">+\infty</script>, on a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-699-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x223C;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8598" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8599" class="mjx-mrow"><span id="MJXc-Node-8600" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8601" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-8602" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8603" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8604" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8605" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">∼</span></span><span id="MJXc-Node-8606" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.272em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.272em; top: -1.368em;"><span id="MJXc-Node-8607" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.272em; bottom: -0.722em;"><span id="MJXc-Node-8608" class="mjx-mrow"><span id="MJXc-Node-8609" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-8610" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.272em;"></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-8611" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>x</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∼</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac><mo>,</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-699">xf(x)\sim\frac{1}{2x},</script>
fonction qui n'est pas intégrable. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-700-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8612" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8613" class="mjx-mrow"><span id="MJXc-Node-8614" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-700">Y</script> n'admet pas d'espérance.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>25</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Carré de la loi uniforme<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2069"> [Signaler une erreur]</a><a id="exo25" href="javascript:ajouteexo(2069,25,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2069&#39;,&#39;imageenonce2069&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2069"></a><div class="inner" id="innerenonce2069">Soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-701-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8615" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8616" class="mjx-mrow"><span id="MJXc-Node-8617" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-701">X</script> une variable aléatoire suivant une loi uniforme sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-702-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8618" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8619" class="mjx-mrow"><span id="MJXc-Node-8620" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-8621" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span><span id="MJXc-Node-8622" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-8623" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">b</span></span><span id="MJXc-Node-8624" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-702">[a,b]</script>, avec <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-703-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8625" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8626" class="mjx-mrow"><span id="MJXc-Node-8627" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-8628" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">&lt;</span></span><span id="MJXc-Node-8629" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span><span id="MJXc-Node-8630" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">&lt;</span></span><span id="MJXc-Node-8631" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">b</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>a</mi><mo>&lt;</mo><mi>b</mi></math></span></span><script type="math/tex" id="MathJax-Element-703">0<a<b</script>.
Donner la fonction de répartition, la densité, l'espérance et la variance de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-704-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8632" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8633" class="mjx-mrow"><span id="MJXc-Node-8634" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-8635" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-8636" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-8637" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.115em; padding-right: 0.071em;"><span id="MJXc-Node-8638" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><msup><mi>X</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-704">Y=X^2</script>.</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2069&#39;,&#39;imageindication2069&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2069"></a><div class="inner" id="innerindication2069" style="display:none;">Suivre les définitions.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2069&#39;,&#39;imagecorrige2069&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2069"></a><div class="inner" id="innercorrige2069" style="display:none;">On calcule la fonction de répartion <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-705-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8639" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8640" class="mjx-mrow"><span id="MJXc-Node-8641" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-8642" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8643" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>Y</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-705">F_Y</script> de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-706-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8644" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8645" class="mjx-mrow"><span id="MJXc-Node-8646" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-706">Y</script>. Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-707-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8647" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8648" class="mjx-mrow"><span id="MJXc-Node-8649" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-8650" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&lt;</span></span><span id="MJXc-Node-8651" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>&lt;</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-707">y<0</script>, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-708-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8652" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8653" class="mjx-mrow"><span id="MJXc-Node-8654" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-8655" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8656" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span id="MJXc-Node-8657" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8658" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-8659" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8660" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8661" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8662" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8663" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-8664" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-8665" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-8666" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8667" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8668" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-708">F_Y(y)=P(Y\leq y)=0.</script>
Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-709-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8669" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8670" class="mjx-mrow"><span id="MJXc-Node-8671" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-8672" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-8673" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>≥</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-709">y\geq 0</script>, alors
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-710-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msqrt&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/msqrt&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;msqrt&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/msqrt&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;msqrt&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/msqrt&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msqrt&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/msqrt&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8674" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8675" class="mjx-mrow"><span id="MJXc-Node-8676" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-8677" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8678" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span id="MJXc-Node-8679" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8680" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-8681" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8682" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8683" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8684" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8685" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-8686" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-8687" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-8688" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8689" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8690" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8691" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8692" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8693" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.091em; border-top: 1.3px solid;"><span id="MJXc-Node-8694" class="mjx-mrow"><span id="MJXc-Node-8695" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span></span></span></span></span><span id="MJXc-Node-8696" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-8697" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-8698" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-8699" class="mjx-msqrt MJXc-space3"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.091em; border-top: 1.3px solid;"><span id="MJXc-Node-8700" class="mjx-mrow"><span id="MJXc-Node-8701" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span></span></span></span></span><span id="MJXc-Node-8702" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8703" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8704" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8705" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8706" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-8707" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-8708" class="mjx-msqrt MJXc-space3"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.091em; border-top: 1.3px solid;"><span id="MJXc-Node-8709" class="mjx-mrow"><span id="MJXc-Node-8710" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span></span></span></span></span><span id="MJXc-Node-8711" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8712" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8713" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-8714" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8715" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span id="MJXc-Node-8716" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8717" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.091em; border-top: 1.3px solid;"><span id="MJXc-Node-8718" class="mjx-mrow"><span id="MJXc-Node-8719" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span></span></span></span></span><span id="MJXc-Node-8720" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8721" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mo>−</mo><msqrt><mi>y</mi></msqrt><mo>≤</mo><mi>X</mi><mo>≤</mo><msqrt><mi>y</mi></msqrt><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><msqrt><mi>y</mi></msqrt><mo stretchy="false">)</mo><mo>=</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><msqrt><mi>y</mi></msqrt><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-710">F_Y(y)=P(Y\leq y)=P(-\sqrt y\leq X\leq \sqrt y)=P(X\leq \sqrt y)=F_X(\sqrt y).</script>
Connaissant la fonction de répartition de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-711-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8722" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8723" class="mjx-mrow"><span id="MJXc-Node-8724" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-711">X</script>, on en déduit
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-712-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable columnalign=&quot;left left&quot; rowspacing=&quot;.2em&quot; columnspacing=&quot;1em&quot; displaystyle=&quot;false&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;si&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msqrt&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/msqrt&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;si&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;si&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; symmetric=&quot;true&quot;&gt;&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8725" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8726" class="mjx-mrow"><span id="MJXc-Node-8727" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-8728" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8729" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span id="MJXc-Node-8730" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8731" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-8732" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8733" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8734" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-8735" class="mjx-mo" style="vertical-align: -0.968em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.703em; padding-bottom: 0.295em;">⎧</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.149em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪
⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎨</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.149em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪
⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="margin-top: -0.231em; padding-bottom: 1.17em;">⎩</span></span></span><span id="MJXc-Node-8736" class="mjx-mtable" style="vertical-align: -1.892em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-8737" class="mjx-mtr" style="height: 1.182em;"><span id="MJXc-Node-8738" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: left; width: 2.245em;"><span id="MJXc-Node-8739" class="mjx-mrow" style="margin-top: -0.148em;"><span id="MJXc-Node-8740" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8741" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 5.377em;"><span id="MJXc-Node-8742" class="mjx-mrow" style="margin-top: -0.148em;"><span id="MJXc-Node-8743" class="mjx-texatom"><span id="MJXc-Node-8744" class="mjx-mrow"><span id="MJXc-Node-8745" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">si&nbsp;</span></span></span></span><span id="MJXc-Node-8746" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-8747" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&lt;</span></span><span id="MJXc-Node-8748" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-8749" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8750" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-8751" class="mjx-mtr" style="height: 1.921em;"><span id="MJXc-Node-8752" class="mjx-mtd" style="padding: 0.1em 0.5em 0px 0px; text-align: left;"><span id="MJXc-Node-8753" class="mjx-mrow"><span id="MJXc-Node-8754" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.005em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.836em; top: -1.712em;"><span id="MJXc-Node-8755" class="mjx-mrow" style=""><span id="MJXc-Node-8756" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.139em; border-top: 1px solid;"><span id="MJXc-Node-8757" class="mjx-mrow"><span id="MJXc-Node-8758" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span></span></span></span></span><span id="MJXc-Node-8759" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8760" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.836em; bottom: -0.721em;"><span id="MJXc-Node-8761" class="mjx-mrow" style=""><span id="MJXc-Node-8762" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-8763" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8764" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.005em;"></span></span><span class="mjx-vsize" style="height: 1.721em; vertical-align: -0.51em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8765" class="mjx-mtd" style="padding: 0.1em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-8766" class="mjx-mrow" style="margin-top: 0.211em;"><span id="MJXc-Node-8767" class="mjx-texatom"><span id="MJXc-Node-8768" class="mjx-mrow"><span id="MJXc-Node-8769" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">si&nbsp;</span></span></span></span><span id="MJXc-Node-8770" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-8771" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">∈</span></span><span id="MJXc-Node-8772" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-8773" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8774" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8775" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-8776" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8777" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-8778" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8779" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-8780" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-8781" class="mjx-mtr" style="height: 1.182em;"><span id="MJXc-Node-8782" class="mjx-mtd" style="padding: 0.1em 0.5em 0px 0px; text-align: left;"><span id="MJXc-Node-8783" class="mjx-mrow" style="margin-top: -0.148em;"><span id="MJXc-Node-8784" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8785" class="mjx-mtd" style="padding: 0.1em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-8786" class="mjx-mrow" style="margin-top: -0.148em;"><span id="MJXc-Node-8787" class="mjx-texatom"><span id="MJXc-Node-8788" class="mjx-mrow"><span id="MJXc-Node-8789" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">si&nbsp;</span></span></span></span><span id="MJXc-Node-8790" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-8791" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-8792" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-8793" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8794" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-8795" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-8796" class="mjx-mo" style="width: 0.12em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mn>0</mn></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>si&nbsp;</mtext></mrow><mi>y</mi><mo>&lt;</mo><msup><mi>a</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mfrac><mrow><msqrt><mi>y</mi></msqrt><mo>−</mo><mi>a</mi></mrow><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>si&nbsp;</mtext></mrow><mi>y</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>a</mi><mn>2</mn></msup><mo>,</mo><msup><mi>b</mi><mn>2</mn></msup><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>si&nbsp;</mtext></mrow><mi>y</mi><mo>&gt;</mo><msup><mi>b</mi><mn>2</mn></msup><mo>.</mo></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-712">F_Y(y)=\begin{cases}
0&\textrm{si }y<a^2\\
\frac{\sqrt y-a}{b-a}&\textrm{si }y\in[a^2,b^2]\\
1&\textrm{si }y>b^2.
\end{cases}
</script>
Cette fonction de répartition est dérivable sauf en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-713-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8797" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8798" class="mjx-mrow"><span id="MJXc-Node-8799" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8800" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8801" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-713">a^2</script> et en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-714-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8802" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8803" class="mjx-mrow"><span id="MJXc-Node-8804" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8805" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8806" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-714">b^2</script>. La dérivée donne la densité.
On en déduit que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-715-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8807" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8808" class="mjx-mrow"><span id="MJXc-Node-8809" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-715">Y</script> admet une densité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-716-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8810" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8811" class="mjx-mrow"><span id="MJXc-Node-8812" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8813" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8814" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>Y</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-716">p_Y</script> donnée par :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-717-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable columnalign=&quot;left left&quot; rowspacing=&quot;.2em&quot; columnspacing=&quot;1em&quot; displaystyle=&quot;false&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;si&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msqrt&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/msqrt&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;si&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;si&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; symmetric=&quot;true&quot;&gt;&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8815" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8816" class="mjx-mrow"><span id="MJXc-Node-8817" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8818" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8819" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span id="MJXc-Node-8820" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8821" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-8822" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8823" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8824" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-8825" class="mjx-mo" style="vertical-align: -0.946em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.703em; padding-bottom: 0.295em;">⎧</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.138em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪
⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎨</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.138em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪
⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="margin-top: -0.231em; padding-bottom: 1.17em;">⎩</span></span></span><span id="MJXc-Node-8826" class="mjx-mtable" style="vertical-align: -1.87em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-8827" class="mjx-mtr" style="height: 1.182em;"><span id="MJXc-Node-8828" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: left; width: 3.452em;"><span id="MJXc-Node-8829" class="mjx-mrow" style="margin-top: -0.148em;"><span id="MJXc-Node-8830" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8831" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 5.377em;"><span id="MJXc-Node-8832" class="mjx-mrow" style="margin-top: -0.148em;"><span id="MJXc-Node-8833" class="mjx-texatom"><span id="MJXc-Node-8834" class="mjx-mrow"><span id="MJXc-Node-8835" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">si&nbsp;</span></span></span></span><span id="MJXc-Node-8836" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-8837" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&lt;</span></span><span id="MJXc-Node-8838" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-8839" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8840" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-8841" class="mjx-mtr" style="height: 1.878em;"><span id="MJXc-Node-8842" class="mjx-mtd" style="padding: 0.1em 0.5em 0px 0px; text-align: left;"><span id="MJXc-Node-8843" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-8844" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.212em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 4.543em; top: -1.318em;"><span id="MJXc-Node-8845" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 4.543em; bottom: -1.054em;"><span id="MJXc-Node-8846" class="mjx-mrow" style=""><span id="MJXc-Node-8847" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-8848" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8849" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-8850" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8851" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-8852" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8853" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.139em; border-top: 1px solid;"><span id="MJXc-Node-8854" class="mjx-mrow"><span id="MJXc-Node-8855" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.212em;"></span></span><span class="mjx-vsize" style="height: 1.678em; vertical-align: -0.745em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8856" class="mjx-mtd" style="padding: 0.1em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-8857" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-8858" class="mjx-texatom"><span id="MJXc-Node-8859" class="mjx-mrow"><span id="MJXc-Node-8860" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">si&nbsp;</span></span></span></span><span id="MJXc-Node-8861" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-8862" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">∈</span></span><span id="MJXc-Node-8863" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-8864" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8865" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8866" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-8867" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8868" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-8869" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8870" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-8871" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-8872" class="mjx-mtr" style="height: 1.182em;"><span id="MJXc-Node-8873" class="mjx-mtd" style="padding: 0.1em 0.5em 0px 0px; text-align: left;"><span id="MJXc-Node-8874" class="mjx-mrow" style="margin-top: -0.148em;"><span id="MJXc-Node-8875" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8876" class="mjx-mtd" style="padding: 0.1em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-8877" class="mjx-mrow" style="margin-top: -0.148em;"><span id="MJXc-Node-8878" class="mjx-texatom"><span id="MJXc-Node-8879" class="mjx-mrow"><span id="MJXc-Node-8880" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">si&nbsp;</span></span></span></span><span id="MJXc-Node-8881" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-8882" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-8883" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-8884" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8885" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-8886" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-8887" class="mjx-mo" style="width: 0.12em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mn>0</mn></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>si&nbsp;</mtext></mrow><mi>y</mi><mo>&lt;</mo><msup><mi>a</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mrow><mn>2</mn><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo><msqrt><mi>y</mi></msqrt></mrow></mfrac></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>si&nbsp;</mtext></mrow><mi>y</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>a</mi><mn>2</mn></msup><mo>,</mo><msup><mi>b</mi><mn>2</mn></msup><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>si&nbsp;</mtext></mrow><mi>y</mi><mo>&gt;</mo><msup><mi>b</mi><mn>2</mn></msup><mo>.</mo></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-717">p_Y(y)=\begin{cases}
0&\textrm{si }y<a^2\\
\frac{1}{2(b-a)\sqrt y}&\textrm{si }y\in[a^2,b^2]\\
0&\textrm{si }y>b^2.
\end{cases}</script>
Pour calculer l'espérance de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-718-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8888" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8889" class="mjx-mrow"><span id="MJXc-Node-8890" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-718">Y</script> et celle de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-719-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8891" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8892" class="mjx-mrow"><span id="MJXc-Node-8893" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.182em;"><span id="MJXc-Node-8894" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.405em; padding-right: 0.071em;"><span id="MJXc-Node-8895" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Y</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-719">Y^2</script>, il est préférable de se souvenir que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-720-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-8896" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8897" class="mjx-mrow"><span id="MJXc-Node-8898" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-8899" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8900" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-8901" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.115em; padding-right: 0.071em;"><span id="MJXc-Node-8902" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><msup><mi>X</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-720">Y=X^2</script>.
On en déduit :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-721-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/msubsup&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8903" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8904" class="mjx-mrow"><span id="MJXc-Node-8905" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-8906" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8907" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-8908" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8909" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8910" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-8911" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8912" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-8913" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0.115em; padding-right: 0.071em;"><span id="MJXc-Node-8914" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-8915" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8916" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8917" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-8918" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.375em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-8920" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-8919" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span></span><span id="MJXc-Node-8921" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 2.38em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.38em; top: -1.528em;"><span id="MJXc-Node-8922" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8923" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8924" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span><span class="mjx-denominator" style="width: 2.38em; bottom: -0.818em;"><span id="MJXc-Node-8925" class="mjx-mrow"><span id="MJXc-Node-8926" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-8927" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8928" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.38em;"></span></span><span class="mjx-vsize" style="height: 2.346em; vertical-align: -0.818em;"></span></span><span id="MJXc-Node-8929" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-8930" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8931" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8932" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.658em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.658em; top: -1.527em;"><span id="MJXc-Node-8933" class="mjx-mrow"><span id="MJXc-Node-8934" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8935" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8936" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-8937" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8938" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-8939" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8940" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span><span class="mjx-denominator" style="width: 3.658em; bottom: -1.042em;"><span id="MJXc-Node-8941" class="mjx-mrow"><span id="MJXc-Node-8942" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-8943" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8944" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-8945" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8946" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-8947" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.658em;"></span></span><span class="mjx-vsize" style="height: 2.57em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-8948" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><msup><mi>X</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mrow><msup><mi>b</mi><mn>3</mn></msup><mo>−</mo><msup><mi>a</mi><mn>3</mn></msup></mrow><mrow><mn>3</mn><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>,</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-721">E(Y)=E(X^2)=\int_a^b\frac{x^2}{b-a}dx=\frac{b^3-a^3}{3(b-a)},</script>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-722-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/msubsup&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8949" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8950" class="mjx-mrow"><span id="MJXc-Node-8951" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-8952" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8953" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.182em;"><span id="MJXc-Node-8954" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0.405em; padding-right: 0.071em;"><span id="MJXc-Node-8955" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-8956" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8957" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8958" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-8959" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8960" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-8961" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0.115em; padding-right: 0.071em;"><span id="MJXc-Node-8962" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span id="MJXc-Node-8963" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8964" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8965" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-8966" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.375em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-8968" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-8967" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span></span><span id="MJXc-Node-8969" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 2.38em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.38em; top: -1.536em;"><span id="MJXc-Node-8970" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8971" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8972" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span><span class="mjx-denominator" style="width: 2.38em; bottom: -0.818em;"><span id="MJXc-Node-8973" class="mjx-mrow"><span id="MJXc-Node-8974" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-8975" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8976" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.38em;"></span></span><span class="mjx-vsize" style="height: 2.354em; vertical-align: -0.818em;"></span></span><span id="MJXc-Node-8977" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-8978" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8979" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8980" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.658em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.658em; top: -1.528em;"><span id="MJXc-Node-8981" class="mjx-mrow"><span id="MJXc-Node-8982" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-8983" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8984" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span id="MJXc-Node-8985" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8986" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-8987" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-8988" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span></span></span><span class="mjx-denominator" style="width: 3.658em; bottom: -1.042em;"><span id="MJXc-Node-8989" class="mjx-mrow"><span id="MJXc-Node-8990" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-8991" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8992" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-8993" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8994" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-8995" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.658em;"></span></span><span class="mjx-vsize" style="height: 2.57em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-8996" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><msup><mi>Y</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><msup><mi>X</mi><mn>4</mn></msup><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mfrac><msup><mi>x</mi><mn>4</mn></msup><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mrow><msup><mi>b</mi><mn>5</mn></msup><mo>−</mo><msup><mi>a</mi><mn>5</mn></msup></mrow><mrow><mn>5</mn><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-722">E(Y^2)=E(X^4)=\int_a^b\frac{x^4}{b-a}dx=\frac{b^5-a^5}{5(b-a)}.</script>
Enfin, un calcul un peu pénible montre que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-723-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;Var&lt;/mtext&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;msup&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;msup&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;45&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-8997" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8998" class="mjx-mrow"><span id="MJXc-Node-8999" class="mjx-texatom"><span id="MJXc-Node-9000" class="mjx-mrow"><span id="MJXc-Node-9001" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">Var</span></span></span></span><span id="MJXc-Node-9002" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9003" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-9004" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9005" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9006" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-9007" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9008" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.182em;"><span id="MJXc-Node-9009" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0.405em; padding-right: 0.071em;"><span id="MJXc-Node-9010" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-9011" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9012" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9013" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-9014" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9015" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-9016" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-9017" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-9018" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-9019" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9020" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 10.964em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 10.964em; top: -1.619em;"><span id="MJXc-Node-9021" class="mjx-mrow"><span id="MJXc-Node-9022" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9023" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-9024" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9025" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-9026" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-9027" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-9028" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-9029" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9030" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-9031" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-9032" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-9033" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-9034" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9035" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-9036" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-9037" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-9038" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9039" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-9040" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-9041" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-9042" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-9043" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 10.964em; bottom: -0.741em;"><span id="MJXc-Node-9044" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">45</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 10.964em;"></span></span><span class="mjx-vsize" style="height: 2.36em; vertical-align: -0.741em;"></span></span><span id="MJXc-Node-9045" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mtext>Var</mtext></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><msup><mi>Y</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>−</mo><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>a</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo stretchy="false">(</mo><mn>4</mn><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>a</mi><mi>b</mi><mo>+</mo><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mn>45</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-723">\textrm{Var}(Y)=E(Y^2)-E(Y)^2=\frac{(b-a)^2(4b^2+7ab+4a^2)}{45}.</script></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>26</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Loi uniforme, moyenne et écart-type<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2314"> [Signaler une erreur]</a><a id="exo26" href="javascript:ajouteexo(2314,26,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2314&#39;,&#39;imageenonce2314&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2314"></a><div class="inner" id="innerenonce2314">Soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-724-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9046" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9047" class="mjx-mrow"><span id="MJXc-Node-9048" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-724">X</script> une variable aléatoire suivant une loi uniforme sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-725-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9049" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9050" class="mjx-mrow"><span id="MJXc-Node-9051" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-9052" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span><span id="MJXc-Node-9053" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-9054" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">b</span></span><span id="MJXc-Node-9055" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-725">[a,b]</script>. On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-726-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9056" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9057" class="mjx-mrow"><span id="MJXc-Node-9058" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">m</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></span></span><script type="math/tex" id="MathJax-Element-726">m</script> sa moyenne et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-727-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9059" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9060" class="mjx-mrow"><span id="MJXc-Node-9061" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em; padding-right: 0.001em;">σ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi></math></span></span><script type="math/tex" id="MathJax-Element-727">\sigma</script> son écart-type. Calculer la probabilité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-728-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9062" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9063" class="mjx-mrow"><span id="MJXc-Node-9064" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9065" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-9066" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9067" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-9068" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-9069" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">m</span></span><span id="MJXc-Node-9070" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-9071" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em; padding-right: 0.001em;">σ</span></span><span id="MJXc-Node-9072" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-9073" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">m</span></span><span id="MJXc-Node-9074" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-9075" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em; padding-right: 0.001em;">σ</span></span><span id="MJXc-Node-9076" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span><span id="MJXc-Node-9077" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>m</mi><mo>−</mo><mi>σ</mi><mo>,</mo><mi>m</mi><mo>+</mo><mi>σ</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-728">P(X\in [m-\sigma,m+\sigma])</script>.</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2314&#39;,&#39;imageindication2314&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2314"></a><div class="inner" id="innerindication2314" style="display:none;"></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2314&#39;,&#39;imagecorrige2314&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2314"></a><div class="inner" id="innercorrige2314" style="display:none;">Rappelons que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-729-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9078" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9079" class="mjx-mrow"><span id="MJXc-Node-9080" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-9081" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9082" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.369em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.936em; top: -1.428em;"><span id="MJXc-Node-9083" class="mjx-mrow" style=""><span id="MJXc-Node-9084" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-9085" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9086" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.936em; bottom: -0.611em;"><span id="MJXc-Node-9087" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.369em;"></span></span><span class="mjx-vsize" style="height: 1.442em; vertical-align: -0.432em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-729">m=\frac{a+b}2</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-730-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msqrt&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msqrt&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9088" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9089" class="mjx-mrow"><span id="MJXc-Node-9090" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span><span id="MJXc-Node-9091" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9092" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.438em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.033em; top: -1.428em;"><span id="MJXc-Node-9093" class="mjx-mrow" style=""><span id="MJXc-Node-9094" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-9095" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9096" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.033em; bottom: -1.005em;"><span id="MJXc-Node-9097" class="mjx-mrow" style=""><span id="MJXc-Node-9098" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-9099" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.119em; border-top: 1px solid;"><span id="MJXc-Node-9100" class="mjx-mrow"><span id="MJXc-Node-9101" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.438em;"></span></span><span class="mjx-vsize" style="height: 1.721em; vertical-align: -0.711em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mo>=</mo><mfrac><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-730">\sigma=\frac{b-a}{2\sqrt 3}</script> et que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-731-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9102" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9103" class="mjx-mrow"><span id="MJXc-Node-9104" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-731">X</script> admet pour densité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-732-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn mathvariant=&quot;bold&quot;&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9105" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9106" class="mjx-mrow"><span id="MJXc-Node-9107" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-9108" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9109" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-9110" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9111" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9112" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.369em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.936em; top: -1.318em;"><span id="MJXc-Node-9113" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.936em; bottom: -0.721em;"><span id="MJXc-Node-9114" class="mjx-mrow" style=""><span id="MJXc-Node-9115" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-9116" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9117" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.369em;"></span></span><span class="mjx-vsize" style="height: 1.442em; vertical-align: -0.51em;"></span></span><span id="MJXc-Node-9118" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-9119" class="mjx-texatom"><span id="MJXc-Node-9120" class="mjx-mrow"><span id="MJXc-Node-9121" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.275em; padding-right: 0.071em;"><span id="MJXc-Node-9122" class="mjx-texatom" style=""><span id="MJXc-Node-9123" class="mjx-mrow"><span id="MJXc-Node-9124" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-9125" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-9126" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-9127" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-9128" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span></span></span><span id="MJXc-Node-9129" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9130" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-9131" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac><msub><mrow class="MJX-TeXAtom-ORD"><mn mathvariant="bold">1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-732">f(x)=\frac{1}{b-a}\mathbf 1_{[a,b]}(x)</script>. Remarquons aussi que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-733-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msqrt&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msqrt&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9132" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9133" class="mjx-mrow"><span id="MJXc-Node-9134" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-9135" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9136" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span><span id="MJXc-Node-9137" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9138" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.38em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.38em; top: -1.396em;"><span id="MJXc-Node-9139" class="mjx-mrow"><span id="MJXc-Node-9140" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-9141" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9142" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span></span><span class="mjx-denominator" style="width: 2.38em; bottom: -0.711em;"><span id="MJXc-Node-9143" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.38em;"></span></span><span class="mjx-vsize" style="height: 2.106em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-9144" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9145" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 2.38em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.38em; top: -1.396em;"><span id="MJXc-Node-9146" class="mjx-mrow"><span id="MJXc-Node-9147" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-9148" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9149" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="mjx-denominator" style="width: 2.38em; bottom: -1.102em;"><span id="MJXc-Node-9150" class="mjx-mrow"><span id="MJXc-Node-9151" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-9152" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.119em; border-top: 1.3px solid;"><span id="MJXc-Node-9153" class="mjx-mrow"><span id="MJXc-Node-9154" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.38em;"></span></span><span class="mjx-vsize" style="height: 2.498em; vertical-align: -1.102em;"></span></span><span id="MJXc-Node-9155" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-9156" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.38em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.38em; top: -1.396em;"><span id="MJXc-Node-9157" class="mjx-mrow"><span id="MJXc-Node-9158" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-9159" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9160" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span></span><span class="mjx-denominator" style="width: 2.38em; bottom: -0.711em;"><span id="MJXc-Node-9161" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.38em;"></span></span><span class="mjx-vsize" style="height: 2.106em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-9162" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9163" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 2.38em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.38em; top: -1.396em;"><span id="MJXc-Node-9164" class="mjx-mrow"><span id="MJXc-Node-9165" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-9166" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9167" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="mjx-denominator" style="width: 2.38em; bottom: -0.711em;"><span id="MJXc-Node-9168" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.38em;"></span></span><span class="mjx-vsize" style="height: 2.106em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-9169" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-9170" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>m</mi><mo>+</mo><mi>σ</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow></mfrac><mo>≤</mo><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow><mn>2</mn></mfrac><mo>≤</mo><mi>b</mi></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-733">m+\sigma=\frac{a+b}2+\frac{b-a}{2\sqrt 3}\leq \frac{a+b}2+\frac{b-a}2\leq b</script>
et de même <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-734-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9171" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9172" class="mjx-mrow"><span id="MJXc-Node-9173" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-9174" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9175" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span><span id="MJXc-Node-9176" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-9177" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>−</mo><mi>σ</mi><mo>≥</mo><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-734">m-\sigma\geq a</script>. Le calcul de la probabilité demandée suit alors la méthode usuelle pour une loi à densité :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-735-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0 thickmathspace&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msqrt&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msqrt&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9178" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9179" class="mjx-mrow"><span id="MJXc-Node-9180" class="mjx-mtable" style="vertical-align: -4.891em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-9181" class="mjx-mtr" style="height: 2.69em;"><span id="MJXc-Node-9182" class="mjx-mtd" style="padding: 0px; text-align: right; width: 9.949em;"><span id="MJXc-Node-9183" class="mjx-mrow" style="margin-top: 0.543em;"><span id="MJXc-Node-9184" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9185" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9186" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9187" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">∈</span></span><span id="MJXc-Node-9188" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-9189" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-9190" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9191" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span><span id="MJXc-Node-9192" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-9193" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-9194" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9195" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span><span id="MJXc-Node-9196" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span><span id="MJXc-Node-9197" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-9198" class="mjx-mtd" style="padding: 0px 0.139em 0px 0px; width: 1.056em;"><span id="MJXc-Node-9199" class="mjx-mrow" style="margin-top: 0.543em;"><span id="MJXc-Node-9200" class="mjx-mi"></span><span id="MJXc-Node-9201" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-9202" class="mjx-mtd" style="padding: 0px 0px 0px 0.139em; text-align: left; width: 11.188em;"><span id="MJXc-Node-9203" class="mjx-mrow"><span id="MJXc-Node-9204" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.38em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.38em; top: -1.368em;"><span id="MJXc-Node-9205" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 2.38em; bottom: -0.818em;"><span id="MJXc-Node-9206" class="mjx-mrow"><span id="MJXc-Node-9207" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-9208" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9209" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.38em;"></span></span><span class="mjx-vsize" style="height: 2.186em; vertical-align: -0.818em;"></span></span><span id="MJXc-Node-9210" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.388em;"><span id="MJXc-Node-9211" class="mjx-mo" style="vertical-align: 0.001em; padding-right: 0.388em;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 1.112em; padding-bottom: 1.112em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.921em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 2.162em; padding-left: 0.784em; padding-right: 0.071em;"><span id="MJXc-Node-9217" class="mjx-texatom" style=""><span id="MJXc-Node-9218" class="mjx-mrow"><span id="MJXc-Node-9219" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-9220" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9221" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-9212" class="mjx-texatom" style=""><span id="MJXc-Node-9213" class="mjx-mrow"><span id="MJXc-Node-9214" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-9215" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9216" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span></span></span></span></span></span><span id="MJXc-Node-9222" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-9223" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-9224" class="mjx-mtr" style="height: 2.486em;"><span id="MJXc-Node-9225" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-9226" class="mjx-mrow" style="margin-top: 0.368em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-9227" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-9228" class="mjx-mrow" style="margin-top: 0.368em;"><span id="MJXc-Node-9229" class="mjx-mi"></span><span id="MJXc-Node-9230" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-9231" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-9232" class="mjx-mrow"><span id="MJXc-Node-9233" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.38em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.38em; top: -1.368em;"><span id="MJXc-Node-9234" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 2.38em; bottom: -0.818em;"><span id="MJXc-Node-9235" class="mjx-mrow"><span id="MJXc-Node-9236" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-9237" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9238" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.38em;"></span></span><span class="mjx-vsize" style="height: 2.186em; vertical-align: -0.818em;"></span></span><span id="MJXc-Node-9239" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-9240" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9241" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-9242" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9243" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span><span id="MJXc-Node-9244" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9245" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-9246" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9247" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span><span id="MJXc-Node-9248" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-9249" class="mjx-mtr" style="height: 2.486em;"><span id="MJXc-Node-9250" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-9251" class="mjx-mrow" style="margin-top: 0.368em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-9252" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-9253" class="mjx-mrow" style="margin-top: 0.368em;"><span id="MJXc-Node-9254" class="mjx-mi"></span><span id="MJXc-Node-9255" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-9256" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-9257" class="mjx-mrow"><span id="MJXc-Node-9258" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.38em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.38em; top: -1.368em;"><span id="MJXc-Node-9259" class="mjx-mrow"><span id="MJXc-Node-9260" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-9261" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span></span></span><span class="mjx-denominator" style="width: 2.38em; bottom: -0.818em;"><span id="MJXc-Node-9262" class="mjx-mrow"><span id="MJXc-Node-9263" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span><span id="MJXc-Node-9264" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9265" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.38em;"></span></span><span class="mjx-vsize" style="height: 2.186em; vertical-align: -0.818em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-9266" class="mjx-mtr" style="height: 2.62em;"><span id="MJXc-Node-9267" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-9268" class="mjx-mrow" style="margin-top: 0.368em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-9269" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-9270" class="mjx-mrow" style="margin-top: 0.368em;"><span id="MJXc-Node-9271" class="mjx-mi"></span><span id="MJXc-Node-9272" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-9273" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-9274" class="mjx-mrow"><span id="MJXc-Node-9275" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.533em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.533em; top: -1.368em;"><span id="MJXc-Node-9276" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.533em; bottom: -1.102em;"><span id="MJXc-Node-9277" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.119em; border-top: 1.3px solid;"><span id="MJXc-Node-9278" class="mjx-mrow"><span id="MJXc-Node-9279" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.533em;"></span></span><span class="mjx-vsize" style="height: 2.47em; vertical-align: -1.102em;"></span></span><span id="MJXc-Node-9280" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>m</mi><mo>−</mo><mi>σ</mi><mo>,</mo><mi>m</mi><mo>+</mo><mi>σ</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mfrac><mn>1</mn><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mo>−</mo><mi>σ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mo>+</mo><mi>σ</mi></mrow></msubsup><mi>d</mi><mi>t</mi></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mfrac><mn>1</mn><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac><mo>×</mo><mo stretchy="false">(</mo><mi>m</mi><mo>+</mo><mi>σ</mi><mo>−</mo><mi>m</mi><mo>+</mo><mi>σ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mfrac><mrow><mn>2</mn><mi>σ</mi></mrow><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mo>.</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-735">\begin{eqnarray*}
P(X\in [m-\sigma,m+\sigma])&=&\frac{1}{b-a}\int_{m-\sigma}^{m+\sigma}dt\\
&=&\frac{1}{b-a}\times (m+\sigma-m+\sigma)\\
&=&\frac{2\sigma}{b-a}\\
&=&\frac 1{\sqrt 3}.
\end{eqnarray*}</script>
Remarquons que ceci ne dépend pas de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-736-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9281" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9282" class="mjx-mrow"><span id="MJXc-Node-9283" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-736">a</script> et de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-737-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9284" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9285" class="mjx-mrow"><span id="MJXc-Node-9286" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math></span></span><script type="math/tex" id="MathJax-Element-737">b</script>, ce qui est une propriété assez remarquable de la loi uniforme.</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>27</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Temps d'attente<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2184"> [Signaler une erreur]</a><a id="exo27" href="javascript:ajouteexo(2184,27,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2184&#39;,&#39;imageenonce2184&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2184"></a><div class="inner" id="innerenonce2184">A partir de 7heures du matin, les bus passent toutes les quinze minutes à un arrêt précis. Un usager se présente à cet arrêt entre 7h et 7h30. On fait l'hypothèse que l'heure exacte de son arrivée, représentée par le nombre de minutes après 7h, est une variable aléatoire uniformément répartie sur l'intervalle [0,30]. Quelle est la probabilité que l'usager attende moins de cinq minutes le prochain bus? Qu'il l'attende plus de dix minutes?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2184&#39;,&#39;imageindication2184&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2184"></a><div class="inner" id="innerindication2184" style="display:none;">Trouver les heures possibles d'arrivée pour que l'usager attende moins de cinq minutes.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2184&#39;,&#39;imagecorrige2184&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2184"></a><div class="inner" id="innercorrige2184" style="display:none;">L'usager attendra moins de cinq minutes si son heure d'arrivée est comprise entre 7h10 et 7h15, ou entre 7h25 et 7h30. Si on note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-738-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9287" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9288" class="mjx-mrow"><span id="MJXc-Node-9289" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-738">X</script> la variable aléatoire définie par l'énoncé, ceci revient à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-739-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9290" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9291" class="mjx-mrow"><span id="MJXc-Node-9292" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-739">X</script> appartient à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-740-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;15&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9293" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9294" class="mjx-mrow"><span id="MJXc-Node-9295" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-9296" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-9297" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-9298" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">15</span></span><span id="MJXc-Node-9299" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mn>10</mn><mo>,</mo><mn>15</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-740">[10,15]</script> ou <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-741-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9300" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9301" class="mjx-mrow"><span id="MJXc-Node-9302" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-741">X</script> appartient à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-742-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;25&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9303" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9304" class="mjx-mrow"><span id="MJXc-Node-9305" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-9306" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">25</span></span><span id="MJXc-Node-9307" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-9308" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">30</span></span><span id="MJXc-Node-9309" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mn>25</mn><mo>,</mo><mn>30</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-742">[25,30]</script>. La probabilité recherchée est donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-743-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9310" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9311" class="mjx-mrow"><span id="MJXc-Node-9312" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.399em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.978em; top: -1.4em;"><span id="MJXc-Node-9313" class="mjx-mrow" style=""><span id="MJXc-Node-9314" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-9315" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9316" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.978em; bottom: -0.633em;"><span id="MJXc-Node-9317" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">30</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.399em;"></span></span><span class="mjx-vsize" style="height: 1.438em; vertical-align: -0.448em;"></span></span><span id="MJXc-Node-9318" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9319" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-9320" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.632em;"><span id="MJXc-Node-9321" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.379em; vertical-align: -0.447em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>5</mn><mo>+</mo><mn>5</mn></mrow><mn>30</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-743">\frac{5+5}{30}=\frac 13</script>.<br>
Le raisonnement est identique pour déterminer la probabilité qu'il attende plus de dix minutes. Dans ce cas, l'heure d'arrivée doit être comprise entre 7h et 7h05 ou entre 7h15 et 7h20. On trouve à nouveau une probabilité de 1/3.</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>28</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Désintégration radioactive.<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2318"> [Signaler une erreur]</a><a id="exo28" href="javascript:ajouteexo(2318,28,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2318&#39;,&#39;imageenonce2318&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2318"></a><div class="inner" id="innerenonce2318">La durée de vie des atomes de radon suit une loi exponentielle. La probabilité qu'un atome de radon ne soit pas désintégré en 40s
sachant qu'il ne l'est pas en 12s vaut <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-744-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mfrac&gt;&lt;msqrt&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msqrt&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9322" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9323" class="mjx-mrow"><span id="MJXc-Node-9324" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.084em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.533em; top: -1.735em;"><span id="MJXc-Node-9325" class="mjx-msqrt" style=""><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.522em; padding-bottom: 0.522em;">√</span></span><span class="mjx-box" style="padding-top: 0.13em; border-top: 1px solid;"><span id="MJXc-Node-9326" class="mjx-mrow"><span id="MJXc-Node-9327" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.533em; bottom: -0.634em;"><span id="MJXc-Node-9328" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.291em; width: 1.084em;"></span></span><span class="mjx-vsize" style="height: 1.676em; vertical-align: -0.449em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-744">\frac{\sqrt 2}2</script>. Quelle est la probabilité qu'il ne soit pas désintégré avant 76s sachant qu'il ne l'est pas en 20s?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2318&#39;,&#39;imageindication2318&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2318"></a><div class="inner" id="innerindication2318" style="display:none;">Utiliser la propriété d'absence de mémoire des lois exponentielles.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2318&#39;,&#39;imagecorrige2318&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2318"></a><div class="inner" id="innercorrige2318" style="display:none;">Soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-745-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9329" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9330" class="mjx-mrow"><span id="MJXc-Node-9331" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-745">X</script> la variable aléatoire donnant la durée de vie d'un atome de radon. On sait que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-746-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9332" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9333" class="mjx-mrow"><span id="MJXc-Node-9334" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-746">X</script> suit une loi exponentielle de paramètre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-747-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9335" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9336" class="mjx-mrow"><span id="MJXc-Node-9337" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-747">\lambda</script>.
On sait que, pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-748-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9338" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9339" class="mjx-mrow"><span id="MJXc-Node-9340" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-9341" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-9342" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>&gt;</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-748">a>0</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-749-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9343" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9344" class="mjx-mrow"><span id="MJXc-Node-9345" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9346" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9347" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9348" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-9349" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-9350" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9351" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9352" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-9353" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.586em; padding-bottom: 0.586em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.733em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-9355" class="mjx-texatom" style=""><span id="MJXc-Node-9356" class="mjx-mrow"><span id="MJXc-Node-9357" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9358" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-9354" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span></span><span id="MJXc-Node-9359" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span><span id="MJXc-Node-9360" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-9361" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-9362" class="mjx-texatom" style=""><span id="MJXc-Node-9363" class="mjx-mrow"><span id="MJXc-Node-9364" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9365" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span><span id="MJXc-Node-9366" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span></span></span></span><span id="MJXc-Node-9367" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-9368" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-9369" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9370" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-9371" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-9372" class="mjx-texatom" style=""><span id="MJXc-Node-9373" class="mjx-mrow"><span id="MJXc-Node-9374" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9375" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span><span id="MJXc-Node-9376" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>λ</mi><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>λ</mi><mi>t</mi></mrow></msup><mi>d</mi><mi>t</mi><mo>=</mo><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>λ</mi><mi>a</mi></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-749">P(X\geq a)=\int_a^{+\infty}\lambda e^{-\lambda t}dt=e^{-\lambda a}</script>. L'énoncé nous dit que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-750-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;40&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;msqrt&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msqrt&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9377" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9378" class="mjx-mrow"><span id="MJXc-Node-9379" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-9380" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.275em; padding-right: 0.071em;"><span id="MJXc-Node-9381" class="mjx-texatom" style=""><span id="MJXc-Node-9382" class="mjx-mrow"><span id="MJXc-Node-9383" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9384" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9385" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-9386" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">12</span></span><span id="MJXc-Node-9387" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span></span></span><span id="MJXc-Node-9388" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9389" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9390" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-9391" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">40</span></span><span id="MJXc-Node-9392" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9393" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9394" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.533em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.533em; top: -1.617em;"><span id="MJXc-Node-9395" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.13em; border-top: 1.3px solid;"><span id="MJXc-Node-9396" class="mjx-mrow"><span id="MJXc-Node-9397" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 1.533em; bottom: -0.711em;"><span id="MJXc-Node-9398" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.533em;"></span></span><span class="mjx-vsize" style="height: 2.328em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-9399" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mn>12</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mn>40</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-750">P_{(X\geq 12)}(X\geq 40)=\frac{\sqrt 2}2.</script>
On utilise ensuite que la loi exponentielle est sans mémoire pour écrire que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-751-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;msqrt&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msqrt&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9400" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9401" class="mjx-mrow"><span id="MJXc-Node-9402" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9403" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9404" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9405" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-9406" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">28</span></span><span id="MJXc-Node-9407" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9408" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9409" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-9410" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.275em; padding-right: 0.071em;"><span id="MJXc-Node-9411" class="mjx-texatom" style=""><span id="MJXc-Node-9412" class="mjx-mrow"><span id="MJXc-Node-9413" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9414" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9415" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-9416" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">12</span></span><span id="MJXc-Node-9417" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span></span></span><span id="MJXc-Node-9418" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9419" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9420" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-9421" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">28</span></span><span id="MJXc-Node-9422" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9423" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">12</span></span><span id="MJXc-Node-9424" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9425" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9426" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.533em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.533em; top: -1.617em;"><span id="MJXc-Node-9427" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.13em; border-top: 1.3px solid;"><span id="MJXc-Node-9428" class="mjx-mrow"><span id="MJXc-Node-9429" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 1.533em; bottom: -0.711em;"><span id="MJXc-Node-9430" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.533em;"></span></span><span class="mjx-vsize" style="height: 2.328em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-9431" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9432" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-9433" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-9434" class="mjx-texatom" style=""><span id="MJXc-Node-9435" class="mjx-mrow"><span id="MJXc-Node-9436" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9437" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">28</span></span><span id="MJXc-Node-9438" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span></span></span></span><span id="MJXc-Node-9439" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mn>28</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mn>12</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mn>28</mn><mo>+</mo><mn>12</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>=</mo><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>28</mn><mi>λ</mi></mrow></msup><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-751">P(X\geq 28)=P_{(X\geq 12)}(X\geq 28+12)=\frac{\sqrt 2}2=e^{-28 \lambda}.</script>
D'autre part, on cherche
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-752-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;20&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;76&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;56&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;56&lt;/mn&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9440" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9441" class="mjx-mrow"><span id="MJXc-Node-9442" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-9443" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.275em; padding-right: 0.071em;"><span id="MJXc-Node-9444" class="mjx-texatom" style=""><span id="MJXc-Node-9445" class="mjx-mrow"><span id="MJXc-Node-9446" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9447" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9448" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-9449" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">20</span></span><span id="MJXc-Node-9450" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span></span></span><span id="MJXc-Node-9451" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9452" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9453" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-9454" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">76</span></span><span id="MJXc-Node-9455" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9456" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9457" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9458" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9459" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9460" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-9461" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">56</span></span><span id="MJXc-Node-9462" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9463" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9464" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-9465" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-9466" class="mjx-texatom" style=""><span id="MJXc-Node-9467" class="mjx-mrow"><span id="MJXc-Node-9468" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9469" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">56</span></span><span id="MJXc-Node-9470" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span></span></span></span><span id="MJXc-Node-9471" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9472" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-9473" class="mjx-mrow"><span id="MJXc-Node-9474" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span><span id="MJXc-Node-9475" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-9476" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-9477" class="mjx-texatom" style=""><span id="MJXc-Node-9478" class="mjx-mrow"><span id="MJXc-Node-9479" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9480" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">28</span></span><span id="MJXc-Node-9481" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span></span></span></span><span id="MJXc-Node-9482" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.917em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-9483" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-9484" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9485" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-9486" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-9487" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-9488" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mn>20</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mn>76</mn><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mn>56</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>56</mn><mi>λ</mi></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>28</mn><mi>λ</mi></mrow></msup><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-752">P_{(X\geq 20)}(X>76)=P(X\geq 56)=e^{-56\lambda}=\left(e^{-28\lambda}\right)^2=\frac 12.</script></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>29</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Chaîne de fabrication<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2084"> [Signaler une erreur]</a><a id="exo29" href="javascript:ajouteexo(2084,29,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2084&#39;,&#39;imageenonce2084&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2084"></a><div class="inner" id="innerenonce2084">Une usine fabrique des cadres de vélo. Pour qu'une pièce soit terminée, il faut
qu'elle passe par la chaine <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-753-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9489" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9490" class="mjx-mrow"><span id="MJXc-Node-9491" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-753">A</script> puis par la chaine <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-754-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9492" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9493" class="mjx-mrow"><span id="MJXc-Node-9494" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-754">B</script>.
Le temps de passage exprimé en minutes pour un objet sur la chaine
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-755-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9495" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9496" class="mjx-mrow"><span id="MJXc-Node-9497" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-755">A</script> est une variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-756-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9498" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9499" class="mjx-mrow"><span id="MJXc-Node-9500" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.081em;">M</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></span></span><script type="math/tex" id="MathJax-Element-756">M</script> suivant une loi exponentielle de
paramètre 2.
Le temps de passage exprimé en minutes pour un objet sur la chaine
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-757-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9501" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9502" class="mjx-mrow"><span id="MJXc-Node-9503" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-757">B</script> est une variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-758-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9504" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9505" class="mjx-mrow"><span id="MJXc-Node-9506" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-758">N</script> suivant une loi uniforme sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-759-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9507" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9508" class="mjx-mrow"><span id="MJXc-Node-9509" class="mjx-mrow"><span id="MJXc-Node-9510" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-9511" class="mjx-mrow"><span id="MJXc-Node-9512" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-9513" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-9514" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span><span id="MJXc-Node-9515" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mo>]</mo></mrow></math></span></span><script type="math/tex" id="MathJax-Element-759">\left[
0,1\right] </script>
Les variables <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-760-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9516" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9517" class="mjx-mrow"><span id="MJXc-Node-9518" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.081em;">M</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></span></span><script type="math/tex" id="MathJax-Element-760">M</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-761-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9519" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9520" class="mjx-mrow"><span id="MJXc-Node-9521" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-761">N</script> sont indépendantes.
<ol class="enumeratechiffre">
<li> Rappeler l'expression d'une densité de probabilité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-762-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9522" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9523" class="mjx-mrow"><span id="MJXc-Node-9524" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">v</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math></span></span><script type="math/tex" id="MathJax-Element-762">v</script> de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-763-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9525" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9526" class="mjx-mrow"><span id="MJXc-Node-9527" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.081em;">M</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></span></span><script type="math/tex" id="MathJax-Element-763">M</script>
et d'une densité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-764-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9528" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9529" class="mjx-mrow"><span id="MJXc-Node-9530" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">w</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math></span></span><script type="math/tex" id="MathJax-Element-764">w</script> de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-765-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9531" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9532" class="mjx-mrow"><span id="MJXc-Node-9533" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-765">N</script>.
</li><li> On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-766-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9534" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9535" class="mjx-mrow"><span id="MJXc-Node-9536" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.032em;">S</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></span></span><script type="math/tex" id="MathJax-Element-766">S</script> la variable aléatoire représentant le temps total
de fabrication d'une pièce.
Exprimer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-767-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9537" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9538" class="mjx-mrow"><span id="MJXc-Node-9539" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.032em;">S</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></span></span><script type="math/tex" id="MathJax-Element-767">S</script> en fonction de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-768-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9540" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9541" class="mjx-mrow"><span id="MJXc-Node-9542" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.081em;">M</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></span></span><script type="math/tex" id="MathJax-Element-768">M</script> et de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-769-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9543" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9544" class="mjx-mrow"><span id="MJXc-Node-9545" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-769">N</script> et déterminer le temps moyen
de fabrication d'une pièce.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2084&#39;,&#39;imageindication2084&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2084"></a><div class="inner" id="innerindication2084" style="display:none;"><ol class="enumeratechiffre">
<li> C'est du cours!
</li><li> L'espérance est linéaire!
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2084&#39;,&#39;imagecorrige2084&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2084"></a><div class="inner" id="innercorrige2084" style="display:none;"><ol class="enumeratechiffre">
<li> Une densité de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-770-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9546" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9547" class="mjx-mrow"><span id="MJXc-Node-9548" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></span></span><script type="math/tex" id="MathJax-Element-770">M</script> est : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-771-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9549" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9550" class="mjx-mrow"><span id="MJXc-Node-9551" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">v</span></span><span id="MJXc-Node-9552" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-9553" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9554" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-9555" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span id="MJXc-Node-9556" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9557" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-771">v\left( t\right) =0</script> sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-772-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9558" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9559" class="mjx-mrow"><span id="MJXc-Node-9560" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-9561" class="mjx-texatom"><span id="MJXc-Node-9562" class="mjx-mrow"><span id="MJXc-Node-9563" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.47em; padding-bottom: 0.295em;">R</span></span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.615em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-9564" class="mjx-texatom" style=""><span id="MJXc-Node-9565" class="mjx-mrow"><span id="MJXc-Node-9566" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>−</mo></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-772">\Bbb{R}^{-}</script>
et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-773-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9567" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9568" class="mjx-mrow"><span id="MJXc-Node-9569" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">v</span></span><span id="MJXc-Node-9570" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-9571" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9572" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-9573" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span id="MJXc-Node-9574" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9575" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-9576" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-9577" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-9578" class="mjx-texatom" style=""><span id="MJXc-Node-9579" class="mjx-mrow"><span id="MJXc-Node-9580" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9581" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-9582" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>2</mn><mi>t</mi></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-773">v\left( t\right) =2e^{-2t}</script> sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-774-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9583" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9584" class="mjx-mrow"><span id="MJXc-Node-9585" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-9586" class="mjx-texatom"><span id="MJXc-Node-9587" class="mjx-mrow"><span id="MJXc-Node-9588" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.47em; padding-bottom: 0.295em;">R</span></span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.615em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-9589" class="mjx-texatom" style=""><span id="MJXc-Node-9590" class="mjx-mrow"><span id="MJXc-Node-9591" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-774">\Bbb{R}^{+}</script>
une densité de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-775-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9592" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9593" class="mjx-mrow"><span id="MJXc-Node-9594" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-775">N</script> est : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-776-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9595" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9596" class="mjx-mrow"><span id="MJXc-Node-9597" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">w</span></span><span id="MJXc-Node-9598" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-9599" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9600" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-9601" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span id="MJXc-Node-9602" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9603" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-776">w\left( t\right) =1</script> sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-777-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9604" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9605" class="mjx-mrow"><span id="MJXc-Node-9606" class="mjx-mrow"><span id="MJXc-Node-9607" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-9608" class="mjx-mrow"><span id="MJXc-Node-9609" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-9610" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-9611" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span id="MJXc-Node-9612" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mo>]</mo></mrow></math></span></span><script type="math/tex" id="MathJax-Element-777">\left[ 0,1\right] </script>
et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-778-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9613" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9614" class="mjx-mrow"><span id="MJXc-Node-9615" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-778">0</script> ailleurs
</li><li> Le temps total de fabrication est la somme des temps de passage sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-779-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9616" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9617" class="mjx-mrow"><span id="MJXc-Node-9618" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-779">A
</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-780-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9619" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9620" class="mjx-mrow"><span id="MJXc-Node-9621" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-9622" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-780">B.</script>
On a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-781-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9623" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9624" class="mjx-mrow"><span id="MJXc-Node-9625" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-9626" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9627" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-9628" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9629" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mi>N</mi><mo>+</mo><mi>M</mi></math></span></span><script type="math/tex" id="MathJax-Element-781">S=N+M</script>.
On en déduit que le temps moyen de fabrication d'une pièce est : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-782-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1.&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9630" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9631" class="mjx-mrow"><span id="MJXc-Node-9632" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-9633" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-9634" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9635" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-9636" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span id="MJXc-Node-9637" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9638" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-9639" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-9640" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9641" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-9642" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span id="MJXc-Node-9643" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9644" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-9645" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-9646" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9647" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-9648" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span id="MJXc-Node-9649" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9650" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-9651" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-9652" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span><span id="MJXc-Node-9653" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9654" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.399em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.978em; top: -1.4em;"><span id="MJXc-Node-9655" class="mjx-mrow" style=""><span id="MJXc-Node-9656" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-9657" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-9658" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.978em; bottom: -0.611em;"><span id="MJXc-Node-9659" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.399em;"></span></span><span class="mjx-vsize" style="height: 1.422em; vertical-align: -0.432em;"></span></span><span id="MJXc-Node-9660" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9661" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mo>=</mo><mi>E</mi><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow><mo>+</mo><mi>E</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mn>0</mn></mrow><mn>2</mn></mfrac><mo>=</mo><mn>1.</mn></math></span></span><script type="math/tex" id="MathJax-Element-782">E\left(
S\right) =E\left( M\right) +E\left( N\right) =\frac{1}{2}+\frac{1+0}{2}=1.</script>
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>30</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Tailles<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2083"> [Signaler une erreur]</a><a id="exo30" href="javascript:ajouteexo(2083,30,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2083&#39;,&#39;imageenonce2083&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2083"></a><div class="inner" id="innerenonce2083">La taille d'un homme âgé de 25 ans suit une loi normale de moyenne 175cm et d'écart-type 6cm.
<ol class="enumeratechiffre">
<li> Quel est le pourcentage d'hommes ayant une taille supérieure à 1m85?
</li><li> Parmi les hommes mesurant plus de 1m80, quelle proportion mesure plus de 1m92?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2083&#39;,&#39;imageindication2083&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2083"></a><div class="inner" id="innerindication2083" style="display:none;"><ol class="enumeratechiffre">
<li> C'est juste un problème de renormalisation de la loi normale.
</li><li>
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2083&#39;,&#39;imagecorrige2083&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2083"></a><div class="inner" id="innercorrige2083" style="display:none;">Comme usuellement, on note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-783-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03D5;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9662" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9663" class="mjx-mrow"><span id="MJXc-Node-9664" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em;">ϕ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math></span></span><script type="math/tex" id="MathJax-Element-783">\phi</script> la fonction de répartition de la loi normale <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-784-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9665" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9666" class="mjx-mrow"><span id="MJXc-Node-9667" class="mjx-texatom"><span id="MJXc-Node-9668" class="mjx-mrow"><span id="MJXc-Node-9669" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.528em; padding-bottom: 0.353em; padding-right: 0.159em;">N</span></span></span></span><span id="MJXc-Node-9670" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9671" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-9672" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-9673" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-9674" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">N</mi></mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-784">\mathcal N(0,1)</script>.
On note aussi <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-785-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;175&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9675" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9676" class="mjx-mrow"><span id="MJXc-Node-9677" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-9678" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9679" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.355em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 3.33em; top: -1.417em;"><span id="MJXc-Node-9680" class="mjx-mrow" style=""><span id="MJXc-Node-9681" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9682" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9683" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">175</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 3.33em; bottom: -0.633em;"><span id="MJXc-Node-9684" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.355em;"></span></span><span class="mjx-vsize" style="height: 1.45em; vertical-align: -0.448em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mfrac><mrow><mi>X</mi><mo>−</mo><mn>175</mn></mrow><mn>6</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-785">Y=\frac{X-175}6</script>, qui suit une loi <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-786-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9685" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9686" class="mjx-mrow"><span id="MJXc-Node-9687" class="mjx-texatom"><span id="MJXc-Node-9688" class="mjx-mrow"><span id="MJXc-Node-9689" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.528em; padding-bottom: 0.353em; padding-right: 0.159em;">N</span></span></span></span><span id="MJXc-Node-9690" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9691" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-9692" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-9693" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-9694" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">N</mi></mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-786">\mathcal N(0,1)</script>.
<ol class="enumeratechiffre">
<li> On cherche <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-787-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;185&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9695" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9696" class="mjx-mrow"><span id="MJXc-Node-9697" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9698" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9699" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9700" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-9701" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">185</span></span><span id="MJXc-Node-9702" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mn>185</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-787">P(X>185)</script> qu'on va calculer en renormalisant la loi normale :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-788-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;185&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;175&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;185&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;175&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03D5;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2243;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;05.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9703" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9704" class="mjx-mrow"><span id="MJXc-Node-9705" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9706" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9707" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9708" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-9709" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">185</span></span><span id="MJXc-Node-9710" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9711" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9712" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9713" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-9714" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-9715" class="mjx-mrow"><span id="MJXc-Node-9716" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.774em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.774em; top: -1.385em;"><span id="MJXc-Node-9717" class="mjx-mrow"><span id="MJXc-Node-9718" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9719" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9720" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">175</span></span></span></span><span class="mjx-denominator" style="width: 3.774em; bottom: -0.733em;"><span id="MJXc-Node-9721" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.774em;"></span></span><span class="mjx-vsize" style="height: 2.117em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-9722" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-9723" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.422em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.422em; top: -1.378em;"><span id="MJXc-Node-9724" class="mjx-mrow"><span id="MJXc-Node-9725" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">185</span></span><span id="MJXc-Node-9726" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9727" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">175</span></span></span></span><span class="mjx-denominator" style="width: 4.422em; bottom: -0.733em;"><span id="MJXc-Node-9728" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.422em;"></span></span><span class="mjx-vsize" style="height: 2.11em; vertical-align: -0.733em;"></span></span></span><span id="MJXc-Node-9729" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span><span id="MJXc-Node-9730" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9731" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9732" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9733" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-9734" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-9735" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-9736" class="mjx-texatom"><span id="MJXc-Node-9737" class="mjx-mrow"><span id="MJXc-Node-9738" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-9739" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-9740" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9741" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9742" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-9743" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9744" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em;">ϕ</span></span><span id="MJXc-Node-9745" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9746" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-9747" class="mjx-texatom"><span id="MJXc-Node-9748" class="mjx-mrow"><span id="MJXc-Node-9749" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-9750" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-9751" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9752" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.295em;">≃</span></span><span id="MJXc-Node-9753" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-9754" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-9755" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">05.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mn>185</mn><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mi>X</mi><mo>−</mo><mn>175</mn></mrow><mn>6</mn></mfrac><mo>&gt;</mo><mfrac><mrow><mn>185</mn><mo>−</mo><mn>175</mn></mrow><mn>6</mn></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>&gt;</mo><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn><mo>,</mo><mn>05.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-788">P(X>185)=P\left(\frac{X-175}6>\frac{185-175}6\right)=P(Y>5/3)=1-\phi(5/3)\simeq 0,05.</script>
</li><li> C'est le même raisonnement, si ce n'est qu'il fait de plus intervenir une probabilité conditionnelle :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-789-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;192&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;180&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;192&lt;/mn&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;180&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;180&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;192&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;180&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9756" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9757" class="mjx-mrow"><span id="MJXc-Node-9758" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9759" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9760" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9761" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-9762" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">192</span></span><span id="MJXc-Node-9763" class="mjx-texatom"><span id="MJXc-Node-9764" class="mjx-mrow"><span id="MJXc-Node-9765" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-9766" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9767" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-9768" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">180</span></span><span id="MJXc-Node-9769" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9770" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9771" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 10.212em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 10.212em; top: -1.542em;"><span id="MJXc-Node-9772" class="mjx-mrow"><span id="MJXc-Node-9773" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9774" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9775" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9776" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-9777" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">192</span></span><span id="MJXc-Node-9778" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-9779" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9780" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-9781" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">180</span></span><span id="MJXc-Node-9782" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 10.212em; bottom: -1.042em;"><span id="MJXc-Node-9783" class="mjx-mrow"><span id="MJXc-Node-9784" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9785" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9786" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9787" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-9788" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">180</span></span><span id="MJXc-Node-9789" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 10.212em;"></span></span><span class="mjx-vsize" style="height: 2.585em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-9790" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9791" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 5.415em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 5.415em; top: -1.542em;"><span id="MJXc-Node-9792" class="mjx-mrow"><span id="MJXc-Node-9793" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9794" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9795" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9796" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-9797" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">192</span></span><span id="MJXc-Node-9798" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 5.415em; bottom: -1.042em;"><span id="MJXc-Node-9799" class="mjx-mrow"><span id="MJXc-Node-9800" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9801" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9802" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9803" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-9804" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">180</span></span><span id="MJXc-Node-9805" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 5.415em;"></span></span><span class="mjx-vsize" style="height: 2.585em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-9806" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mn>192</mn><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>X</mi><mo>&gt;</mo><mn>180</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mn>192</mn><mo>∩</mo><mi>X</mi><mo>&gt;</mo><mn>180</mn><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mn>180</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mn>192</mn><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mn>180</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-789">P(X>192|X>180)=\frac{P(X>192\cap X>180)}{P(X>180)}=\frac{P(X>192)}{P(X>180)}.</script>
On mène les calculs comme précédemment, et on trouve
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-790-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;192&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;180&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03D5;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;17&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03D5;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x2243;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;01.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9807" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9808" class="mjx-mrow"><span id="MJXc-Node-9809" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9810" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9811" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9812" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-9813" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">192</span></span><span id="MJXc-Node-9814" class="mjx-texatom"><span id="MJXc-Node-9815" class="mjx-mrow"><span id="MJXc-Node-9816" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-9817" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9818" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-9819" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">180</span></span><span id="MJXc-Node-9820" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9821" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9822" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 5.296em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 5.296em; top: -1.542em;"><span id="MJXc-Node-9823" class="mjx-mrow"><span id="MJXc-Node-9824" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-9825" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9826" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em;">ϕ</span></span><span id="MJXc-Node-9827" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9828" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">17</span></span><span id="MJXc-Node-9829" class="mjx-texatom"><span id="MJXc-Node-9830" class="mjx-mrow"><span id="MJXc-Node-9831" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-9832" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-9833" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 5.296em; bottom: -1.042em;"><span id="MJXc-Node-9834" class="mjx-mrow"><span id="MJXc-Node-9835" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-9836" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9837" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em;">ϕ</span></span><span id="MJXc-Node-9838" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9839" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-9840" class="mjx-texatom"><span id="MJXc-Node-9841" class="mjx-mrow"><span id="MJXc-Node-9842" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-9843" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-9844" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 5.296em;"></span></span><span class="mjx-vsize" style="height: 2.585em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-9845" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.295em;">≃</span></span><span id="MJXc-Node-9846" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-9847" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-9848" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">01.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><mn>192</mn><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>X</mi><mo>&gt;</mo><mn>180</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mn>17</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>6</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>−</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>6</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>≃</mo><mn>0</mn><mo>,</mo><mn>01.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-790">P(X>192|X>180)=\frac{1-\phi(17/6)}{1-\phi(5/6)}\simeq 0,01.</script>
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>31</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Javelot!<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2352"> [Signaler une erreur]</a><a id="exo31" href="javascript:ajouteexo(2352,31,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2352&#39;,&#39;imageenonce2352&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2352"></a><div class="inner" id="innerenonce2352">On suppose que la distance en mètres parcourues par un javelot lancé par un athlète A suit une loi normale. Au cours d'un entraînement, on constate que
<ul class="rien">
<li> exactement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-791-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x0025;&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9849" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9850" class="mjx-mrow"><span id="MJXc-Node-9851" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">10</span></span><span id="MJXc-Node-9852" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.393em;">%</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-791">10\%</script> des javelots atteignent plus de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-792-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;75&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9853" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9854" class="mjx-mrow"><span id="MJXc-Node-9855" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">75</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>75</mn></math></span></span><script type="math/tex" id="MathJax-Element-792">75</script> mètres.
</li><li> exactement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-793-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;25&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x0025;&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9856" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9857" class="mjx-mrow"><span id="MJXc-Node-9858" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">25</span></span><span id="MJXc-Node-9859" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.393em;">%</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-793">25\%</script> des javelots atteignent moint de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-794-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;50&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9860" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9861" class="mjx-mrow"><span id="MJXc-Node-9862" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">50</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math></span></span><script type="math/tex" id="MathJax-Element-794">50</script> mètres.
</li></ul>
Calculer la longueur moyenne parcourue par un javelot ainsi que l'écart-type de cette longueur.</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2352&#39;,&#39;imageindication2352&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2352"></a><div class="inner" id="innerindication2352" style="display:none;"></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2352&#39;,&#39;imagecorrige2352&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2352"></a><div class="inner" id="innercorrige2352" style="display:none;">Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-795-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9863" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9864" class="mjx-mrow"><span id="MJXc-Node-9865" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-795">X</script> la variable aléatoire égale au nombre de mètres parcourus par un javelot lancé par l'athlète <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-796-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9866" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9867" class="mjx-mrow"><span id="MJXc-Node-9868" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-796">A</script>. L'énoncé nous dit que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-797-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9869" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9870" class="mjx-mrow"><span id="MJXc-Node-9871" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-797">X</script> suit une loi normale de paramètre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-798-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9872" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9873" class="mjx-mrow"><span id="MJXc-Node-9874" class="mjx-texatom"><span id="MJXc-Node-9875" class="mjx-mrow"><span id="MJXc-Node-9876" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.528em; padding-bottom: 0.353em; padding-right: 0.159em;">N</span></span></span></span><span id="MJXc-Node-9877" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9878" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-9879" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-9880" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span><span id="MJXc-Node-9881" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">N</mi></mrow><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-798">\mathcal N(m,\sigma)</script>, avec <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-799-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9882" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9883" class="mjx-mrow"><span id="MJXc-Node-9884" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></span></span><script type="math/tex" id="MathJax-Element-799">m</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-800-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9885" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9886" class="mjx-mrow"><span id="MJXc-Node-9887" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi></math></span></span><script type="math/tex" id="MathJax-Element-800">\sigma</script> à déterminer, et que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-801-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;75&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;50&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;25.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9888" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9889" class="mjx-mrow"><span id="MJXc-Node-9890" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9891" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9892" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9893" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-9894" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">75</span></span><span id="MJXc-Node-9895" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9896" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9897" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-9898" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-9899" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-9900" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-9901" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.289em; padding-bottom: 0.353em;">&nbsp;</span></span><span id="MJXc-Node-9902" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9903" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9904" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9905" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-9906" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">50</span></span><span id="MJXc-Node-9907" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9908" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9909" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-9910" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-9911" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">25.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mn>75</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mtext>&nbsp;</mtext><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mn>50</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>25.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-801">P(X\geq 75)=0,1,\ P(X\leq 50)=0,25.</script>
On va renormaliser ceci pour se ramener à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-802-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9912" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9913" class="mjx-mrow"><span id="MJXc-Node-9914" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-9915" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9916" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-9917" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9918" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9919" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-9920" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-9921" class="mjx-texatom"><span id="MJXc-Node-9922" class="mjx-mrow"><span id="MJXc-Node-9923" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-9924" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>m</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>σ</mi></math></span></span><script type="math/tex" id="MathJax-Element-802">Y=(X-m)/\sigma</script> qui suit une loi normale centrée réduite. On a donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-803-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;75&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;et&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;50&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;25.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9925" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9926" class="mjx-mrow"><span id="MJXc-Node-9927" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9928" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-9929" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-9930" class="mjx-mrow"><span id="MJXc-Node-9931" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.152em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.152em; top: -1.385em;"><span id="MJXc-Node-9932" class="mjx-mrow"><span id="MJXc-Node-9933" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9934" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9935" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span><span class="mjx-denominator" style="width: 3.152em; bottom: -0.722em;"><span id="MJXc-Node-9936" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.152em;"></span></span><span class="mjx-vsize" style="height: 2.106em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-9937" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-9938" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.3em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.3em; top: -1.378em;"><span id="MJXc-Node-9939" class="mjx-mrow"><span id="MJXc-Node-9940" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">75</span></span><span id="MJXc-Node-9941" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9942" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span><span class="mjx-denominator" style="width: 3.3em; bottom: -0.722em;"><span id="MJXc-Node-9943" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.3em;"></span></span><span class="mjx-vsize" style="height: 2.099em; vertical-align: -0.722em;"></span></span></span><span id="MJXc-Node-9944" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span><span id="MJXc-Node-9945" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9946" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-9947" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-9948" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-9949" class="mjx-texatom"><span id="MJXc-Node-9950" class="mjx-mrow"><span id="MJXc-Node-9951" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">&nbsp;et&nbsp;</span></span></span></span><span id="MJXc-Node-9952" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-9953" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-9954" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-9955" class="mjx-mrow"><span id="MJXc-Node-9956" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.152em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.152em; top: -1.385em;"><span id="MJXc-Node-9957" class="mjx-mrow"><span id="MJXc-Node-9958" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-9959" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9960" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span><span class="mjx-denominator" style="width: 3.152em; bottom: -0.722em;"><span id="MJXc-Node-9961" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.152em;"></span></span><span class="mjx-vsize" style="height: 2.106em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-9962" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-9963" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.3em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.3em; top: -1.368em;"><span id="MJXc-Node-9964" class="mjx-mrow"><span id="MJXc-Node-9965" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">50</span></span><span id="MJXc-Node-9966" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9967" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span><span class="mjx-denominator" style="width: 3.3em; bottom: -0.722em;"><span id="MJXc-Node-9968" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.3em;"></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.722em;"></span></span></span><span id="MJXc-Node-9969" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span><span id="MJXc-Node-9970" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-9971" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-9972" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-9973" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">25.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mi>X</mi><mo>−</mo><mi>m</mi></mrow><mi>σ</mi></mfrac><mo>≥</mo><mfrac><mrow><mn>75</mn><mo>−</mo><mi>m</mi></mrow><mi>σ</mi></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;et&nbsp;</mtext></mrow><mi>P</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mi>X</mi><mo>−</mo><mi>m</mi></mrow><mi>σ</mi></mfrac><mo>≤</mo><mfrac><mrow><mn>50</mn><mo>−</mo><mi>m</mi></mrow><mi>σ</mi></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo><mn>25.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-803">P\left(\frac{X-m}{\sigma}\geq \frac{75-m}{\sigma}\right)=0,1\textrm{ et }
P\left(\frac{X-m}{\sigma}\leq \frac{50-m}{\sigma}\right)=0,25.</script>
Notant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-804-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03D5;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-9974" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9975" class="mjx-mrow"><span id="MJXc-Node-9976" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em;">ϕ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math></span></span><script type="math/tex" id="MathJax-Element-804">\phi</script> la fonction de répartition d'une loi normale centrée réduite, on a donc le système
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-805-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03D5;&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;75&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;&amp;#x03D5;&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;50&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;25&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; symmetric=&quot;true&quot;&gt;&lt;/mo&gt;&lt;/mrow&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27FA;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;&amp;#x03D5;&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;75&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;&amp;#x03D5;&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;50&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;25.&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; symmetric=&quot;true&quot;&gt;&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-9977" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9978" class="mjx-mrow"><span id="MJXc-Node-9979" class="mjx-mrow"><span id="MJXc-Node-9980" class="mjx-mo" style="vertical-align: -0.875em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.703em; padding-bottom: 0.295em;">⎧</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.205em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎨</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.205em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="margin-top: -0.231em; padding-bottom: 1.17em;">⎩</span></span></span><span id="MJXc-Node-9981" class="mjx-mtable" style="vertical-align: -1.799em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-9982" class="mjx-mtr" style="height: 2.049em;"><span id="MJXc-Node-9983" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: right; width: 5.939em;"><span id="MJXc-Node-9984" class="mjx-mrow"><span id="MJXc-Node-9985" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-9986" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9987" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em;">ϕ</span></span><span id="MJXc-Node-9988" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-9989" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">(</span></span><span id="MJXc-Node-9990" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.019em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.856em; top: -1.41em;"><span id="MJXc-Node-9991" class="mjx-mrow" style=""><span id="MJXc-Node-9992" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">75</span></span><span id="MJXc-Node-9993" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-9994" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.856em; bottom: -0.524em;"><span id="MJXc-Node-9995" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.019em;"></span></span><span class="mjx-vsize" style="height: 1.368em; vertical-align: -0.37em;"></span></span><span id="MJXc-Node-9996" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">)</span></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-9997" class="mjx-mtd" style="padding: 0px 0.5em; width: 0.778em;"><span id="MJXc-Node-9998" class="mjx-mrow" style="margin-top: 0.175em;"><span id="MJXc-Node-9999" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10000" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 1.945em;"><span id="MJXc-Node-10001" class="mjx-mrow" style="margin-top: 0.175em;"><span id="MJXc-Node-10002" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10003" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10004" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-10005" class="mjx-mtr" style="height: 2.049em;"><span id="MJXc-Node-10006" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: right;"><span id="MJXc-Node-10007" class="mjx-mrow"><span id="MJXc-Node-10008" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em;">ϕ</span></span><span id="MJXc-Node-10009" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-10010" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">(</span></span><span id="MJXc-Node-10011" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.019em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.856em; top: -1.4em;"><span id="MJXc-Node-10012" class="mjx-mrow" style=""><span id="MJXc-Node-10013" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">50</span></span><span id="MJXc-Node-10014" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10015" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.856em; bottom: -0.524em;"><span id="MJXc-Node-10016" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.019em;"></span></span><span class="mjx-vsize" style="height: 1.36em; vertical-align: -0.37em;"></span></span><span id="MJXc-Node-10017" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">)</span></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10018" class="mjx-mtd" style="padding: 0.2em 0.5em 0px;"><span id="MJXc-Node-10019" class="mjx-mrow" style="margin-top: 0.175em;"><span id="MJXc-Node-10020" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10021" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-10022" class="mjx-mrow" style="margin-top: 0.175em;"><span id="MJXc-Node-10023" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10024" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10025" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">25</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-10026" class="mjx-mo" style="width: 0.12em;"></span></span><span id="MJXc-Node-10027" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-10028" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟺</span></span><span id="MJXc-Node-10029" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-10030" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-10031" class="mjx-mo" style="vertical-align: -0.875em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.703em; padding-bottom: 0.295em;">⎧</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.205em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎨</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.205em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="margin-top: -0.231em; padding-bottom: 1.17em;">⎩</span></span></span><span id="MJXc-Node-10032" class="mjx-mtable" style="vertical-align: -1.799em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-10033" class="mjx-mtr" style="height: 2.049em;"><span id="MJXc-Node-10034" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: right; width: 4.216em;"><span id="MJXc-Node-10035" class="mjx-mrow"><span id="MJXc-Node-10036" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em;">ϕ</span></span><span id="MJXc-Node-10037" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-10038" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">(</span></span><span id="MJXc-Node-10039" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.019em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.856em; top: -1.41em;"><span id="MJXc-Node-10040" class="mjx-mrow" style=""><span id="MJXc-Node-10041" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">75</span></span><span id="MJXc-Node-10042" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10043" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.856em; bottom: -0.524em;"><span id="MJXc-Node-10044" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.019em;"></span></span><span class="mjx-vsize" style="height: 1.368em; vertical-align: -0.37em;"></span></span><span id="MJXc-Node-10045" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">)</span></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10046" class="mjx-mtd" style="padding: 0px 0.5em; width: 0.778em;"><span id="MJXc-Node-10047" class="mjx-mrow" style="margin-top: 0.175em;"><span id="MJXc-Node-10048" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10049" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 2.223em;"><span id="MJXc-Node-10050" class="mjx-mrow" style="margin-top: 0.175em;"><span id="MJXc-Node-10051" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10052" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10053" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-10054" class="mjx-mtr" style="height: 2.049em;"><span id="MJXc-Node-10055" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: right;"><span id="MJXc-Node-10056" class="mjx-mrow"><span id="MJXc-Node-10057" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em;">ϕ</span></span><span id="MJXc-Node-10058" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-10059" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">(</span></span><span id="MJXc-Node-10060" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.019em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.856em; top: -1.4em;"><span id="MJXc-Node-10061" class="mjx-mrow" style=""><span id="MJXc-Node-10062" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">50</span></span><span id="MJXc-Node-10063" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10064" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.856em; bottom: -0.524em;"><span id="MJXc-Node-10065" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.019em;"></span></span><span class="mjx-vsize" style="height: 1.36em; vertical-align: -0.37em;"></span></span><span id="MJXc-Node-10066" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">)</span></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10067" class="mjx-mtd" style="padding: 0.2em 0.5em 0px;"><span id="MJXc-Node-10068" class="mjx-mrow" style="margin-top: 0.175em;"><span id="MJXc-Node-10069" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10070" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-10071" class="mjx-mrow" style="margin-top: 0.175em;"><span id="MJXc-Node-10072" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10073" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10074" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">25.</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-10075" class="mjx-mo" style="width: 0.12em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>{</mo><mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>1</mn><mo>−</mo><mi>ϕ</mi><mrow><mo>(</mo><mfrac><mrow><mn>75</mn><mo>−</mo><mi>m</mi></mrow><mi>σ</mi></mfrac><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mn>0</mn><mo>,</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>ϕ</mi><mrow><mo>(</mo><mfrac><mrow><mn>50</mn><mo>−</mo><mi>m</mi></mrow><mi>σ</mi></mfrac><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mn>0</mn><mo>,</mo><mn>25</mn></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi>ϕ</mi><mrow><mo>(</mo><mfrac><mrow><mn>75</mn><mo>−</mo><mi>m</mi></mrow><mi>σ</mi></mfrac><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mn>0</mn><mo>,</mo><mn>9</mn></mtd></mtr><mtr><mtd><mi>ϕ</mi><mrow><mo>(</mo><mfrac><mrow><mn>50</mn><mo>−</mo><mi>m</mi></mrow><mi>σ</mi></mfrac><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mn>0</mn><mo>,</mo><mn>25.</mn></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-805">\left\{
\begin{array}{rcl}
1-\phi\left(\frac{75-m}{\sigma}\right)&=&0,1\\
\phi\left(\frac{50-m}{\sigma}\right)&=&0,25
\end{array}\right.
\iff
\left\{
\begin{array}{rcl}
\phi\left(\frac{75-m}{\sigma}\right)&=&0,9\\
\phi\left(\frac{50-m}{\sigma}\right)&=&0,25.
\end{array}
\right.
</script>
En s'aidant d'un tableur ou d'une table de la loi normale, on en déduit que, à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-806-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10076" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10077" class="mjx-mrow"><span id="MJXc-Node-10078" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-10079" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-10080" class="mjx-texatom" style=""><span id="MJXc-Node-10081" class="mjx-mrow"><span id="MJXc-Node-10082" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10083" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>2</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-806">10^{-2}</script> près, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-807-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;75&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;50&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;67.&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; symmetric=&quot;true&quot;&gt;&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10084" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10085" class="mjx-mrow"><span id="MJXc-Node-10086" class="mjx-mrow"><span id="MJXc-Node-10087" class="mjx-mo"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.52em; padding-bottom: 1.52em;">{</span></span><span id="MJXc-Node-10088" class="mjx-mtable" style="vertical-align: -1.314em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-10089" class="mjx-mtr" style="height: 1.568em;"><span id="MJXc-Node-10090" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: right; width: 2.259em;"><span id="MJXc-Node-10091" class="mjx-mrow" style="margin-top: -0.003em;"><span id="MJXc-Node-10092" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.019em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.856em; top: -1.41em;"><span id="MJXc-Node-10093" class="mjx-mrow" style=""><span id="MJXc-Node-10094" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">75</span></span><span id="MJXc-Node-10095" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10096" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.856em; bottom: -0.524em;"><span id="MJXc-Node-10097" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.019em;"></span></span><span class="mjx-vsize" style="height: 1.368em; vertical-align: -0.37em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10098" class="mjx-mtd" style="padding: 0px 0.5em; width: 0.778em;"><span id="MJXc-Node-10099" class="mjx-mrow" style="margin-top: -0.003em;"><span id="MJXc-Node-10100" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10101" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 3.001em;"><span id="MJXc-Node-10102" class="mjx-mrow" style="margin-top: -0.003em;"><span id="MJXc-Node-10103" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-10104" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10105" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">28</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-10106" class="mjx-mtr" style="height: 1.56em;"><span id="MJXc-Node-10107" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: right;"><span id="MJXc-Node-10108" class="mjx-mrow" style="margin-top: -0.01em;"><span id="MJXc-Node-10109" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.019em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.856em; top: -1.4em;"><span id="MJXc-Node-10110" class="mjx-mrow" style=""><span id="MJXc-Node-10111" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">50</span></span><span id="MJXc-Node-10112" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10113" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.856em; bottom: -0.524em;"><span id="MJXc-Node-10114" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.019em;"></span></span><span class="mjx-vsize" style="height: 1.36em; vertical-align: -0.37em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10115" class="mjx-mtd" style="padding: 0.2em 0.5em 0px;"><span id="MJXc-Node-10116" class="mjx-mrow" style="margin-top: -0.01em;"><span id="MJXc-Node-10117" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10118" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-10119" class="mjx-mrow" style="margin-top: -0.01em;"><span id="MJXc-Node-10120" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10121" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10122" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10123" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">67.</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-10124" class="mjx-mo" style="width: 0.12em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>{</mo><mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mfrac><mrow><mn>75</mn><mo>−</mo><mi>m</mi></mrow><mi>σ</mi></mfrac></mtd><mtd><mo>=</mo></mtd><mtd><mn>1</mn><mo>,</mo><mn>28</mn></mtd></mtr><mtr><mtd><mfrac><mrow><mn>50</mn><mo>−</mo><mi>m</mi></mrow><mi>σ</mi></mfrac></mtd><mtd><mo>=</mo></mtd><mtd><mo>−</mo><mn>0</mn><mo>,</mo><mn>67.</mn></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-807">\left\{\begin{array}{rcl}
\frac{75-m}{\sigma}&=&1,28\\
\frac{50-m}{\sigma}&=&-0,67.
\end{array}
\right.</script>
On résout le système et on trouve <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-808-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;58&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;59&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10125" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10126" class="mjx-mrow"><span id="MJXc-Node-10127" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-10128" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10129" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">58</span></span><span id="MJXc-Node-10130" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10131" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">59</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>58</mn><mo>,</mo><mn>59</mn></math></span></span><script type="math/tex" id="MathJax-Element-808">m=58,59</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-809-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;82&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10132" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10133" class="mjx-mrow"><span id="MJXc-Node-10134" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em; padding-right: 0.001em;">σ</span></span><span id="MJXc-Node-10135" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10136" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">12</span></span><span id="MJXc-Node-10137" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10138" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">82</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mo>=</mo><mn>12</mn><mo>,</mo><mn>82</mn></math></span></span><script type="math/tex" id="MathJax-Element-809">\sigma=12,82</script>. Le javelot parcourt en moyenne 58,59 mètres et l'écart-type est de 12,82.</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>32</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - C'est le pied!<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2181"> [Signaler une erreur]</a><a id="exo32" href="javascript:ajouteexo(2181,32,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2181&#39;,&#39;imageenonce2181&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2181"></a><div class="inner" id="innerenonce2181">On a observé que la longueur d'un pied adulte, en cm, suivait une loi normale <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-810-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10139" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10140" class="mjx-mrow"><span id="MJXc-Node-10141" class="mjx-texatom"><span id="MJXc-Node-10142" class="mjx-mrow"><span id="MJXc-Node-10143" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.522em; padding-bottom: 0.393em; padding-right: 0.159em;">N</span></span></span></span><span id="MJXc-Node-10144" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-10145" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">26</span></span><span id="MJXc-Node-10146" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-10147" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">36</span></span><span id="MJXc-Node-10148" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">N</mi></mrow><mo stretchy="false">(</mo><mn>26</mn><mo>,</mo><mn>36</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-810">\mathcal N(26,36)</script>. Une entreprise décide de fabriquer des chaussettes, en proposant trois tailles.
<ol class="enumeratechiffre">
<li> Déterminer un intervalle centré en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-811-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10149" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10150" class="mjx-mrow"><span id="MJXc-Node-10151" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">26</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>26</mn></math></span></span><script type="math/tex" id="MathJax-Element-811">26</script> qui concentre au moins <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-812-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;95&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x0025;&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10152" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10153" class="mjx-mrow"><span id="MJXc-Node-10154" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">95</span></span><span id="MJXc-Node-10155" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.393em;">%</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>95</mn><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-812">95\%</script> des tailles.
</li><li> Diviser l'intervalle obtenu en trois intervalles égaux, qui détermineront les trois tailles.
</li><li> Déterminer quelle part de production on doit réserver à chacune des tailles.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2181&#39;,&#39;imageindication2181&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2181"></a><div class="inner" id="innerindication2181" style="display:none;"><ol class="enumeratechiffre">
<li> C'est du cours.
</li><li>
</li><li> La production doit être proportionnelle au nombre de pieds présentant cette taille. Attention, il faut que la somme des parts soit égale à 1!!
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2181&#39;,&#39;imagecorrige2181&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2181"></a><div class="inner" id="innercorrige2181" style="display:none;"><ol class="enumeratechiffre">
<li> C'est du cours (rappelons que 36 désigne ici la variance). Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-813-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10156" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10157" class="mjx-mrow"><span id="MJXc-Node-10158" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-813">X</script> désigne la variable aléatoire égale à la longueur d'un pied, on sait que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-814-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;96&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;96&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2261;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;95.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10159" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10160" class="mjx-mrow"><span id="MJXc-Node-10161" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10162" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10163" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">26</span></span><span id="MJXc-Node-10164" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10165" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-10166" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10167" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">96</span></span><span id="MJXc-Node-10168" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-10169" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-10170" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-10171" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10172" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-10173" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">26</span></span><span id="MJXc-Node-10174" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10175" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-10176" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10177" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">96</span></span><span id="MJXc-Node-10178" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-10179" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-10180" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-10181" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.295em;">≡</span></span><span id="MJXc-Node-10182" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10183" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10184" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">95.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mn>26</mn><mo>−</mo><mn>1</mn><mo>,</mo><mn>96</mn><mo>×</mo><mn>6</mn><mo>≤</mo><mi>X</mi><mo>≤</mo><mn>26</mn><mo>+</mo><mn>1</mn><mo>,</mo><mn>96</mn><mo>×</mo><mn>6</mn><mo stretchy="false">)</mo><mo>≡</mo><mn>0</mn><mo>,</mo><mn>95.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-814">P(26-1,96\times 6\leq X\leq 26+1,96\times 6)\equiv 0,95.</script>
Comme on veut un intervalle qui contienne au moins 95\% des tailles, et pour simplifier les calculs dans la suite, on va utiliser l'intervalle <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-815-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;38&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10185" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10186" class="mjx-mrow"><span id="MJXc-Node-10187" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-10188" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">26</span></span><span id="MJXc-Node-10189" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10190" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-10191" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-10192" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-10193" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10194" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">26</span></span><span id="MJXc-Node-10195" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10196" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-10197" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-10198" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-10199" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span><span id="MJXc-Node-10200" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10201" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-10202" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">14</span></span><span id="MJXc-Node-10203" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10204" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">38</span></span><span id="MJXc-Node-10205" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mn>26</mn><mo>−</mo><mn>2</mn><mo>×</mo><mn>6</mn><mo>,</mo><mn>26</mn><mo>+</mo><mn>2</mn><mo>×</mo><mn>6</mn><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mn>14</mn><mo>,</mo><mn>38</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-815">[26-2\times 6,26+2\times 6]=[14,38]</script>.
</li><li> Les trois intervalles sont donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-816-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;22&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10206" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10207" class="mjx-mrow"><span id="MJXc-Node-10208" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-10209" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">14</span></span><span id="MJXc-Node-10210" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10211" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">22</span></span><span id="MJXc-Node-10212" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mn>14</mn><mo>,</mo><mn>22</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-816">[14,22]</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-817-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;22&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10213" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10214" class="mjx-mrow"><span id="MJXc-Node-10215" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-10216" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">22</span></span><span id="MJXc-Node-10217" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10218" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">30</span></span><span id="MJXc-Node-10219" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mn>22</mn><mo>,</mo><mn>30</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-817">[22,30]</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-818-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;38&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10220" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10221" class="mjx-mrow"><span id="MJXc-Node-10222" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">[</span></span><span id="MJXc-Node-10223" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">30</span></span><span id="MJXc-Node-10224" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10225" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">38</span></span><span id="MJXc-Node-10226" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mn>30</mn><mo>,</mo><mn>38</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-818">[30,38]</script>.
</li><li> On veut bien sûr que la production soit proportionnelle au nombre de pieds présentant cette taille. Mais il faut faire attention que l'on ne raisonne pas sur tous les pieds, mais sur les pieds ayant une taille entre 14cm et 38cm. \`A l'aide d'un tableur, on trouve :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-819-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0 thickmathspace&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;22&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;&amp;#x2243;&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;230&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;22&lt;/mn&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;&amp;#x2243;&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;495&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;38&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;&amp;#x2243;&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;230&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;38&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;&amp;#x2243;&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;954.&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10227" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10228" class="mjx-mrow"><span id="MJXc-Node-10229" class="mjx-mtable" style="vertical-align: -2.35em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-10230" class="mjx-mtr" style="height: 1.225em;"><span id="MJXc-Node-10231" class="mjx-mtd" style="padding: 0px; text-align: right; width: 7.048em;"><span id="MJXc-Node-10232" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-10233" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10234" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10235" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">14</span></span><span id="MJXc-Node-10236" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-10237" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10238" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-10239" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">22</span></span><span id="MJXc-Node-10240" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10241" class="mjx-mtd" style="padding: 0px 0.139em 0px 0px; width: 1.056em;"><span id="MJXc-Node-10242" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-10243" class="mjx-mi"></span><span id="MJXc-Node-10244" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.295em;">≃</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10245" class="mjx-mtd" style="padding: 0px 0px 0px 0.139em; text-align: left; width: 2.723em;"><span id="MJXc-Node-10246" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-10247" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10248" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10249" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">230</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-10250" class="mjx-mtr" style="height: 1.375em;"><span id="MJXc-Node-10251" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-10252" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-10253" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10254" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10255" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">22</span></span><span id="MJXc-Node-10256" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-10257" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10258" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-10259" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">30</span></span><span id="MJXc-Node-10260" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10261" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-10262" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-10263" class="mjx-mi"></span><span id="MJXc-Node-10264" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.295em;">≃</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10265" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-10266" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-10267" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10268" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10269" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">495</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-10270" class="mjx-mtr" style="height: 1.375em;"><span id="MJXc-Node-10271" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-10272" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-10273" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10274" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10275" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">30</span></span><span id="MJXc-Node-10276" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-10277" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10278" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-10279" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">38</span></span><span id="MJXc-Node-10280" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10281" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-10282" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-10283" class="mjx-mi"></span><span id="MJXc-Node-10284" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.295em;">≃</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10285" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-10286" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-10287" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10288" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10289" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">230</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-10290" class="mjx-mtr" style="height: 1.225em;"><span id="MJXc-Node-10291" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-10292" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-10293" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10294" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10295" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">14</span></span><span id="MJXc-Node-10296" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-10297" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10298" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-10299" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">38</span></span><span id="MJXc-Node-10300" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10301" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-10302" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-10303" class="mjx-mi"></span><span id="MJXc-Node-10304" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.295em;">≃</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-10305" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-10306" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-10307" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10308" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10309" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">954.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mn>14</mn><mo>≤</mo><mi>X</mi><mo>≤</mo><mn>22</mn><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>≃</mo></mtd><mtd><mn>0</mn><mo>,</mo><mn>230</mn></mtd></mtr><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mn>22</mn><mo>≤</mo><mi>X</mi><mo>≤</mo><mn>30</mn><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>≃</mo></mtd><mtd><mn>0</mn><mo>,</mo><mn>495</mn></mtd></mtr><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mn>30</mn><mo>≤</mo><mi>X</mi><mo>≤</mo><mn>38</mn><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>≃</mo></mtd><mtd><mn>0</mn><mo>,</mo><mn>230</mn></mtd></mtr><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mn>14</mn><mo>≤</mo><mi>X</mi><mo>≤</mo><mn>38</mn><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>≃</mo></mtd><mtd><mn>0</mn><mo>,</mo><mn>954.</mn></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-819">\begin{eqnarray*}
P(14\leq X\leq 22)&\simeq&0,230\\
P(22\leq X\leq 30)&\simeq&0,495\\
P(30\leq X\leq 38)&\simeq&0,230\\
P(14\leq X\leq 38)&\simeq&0,954.
\end{eqnarray*}</script>
La part de production pour la taille 14-22 est donc 0,230/0,954, soit à peu près 24,1\% (de même pour la taille 30-38). La part de production pour la taille 22-30 est 0,495/0,954, soit à peu près 51,9\%.<br>
Un dernier conseil : si un jour vous souhaitez fabriquer des chaussettes, ne croyez pas trop aux données de cet exercice!
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>33</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Somme<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2072"> [Signaler une erreur]</a><a id="exo33" href="javascript:ajouteexo(2072,33,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2072&#39;,&#39;imageenonce2072&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2072"></a><div class="inner" id="innerenonce2072">Un grossiste fournit en viande hachée trois cantines. Il reçoit chaque matin leurs commandes. Ce sont des variables aléatoires indépendantes
suivant des lois normales d'espérance respective 55 kg, 65 kg et 30kg, et d'écart-type respectif 4 kg, 10 kg et 3 kg. Calculer la quantité de viande dont
le grossiste dois disposer pour que le risque de ne pouvoir satisfaire la demande soit inférieur à 5\%.</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2072&#39;,&#39;imageindication2072&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2072"></a><div class="inner" id="innerindication2072" style="display:none;">La somme de trois variables aléatoires indépendantes suivant une loi normale suit une loi normale.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2072&#39;,&#39;imagecorrige2072&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2072"></a><div class="inner" id="innercorrige2072" style="display:none;">Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-820-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10310" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10311" class="mjx-mrow"><span id="MJXc-Node-10312" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-10313" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-10314" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-820">X_i</script> la variable aléatoire correspondant à la quantité commandée par la <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-821-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10315" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10316" class="mjx-mrow"><span id="MJXc-Node-10317" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-821">i</script>-ème cantine. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-822-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10318" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10319" class="mjx-mrow"><span id="MJXc-Node-10320" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-10321" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-10322" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-822">X_1</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-823-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10323" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10324" class="mjx-mrow"><span id="MJXc-Node-10325" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-10326" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-10327" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-823">X_2</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-824-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10328" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10329" class="mjx-mrow"><span id="MJXc-Node-10330" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-10331" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-10332" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-824">X_3</script>
sont indépendantes, suivent des lois normales, avec <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-825-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x223C;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;55&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10333" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10334" class="mjx-mrow"><span id="MJXc-Node-10335" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-10336" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-10337" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-10338" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">∼</span></span><span id="MJXc-Node-10339" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-10340" class="mjx-mrow"><span id="MJXc-Node-10341" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.528em; padding-bottom: 0.353em; padding-right: 0.159em;">N</span></span></span></span><span id="MJXc-Node-10342" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10343" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">55</span></span><span id="MJXc-Node-10344" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10345" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">16</span></span><span id="MJXc-Node-10346" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>1</mn></msub><mo>∼</mo><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">N</mi></mrow><mo stretchy="false">(</mo><mn>55</mn><mo>,</mo><mn>16</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-825">X_1\sim \mathcal N(55,16)</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-826-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x223C;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;65&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;100&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10347" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10348" class="mjx-mrow"><span id="MJXc-Node-10349" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-10350" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-10351" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-10352" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">∼</span></span><span id="MJXc-Node-10353" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-10354" class="mjx-mrow"><span id="MJXc-Node-10355" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.528em; padding-bottom: 0.353em; padding-right: 0.159em;">N</span></span></span></span><span id="MJXc-Node-10356" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10357" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">65</span></span><span id="MJXc-Node-10358" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10359" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">100</span></span><span id="MJXc-Node-10360" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>2</mn></msub><mo>∼</mo><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">N</mi></mrow><mo stretchy="false">(</mo><mn>65</mn><mo>,</mo><mn>100</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-826">X_2\sim \mathcal N(65,100)</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-827-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x223C;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10361" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10362" class="mjx-mrow"><span id="MJXc-Node-10363" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-10364" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-10365" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-10366" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">∼</span></span><span id="MJXc-Node-10367" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-10368" class="mjx-mrow"><span id="MJXc-Node-10369" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.528em; padding-bottom: 0.353em; padding-right: 0.159em;">N</span></span></span></span><span id="MJXc-Node-10370" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10371" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">30</span></span><span id="MJXc-Node-10372" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10373" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-10374" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>3</mn></msub><mo>∼</mo><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">N</mi></mrow><mo stretchy="false">(</mo><mn>30</mn><mo>,</mo><mn>9</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-827">X_3\sim\mathcal N(30,9)</script>.
Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-828-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10375" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10376" class="mjx-mrow"><span id="MJXc-Node-10377" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-828">Y</script> la quantité totale commandée au grossiste, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-829-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10378" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10379" class="mjx-mrow"><span id="MJXc-Node-10380" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-10381" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10382" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-10383" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-10384" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-10385" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10386" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-10387" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-10388" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-10389" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10390" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-10391" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-10392" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><msub><mi>X</mi><mn>2</mn></msub><mo>+</mo><msub><mi>X</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-829">Y=X_1+X_2+X_3</script>. Alors <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-830-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10393" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10394" class="mjx-mrow"><span id="MJXc-Node-10395" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-830">Y</script> suit une loi normale, d'espérance la somme des espérances
et de variance la somme des variances. Autrement dit, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-831-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10396" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10397" class="mjx-mrow"><span id="MJXc-Node-10398" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-831">Y</script> suit une loi <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-832-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;150&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;125&lt;/mn&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10399" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10400" class="mjx-mrow"><span id="MJXc-Node-10401" class="mjx-texatom"><span id="MJXc-Node-10402" class="mjx-mrow"><span id="MJXc-Node-10403" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.528em; padding-bottom: 0.353em; padding-right: 0.159em;">N</span></span></span></span><span id="MJXc-Node-10404" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10405" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">150</span></span><span id="MJXc-Node-10406" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10407" class="mjx-texatom MJXc-space1"><span id="MJXc-Node-10408" class="mjx-mrow"><span id="MJXc-Node-10409" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">125</span></span></span></span><span id="MJXc-Node-10410" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">N</mi></mrow><mo stretchy="false">(</mo><mn>150</mn><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mn>125</mn></mrow><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-832">\mathcal N(150,{125})</script>. Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-833-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10411" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10412" class="mjx-mrow"><span id="MJXc-Node-10413" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-833">A</script> la quantité de viande dont il doit disposer
pour que le risque de ne pouvoir satisfaire la demande soit inférieure à 0,05. On a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-834-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;05&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10414" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10415" class="mjx-mrow"><span id="MJXc-Node-10416" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10417" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10418" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-10419" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-10420" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-10421" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-10422" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-10423" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10424" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10425" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">05</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>&gt;</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≤</mo><mn>0</mn><mo>,</mo><mn>05</mn></math></span></span><script type="math/tex" id="MathJax-Element-834">P(Y>A)\leq 0,05</script>.
Pour déterminer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-835-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10426" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10427" class="mjx-mrow"><span id="MJXc-Node-10428" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-835">A</script> vérifiant cette propriété, on va renormaliser <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-836-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10429" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10430" class="mjx-mrow"><span id="MJXc-Node-10431" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-836">Y</script> et s'aider d'une table de la loi normale centrée réduite.
Posons en effet <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-837-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;150&lt;/mn&gt;&lt;/mrow&gt;&lt;msqrt&gt;&lt;mn&gt;125&lt;/mn&gt;&lt;/msqrt&gt;&lt;/mfrac&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10432" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10433" class="mjx-mrow"><span id="MJXc-Node-10434" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-10435" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10436" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.292em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 3.241em; top: -1.417em;"><span id="MJXc-Node-10437" class="mjx-mrow" style=""><span id="MJXc-Node-10438" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-10439" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10440" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">150</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 3.241em; bottom: -1.005em;"><span id="MJXc-Node-10441" class="mjx-msqrt" style=""><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.119em; border-top: 1px solid;"><span id="MJXc-Node-10442" class="mjx-mrow"><span id="MJXc-Node-10443" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">125</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.292em;"></span></span><span class="mjx-vsize" style="height: 1.713em; vertical-align: -0.711em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo>=</mo><mfrac><mrow><mi>Y</mi><mo>−</mo><mn>150</mn></mrow><msqrt><mn>125</mn></msqrt></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-837">Z=\frac{Y-150}{\sqrt{125}}</script>. Alors <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-838-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10444" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10445" class="mjx-mrow"><span id="MJXc-Node-10446" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-838">Y</script> suit une loi normale centrée réduite. De plus,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-839-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27FA;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;msqrt&gt;&lt;mn&gt;125&lt;/mn&gt;&lt;/msqrt&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;150&lt;/mn&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27FA;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;150&lt;/mn&gt;&lt;/mrow&gt;&lt;msqrt&gt;&lt;mn&gt;125&lt;/mn&gt;&lt;/msqrt&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10447" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10448" class="mjx-mrow"><span id="MJXc-Node-10449" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-10450" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-10451" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-10452" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-10453" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟺</span></span><span id="MJXc-Node-10454" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-10455" class="mjx-msqrt MJXc-space3"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.071em; border-top: 1.3px solid;"><span id="MJXc-Node-10456" class="mjx-mrow"><span id="MJXc-Node-10457" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">125</span></span></span></span></span></span><span id="MJXc-Node-10458" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-10459" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10460" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">150</span></span><span id="MJXc-Node-10461" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-10462" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-10463" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-10464" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟺</span></span><span id="MJXc-Node-10465" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-10466" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-10467" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-10468" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.672em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.672em; top: -1.418em;"><span id="MJXc-Node-10469" class="mjx-mrow"><span id="MJXc-Node-10470" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-10471" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10472" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">150</span></span></span></span><span class="mjx-denominator" style="width: 3.672em; bottom: -1.102em;"><span id="MJXc-Node-10473" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.119em; border-top: 1.3px solid;"><span id="MJXc-Node-10474" class="mjx-mrow"><span id="MJXc-Node-10475" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">125</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.672em;"></span></span><span class="mjx-vsize" style="height: 2.52em; vertical-align: -1.102em;"></span></span><span id="MJXc-Node-10476" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>Y</mi><mo>&gt;</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><msqrt><mn>125</mn></msqrt><mi>Z</mi><mo>+</mo><mn>150</mn><mo>&gt;</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mi>Z</mi><mo>&gt;</mo><mfrac><mrow><mi>A</mi><mo>−</mo><mn>150</mn></mrow><msqrt><mn>125</mn></msqrt></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-839">Y>A\iff \sqrt{125}Z+150>A\iff Z>\frac{A-150}{\sqrt{125}}.</script>
Or, une lecture de la table de la loi normale indique que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-840-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;65&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;05.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10477" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10478" class="mjx-mrow"><span id="MJXc-Node-10479" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10480" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10481" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.04em;">Z</span></span><span id="MJXc-Node-10482" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-10483" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-10484" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10485" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">65</span></span><span id="MJXc-Node-10486" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-10487" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&lt;</span></span><span id="MJXc-Node-10488" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10489" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10490" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">05.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>&gt;</mo><mn>1</mn><mo>,</mo><mn>65</mn><mo stretchy="false">)</mo><mo>&lt;</mo><mn>0</mn><mo>,</mo><mn>05.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-840">P(Z>1,65)<0,05.</script>
Ainsi, il suffit que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-841-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;65&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;msqrt&gt;&lt;mn&gt;125&lt;/mn&gt;&lt;/msqrt&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;150&lt;/mn&gt;&lt;mo&gt;&amp;#x2243;&lt;/mo&gt;&lt;mn&gt;168&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10491" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10492" class="mjx-mrow"><span id="MJXc-Node-10493" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-10494" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-10495" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-10496" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10497" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">65</span></span><span id="MJXc-Node-10498" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-10499" class="mjx-msqrt MJXc-space2"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.071em; border-top: 1.3px solid;"><span id="MJXc-Node-10500" class="mjx-mrow"><span id="MJXc-Node-10501" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">125</span></span></span></span></span></span><span id="MJXc-Node-10502" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10503" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">150</span></span><span id="MJXc-Node-10504" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.295em;">≃</span></span><span id="MJXc-Node-10505" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">168</span></span><span id="MJXc-Node-10506" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10507" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>A</mi><mo>≥</mo><mn>1</mn><mo>,</mo><mn>65</mn><mo>×</mo><msqrt><mn>125</mn></msqrt><mo>+</mo><mn>150</mn><mo>≃</mo><mn>168</mn><mo>,</mo><mn>4.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-841">A\geq 1,65\times\sqrt{125}+150\simeq 168,4.</script>
Le grossiste doit donc disposer d'environ <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-842-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;168&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10508" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10509" class="mjx-mrow"><span id="MJXc-Node-10510" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">168</span></span><span id="MJXc-Node-10511" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10512" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>168</mn><mo>,</mo><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-842">168,4</script> kg de viande.</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>34</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - L'ascenseur<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2298"> [Signaler une erreur]</a><a id="exo34" href="javascript:ajouteexo(2298,34,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2298&#39;,&#39;imageenonce2298&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce2298"></a><div class="inner" id="innerenonce2298">La capacité des ascenseurs est déterminée par le fait que la
masse d’une personne suit une loi normale de moyenne 75 kg et d’écart-type
5 kg. Dans un ascenseur du type WH1 le nombre maximum de personnes est
de 9. Un voyant lumineux affiche qu’il y a surpoids pour une masse supérieure
à 700 kg, dans ce cas l’ascenseur ne démarre pas.
<ol class="enumeratechiffre">
<li> Calculer la probabilité qu’il y ait surpoids, quand un groupe de 9 personnes
monte dans l’ascenseur.
</li><li> Une enquête récente a montré qu’aux USA, le poids moyen est de 76 kg
et l’écart-type de 6 kg. Le building de la World-Company, à New York,
est équipé d’un ascenseur du type WH1. Calculer la probabilité qu’il y
ait surpoids, quand un groupe de 9 personnes monte dans cet ascenseur.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2298&#39;,&#39;imageindication2298&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication2298"></a><div class="inner" id="innerindication2298" style="display:none;">La somme de variables aléatoires suivant une loi normale suit une loi normale.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2298&#39;,&#39;imagecorrige2298&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige2298"></a><div class="inner" id="innercorrige2298" style="display:none;"><ol class="enumeratechiffre">
<li> Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-843-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10513" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10514" class="mjx-mrow"><span id="MJXc-Node-10515" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-10516" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-10517" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-843">X_i</script> le poids de la <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-844-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10518" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10519" class="mjx-mrow"><span id="MJXc-Node-10520" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-844">i</script>-ème personne. Par hypothèse, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-845-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10521" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10522" class="mjx-mrow"><span id="MJXc-Node-10523" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-10524" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-10525" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-845">X_i</script> suit une loi <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-846-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;75&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10526" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10527" class="mjx-mrow"><span id="MJXc-Node-10528" class="mjx-texatom"><span id="MJXc-Node-10529" class="mjx-mrow"><span id="MJXc-Node-10530" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.528em; padding-bottom: 0.353em; padding-right: 0.159em;">N</span></span></span></span><span id="MJXc-Node-10531" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10532" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">75</span></span><span id="MJXc-Node-10533" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10534" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-10535" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">N</mi></mrow><mo stretchy="false">(</mo><mn>75</mn><mo>,</mo><mn>5</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-846">\mathcal N(75,5)</script>.
Si l’on suppose que le poids de chaque personne est indépendant du
poids des autres personnes présentes dans l’ascenseur alors les <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-847-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10536" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10537" class="mjx-mrow"><span id="MJXc-Node-10538" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-10539" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-10540" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-847">X_i</script> sont
indépendants et
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-848-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;&amp;#xB7;&lt;/mo&gt;&lt;mo&gt;&amp;#xB7;&lt;/mo&gt;&lt;mo&gt;&amp;#xB7;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x223C;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot; stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#xD7;&lt;/mo&gt;&lt;mn&gt;75&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msqrt&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/msqrt&gt;&lt;mo&gt;&amp;#xD7;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27F9;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;&amp;#x223C;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;675&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;15&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10541" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10542" class="mjx-mrow"><span id="MJXc-Node-10543" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-10544" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10545" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-10546" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-10547" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-10548" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10549" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋅</span></span><span id="MJXc-Node-10550" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋅</span></span><span id="MJXc-Node-10551" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋅</span></span><span id="MJXc-Node-10552" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-10553" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-10554" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span></span></span><span id="MJXc-Node-10555" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">∼</span></span><span id="MJXc-Node-10556" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-10557" class="mjx-mrow"><span id="MJXc-Node-10558" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span></span></span><span id="MJXc-Node-10559" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-10560" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-10561" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">75</span></span><span id="MJXc-Node-10562" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10563" class="mjx-msqrt MJXc-space1"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">√</span></span><span class="mjx-box" style="padding-top: 0.071em; border-top: 1.3px solid;"><span id="MJXc-Node-10564" class="mjx-mrow"><span id="MJXc-Node-10565" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span></span></span></span></span><span id="MJXc-Node-10566" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-10567" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-10568" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-10569" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-10570" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟹</span></span><span id="MJXc-Node-10571" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-10572" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-10573" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">∼</span></span><span id="MJXc-Node-10574" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-10575" class="mjx-mrow"><span id="MJXc-Node-10576" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.528em; padding-bottom: 0.353em; padding-right: 0.159em;">N</span></span></span></span><span id="MJXc-Node-10577" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10578" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">675</span></span><span id="MJXc-Node-10579" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10580" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">15</span></span><span id="MJXc-Node-10581" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-10582" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>S</mi><mo>=</mo><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><mo>·</mo><mo>·</mo><mo>·</mo><msub><mi>X</mi><mn>9</mn></msub><mo>∼</mo><mrow class="MJX-TeXAtom-ORD"><mo class="MJX-tex-caligraphic" mathvariant="script" stretchy="false">(</mo></mrow><mn>9</mn><mo>×</mo><mn>75</mn><mo>,</mo><msqrt><mn>9</mn></msqrt><mo>×</mo><mn>5</mn><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">⟹</mo><mspace width="thickmathspace"></mspace><mi>S</mi><mo>∼</mo><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">N</mi></mrow><mo stretchy="false">(</mo><mn>675</mn><mo>,</mo><mn>15</mn><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-848">S = X_1 + · · · X_9 \sim \mathcal (9 × 75,
\sqrt 9 × 5)\implies S\sim\mathcal N(675,15).</script>
On cherche à calculer la probabilité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-849-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;700&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10583" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10584" class="mjx-mrow"><span id="MJXc-Node-10585" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10586" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10587" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-10588" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-10589" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">700</span></span><span id="MJXc-Node-10590" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>S</mi><mo>&gt;</mo><mn>700</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-849">P(S > 700)</script>.
Pour cela, on renormalise <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-850-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10591" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10592" class="mjx-mrow"><span id="MJXc-Node-10593" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></span></span><script type="math/tex" id="MathJax-Element-850">S</script> pour se ramener à une loi normale centrée réduite. On a donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-851-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;700&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;675&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;15&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;700&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;675&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;15&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10594" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10595" class="mjx-mrow"><span id="MJXc-Node-10596" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10597" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10598" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-10599" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-10600" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">700</span></span><span id="MJXc-Node-10601" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-10602" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10603" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10604" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-10605" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-10606" class="mjx-mrow"><span id="MJXc-Node-10607" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.567em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.567em; top: -1.407em;"><span id="MJXc-Node-10608" class="mjx-mrow"><span id="MJXc-Node-10609" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-10610" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10611" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">675</span></span></span></span><span class="mjx-denominator" style="width: 3.567em; bottom: -0.733em;"><span id="MJXc-Node-10612" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">15</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.567em;"></span></span><span class="mjx-vsize" style="height: 2.139em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-10613" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-10614" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.422em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.422em; top: -1.378em;"><span id="MJXc-Node-10615" class="mjx-mrow"><span id="MJXc-Node-10616" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">700</span></span><span id="MJXc-Node-10617" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10618" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">675</span></span></span></span><span class="mjx-denominator" style="width: 4.422em; bottom: -0.733em;"><span id="MJXc-Node-10619" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">15</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.422em;"></span></span><span class="mjx-vsize" style="height: 2.11em; vertical-align: -0.733em;"></span></span></span><span id="MJXc-Node-10620" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span><span id="MJXc-Node-10621" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10622" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10623" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10624" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-10625" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-10626" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-10627" class="mjx-texatom"><span id="MJXc-Node-10628" class="mjx-mrow"><span id="MJXc-Node-10629" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-10630" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-10631" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>S</mi><mo>&gt;</mo><mn>700</mn><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mi>S</mi><mo>−</mo><mn>675</mn></mrow><mn>15</mn></mfrac><mo>&gt;</mo><mfrac><mrow><mn>700</mn><mo>−</mo><mn>675</mn></mrow><mn>15</mn></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>&gt;</mo><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-851">P(S>700)=P\left(\frac{S-675}{15}>\frac{700-675}{15}\right)=P(Y>5/3)</script>
où <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-852-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10632" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10633" class="mjx-mrow"><span id="MJXc-Node-10634" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-852">Y</script> suit une loi <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-853-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10635" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10636" class="mjx-mrow"><span id="MJXc-Node-10637" class="mjx-texatom"><span id="MJXc-Node-10638" class="mjx-mrow"><span id="MJXc-Node-10639" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.528em; padding-bottom: 0.353em; padding-right: 0.159em;">N</span></span></span></span><span id="MJXc-Node-10640" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10641" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10642" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10643" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-10644" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">N</mi></mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-853">\mathcal N(0,1)</script>. En notant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-854-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03D5;&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10645" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10646" class="mjx-mrow"><span id="MJXc-Node-10647" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em;">ϕ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math></span></span><script type="math/tex" id="MathJax-Element-854">\phi</script> la fonction de répartition de la loi normale centrée réduite, la probabilité recherchée est
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-855-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03D5;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2243;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;05.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10648" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10649" class="mjx-mrow"><span id="MJXc-Node-10650" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-10651" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10652" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em;">ϕ</span></span><span id="MJXc-Node-10653" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10654" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-10655" class="mjx-texatom"><span id="MJXc-Node-10656" class="mjx-mrow"><span id="MJXc-Node-10657" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-10658" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-10659" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-10660" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.295em;">≃</span></span><span id="MJXc-Node-10661" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10662" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10663" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">05.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>1</mn><mo>−</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn><mo>,</mo><mn>05.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-855">1-\phi(5/3)\simeq 0,05.</script>
</li><li> Un calcul complètement similaire donne cette fois une probabilité de 0,22. On constate donc que malgré un « faible » écart dans les valeurs de départ, on a quadruplé la probabilté de mettre la
machine en panne...
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>35</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Loi jointe uniforme<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1277"> [Signaler une erreur]</a><a id="exo35" href="javascript:ajouteexo(1277,35,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1277&#39;,&#39;imageenonce1277&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce1277"></a><div class="inner" id="innerenonce1277">Soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-856-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10664" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10665" class="mjx-mrow"><span id="MJXc-Node-10666" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-10667" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10668" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-10669" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-10670" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-856">(X,Y)</script> un couple de variables aléatoires suivant une loi uniforme sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-857-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;msup&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10671" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10672" class="mjx-mrow"><span id="MJXc-Node-10673" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">{</span></span><span id="MJXc-Node-10674" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-10675" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-10676" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.328em;">…</span></span><span id="MJXc-Node-10677" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-10678" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-10679" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-10680" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">}</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-10681" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><msup><mo fence="false" stretchy="false">}</mo><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-857">\{0,\dots,n\}^2</script>.
<ol class="enumeratechiffre">
<li> Déterminer la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-858-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10682" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10683" class="mjx-mrow"><span id="MJXc-Node-10684" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-858">X</script>, la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-859-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10685" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10686" class="mjx-mrow"><span id="MJXc-Node-10687" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-859">Y</script>, la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-860-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10688" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10689" class="mjx-mrow"><span id="MJXc-Node-10690" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10691" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-10692" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>+</mo><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-860">X+Y</script>.
</li><li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-861-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10693" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10694" class="mjx-mrow"><span id="MJXc-Node-10695" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-861">X</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-862-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10696" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10697" class="mjx-mrow"><span id="MJXc-Node-10698" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-862">Y</script> sont-elles indépendantes?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1277&#39;,&#39;imageindication1277&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication1277"></a><div class="inner" id="innerindication1277" style="display:none;"><ol class="enumeratechiffre">
<li> Utiliser les méthodes classiques. Pour la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-863-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10699" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10700" class="mjx-mrow"><span id="MJXc-Node-10701" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10702" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10703" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>+</mo><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-863">X+Y</script>, on pourra séparer les cas <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-864-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10704" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10705" class="mjx-mrow"><span id="MJXc-Node-10706" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10707" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10708" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-10709" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-10710" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>+</mo><mi>Y</mi><mo>≤</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-864">X+Y\leq n</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-865-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10711" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10712" class="mjx-mrow"><span id="MJXc-Node-10713" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10714" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10715" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-10716" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-10717" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>+</mo><mi>Y</mi><mo>≥</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-865">X+Y\geq n</script>.
</li><li> Utiliser la définition.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1277&#39;,&#39;imagecorrige1277&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige1277"></a><div class="inner" id="innercorrige1277" style="display:none;"><ol class="enumeratechiffre">
<li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-866-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10718" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10719" class="mjx-mrow"><span id="MJXc-Node-10720" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-866">X</script> est à valeurs dans <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-867-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10721" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10722" class="mjx-mrow"><span id="MJXc-Node-10723" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-10724" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10725" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10726" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-10727" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10728" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-10729" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-867">\{0,\dots,n\}</script>. On a, pour tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-868-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10730" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10731" class="mjx-mrow"><span id="MJXc-Node-10732" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-868">k</script> de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-869-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10733" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10734" class="mjx-mrow"><span id="MJXc-Node-10735" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-10736" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10737" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10738" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-10739" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10740" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-10741" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-869">\{0,\dots,n\}</script>,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-870-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/munderover&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/munderover&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10742" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10743" class="mjx-mrow"><span id="MJXc-Node-10744" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10745" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10746" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10747" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10748" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-10749" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-10750" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10751" class="mjx-munderover MJXc-space3"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.247em; padding-top: 0.141em; padding-left: 0.721em;"><span id="MJXc-Node-10758" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-op"><span id="MJXc-Node-10752" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.21em;"><span id="MJXc-Node-10753" class="mjx-texatom" style=""><span id="MJXc-Node-10754" class="mjx-mrow"><span id="MJXc-Node-10755" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-10756" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10757" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span></span></span></span><span id="MJXc-Node-10759" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10760" class="mjx-texatom"><span id="MJXc-Node-10761" class="mjx-mrow"><span id="MJXc-Node-10762" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-10763" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10764" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10765" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10766" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-10767" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-10768" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10769" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10770" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-10771" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10772" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-10773" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-10774" class="mjx-texatom"><span id="MJXc-Node-10775" class="mjx-mrow"><span id="MJXc-Node-10776" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span><span id="MJXc-Node-10777" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10778" class="mjx-munderover MJXc-space3"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.247em; padding-top: 0.141em; padding-left: 0.721em;"><span id="MJXc-Node-10785" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-op"><span id="MJXc-Node-10779" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.21em;"><span id="MJXc-Node-10780" class="mjx-texatom" style=""><span id="MJXc-Node-10781" class="mjx-mrow"><span id="MJXc-Node-10782" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-10783" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10784" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span></span></span></span><span id="MJXc-Node-10786" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 3.689em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.689em; top: -1.368em;"><span id="MJXc-Node-10787" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 3.689em; bottom: -1.045em;"><span id="MJXc-Node-10788" class="mjx-mrow"><span id="MJXc-Node-10789" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10790" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-10791" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10792" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-10793" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-10794" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-10795" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.689em;"></span></span><span class="mjx-vsize" style="height: 2.412em; vertical-align: -1.045em;"></span></span><span id="MJXc-Node-10796" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10797" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.522em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.522em; top: -1.368em;"><span id="MJXc-Node-10798" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 2.522em; bottom: -0.793em;"><span id="MJXc-Node-10799" class="mjx-mrow"><span id="MJXc-Node-10800" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-10801" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10802" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.522em;"></span></span><span class="mjx-vsize" style="height: 2.16em; vertical-align: -0.793em;"></span></span><span id="MJXc-Node-10803" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>n</mi></munderover><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>n</mi></munderover><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-870">P(X=k)=\sum_{i=0}^n P\big ( (X,Y)=(k,i)\big) =\sum_{i=0}^n \frac{1}{(n+1)^2}=\frac 1{n+1}.</script>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-871-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10804" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10805" class="mjx-mrow"><span id="MJXc-Node-10806" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-871">X</script> suit donc une loi uniforme sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-872-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10807" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10808" class="mjx-mrow"><span id="MJXc-Node-10809" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-10810" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10811" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10812" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-10813" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10814" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-10815" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-872">\{0,\dots,n\}</script>. Par symétrie, il en est de même de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-873-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10816" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10817" class="mjx-mrow"><span id="MJXc-Node-10818" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-873">Y</script>.<br>
La variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-874-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10819" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10820" class="mjx-mrow"><span id="MJXc-Node-10821" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10822" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10823" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>+</mo><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-874">X+Y</script> est à valeurs dans <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-875-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10824" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10825" class="mjx-mrow"><span id="MJXc-Node-10826" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-10827" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-10828" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10829" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-10830" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10831" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-10832" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-10833" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>2</mn><mi>n</mi><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-875">\{0,\dots,2n\}</script> et pour tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-876-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10834" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10835" class="mjx-mrow"><span id="MJXc-Node-10836" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-876">k</script> dans cet intervalle d'entiers, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-877-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/munderover&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10837" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10838" class="mjx-mrow"><span id="MJXc-Node-10839" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10840" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10841" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10842" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10843" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-10844" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10845" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-10846" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-10847" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10848" class="mjx-munderover MJXc-space3"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.247em; padding-top: 0.141em; padding-left: 0.761em;"><span id="MJXc-Node-10855" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span><span class="mjx-op"><span id="MJXc-Node-10849" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.21em;"><span id="MJXc-Node-10850" class="mjx-texatom" style=""><span id="MJXc-Node-10851" class="mjx-mrow"><span id="MJXc-Node-10852" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-10853" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10854" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span></span></span></span><span id="MJXc-Node-10856" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10857" class="mjx-texatom"><span id="MJXc-Node-10858" class="mjx-mrow"><span id="MJXc-Node-10859" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-10860" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10861" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10862" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10863" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-10864" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-10865" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-10866" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10867" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-10868" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10869" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-10870" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-10871" class="mjx-texatom"><span id="MJXc-Node-10872" class="mjx-mrow"><span id="MJXc-Node-10873" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span><span id="MJXc-Node-10874" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>k</mi></munderover><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>i</mi><mo>,</mo><mi>Y</mi><mo>=</mo><mi>k</mi><mo>−</mo><mi>i</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-877">P(X+Y=k)=\sum_{i=0}^k P\big( (X=i,Y=k-i)\big).</script>
Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-878-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10875" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10876" class="mjx-mrow"><span id="MJXc-Node-10877" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-10878" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-10879" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>≤</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-878">k\leq n</script>, ceci devient égal à
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-879-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/munderover&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10880" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10881" class="mjx-mrow"><span id="MJXc-Node-10882" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10883" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10884" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10885" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10886" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-10887" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10888" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-10889" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-10890" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10891" class="mjx-munderover MJXc-space3"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.247em; padding-top: 0.141em; padding-left: 0.761em;"><span id="MJXc-Node-10898" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span><span class="mjx-op"><span id="MJXc-Node-10892" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.21em;"><span id="MJXc-Node-10893" class="mjx-texatom" style=""><span id="MJXc-Node-10894" class="mjx-mrow"><span id="MJXc-Node-10895" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-10896" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10897" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span></span></span></span><span id="MJXc-Node-10899" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 3.689em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.689em; top: -1.368em;"><span id="MJXc-Node-10900" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 3.689em; bottom: -1.045em;"><span id="MJXc-Node-10901" class="mjx-mrow"><span id="MJXc-Node-10902" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10903" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-10904" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10905" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-10906" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-10907" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-10908" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.689em;"></span></span><span class="mjx-vsize" style="height: 2.412em; vertical-align: -1.045em;"></span></span><span id="MJXc-Node-10909" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10910" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.689em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.689em; top: -1.396em;"><span id="MJXc-Node-10911" class="mjx-mrow"><span id="MJXc-Node-10912" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-10913" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10914" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="width: 3.689em; bottom: -1.045em;"><span id="MJXc-Node-10915" class="mjx-mrow"><span id="MJXc-Node-10916" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10917" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-10918" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10919" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-10920" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-10921" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-10922" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.689em;"></span></span><span class="mjx-vsize" style="height: 2.44em; vertical-align: -1.045em;"></span></span><span id="MJXc-Node-10923" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>k</mi></munderover><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-879">P(X+Y=k)=\sum_{i=0}^k \frac{1}{(n+1)^2}=\frac{k+1}{(n+1)^2}.</script>
Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-880-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10924" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10925" class="mjx-mrow"><span id="MJXc-Node-10926" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-10927" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-10928" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>&gt;</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-880">k>n</script>, la somme ne peut aller que jusque <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-881-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10929" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10930" class="mjx-mrow"><span id="MJXc-Node-10931" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-881">n</script> (car <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-882-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10932" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10933" class="mjx-mrow"><span id="MJXc-Node-10934" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-10935" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-10936" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>≤</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-882">i\leq n</script>) et ne peut commencer qu'en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-883-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10937" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10938" class="mjx-mrow"><span id="MJXc-Node-10939" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-10940" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10941" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>−</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-883">k-n</script> (car <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-884-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10942" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10943" class="mjx-mrow"><span id="MJXc-Node-10944" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-10945" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10946" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-10947" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-10948" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>−</mo><mi>i</mi><mo>≤</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-884">k-i\leq n</script>) et donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-885-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/munderover&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10949" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10950" class="mjx-mrow"><span id="MJXc-Node-10951" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-10952" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10953" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-10954" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10955" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-10956" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10957" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-10958" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-10959" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10960" class="mjx-munderover MJXc-space3"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.247em; padding-top: 0.141em; padding-left: 1.211em;"><span id="MJXc-Node-10969" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-op" style="padding-left: 0.346em;"><span id="MJXc-Node-10961" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em;"><span id="MJXc-Node-10962" class="mjx-texatom" style=""><span id="MJXc-Node-10963" class="mjx-mrow"><span id="MJXc-Node-10964" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-10965" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10966" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-10967" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10968" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span></span><span id="MJXc-Node-10970" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 3.689em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.689em; top: -1.368em;"><span id="MJXc-Node-10971" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 3.689em; bottom: -1.045em;"><span id="MJXc-Node-10972" class="mjx-mrow"><span id="MJXc-Node-10973" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10974" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-10975" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10976" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-10977" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-10978" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-10979" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.689em;"></span></span><span class="mjx-vsize" style="height: 2.412em; vertical-align: -1.045em;"></span></span><span id="MJXc-Node-10980" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-10981" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.766em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.766em; top: -1.396em;"><span id="MJXc-Node-10982" class="mjx-mrow"><span id="MJXc-Node-10983" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-10984" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-10985" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-10986" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-10987" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10988" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="width: 4.766em; bottom: -1.045em;"><span id="MJXc-Node-10989" class="mjx-mrow"><span id="MJXc-Node-10990" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-10991" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-10992" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-10993" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-10994" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-10995" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-10996" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.766em;"></span></span><span class="mjx-vsize" style="height: 2.44em; vertical-align: -1.045em;"></span></span><span id="MJXc-Node-10997" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mi>k</mi><mo>−</mo><mi>n</mi></mrow><mi>n</mi></munderover><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-885">P(X+Y=k)=\sum_{i=k-n}^n \frac{1}{(n+1)^2}=\frac{2n-k+1}{(n+1)^2}.</script>
</li><li> Les variables <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-886-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-10998" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-10999" class="mjx-mrow"><span id="MJXc-Node-11000" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-886">X</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-887-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11001" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11002" class="mjx-mrow"><span id="MJXc-Node-11003" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-887">Y</script> sont indépendantes, car pour tout couple <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-888-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11004" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11005" class="mjx-mrow"><span id="MJXc-Node-11006" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11007" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11008" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-11009" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span><span id="MJXc-Node-11010" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-888">(i,j)</script> de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-889-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11011" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11012" class="mjx-mrow"><span id="MJXc-Node-11013" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-11014" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-11015" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-11016" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-11017" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-11018" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11019" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-889">\{0,\dots,n\}</script>, on a bien
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-890-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11020" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11021" class="mjx-mrow"><span id="MJXc-Node-11022" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11023" class="mjx-texatom"><span id="MJXc-Node-11024" class="mjx-mrow"><span id="MJXc-Node-11025" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-11026" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11027" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11028" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11029" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11030" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11031" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-11032" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11033" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11034" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11035" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span><span id="MJXc-Node-11036" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11037" class="mjx-texatom"><span id="MJXc-Node-11038" class="mjx-mrow"><span id="MJXc-Node-11039" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span><span id="MJXc-Node-11040" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11041" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11042" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11043" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11044" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11045" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11046" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11047" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11048" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11049" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11050" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11051" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span><span id="MJXc-Node-11052" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11053" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11054" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.689em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.689em; top: -1.368em;"><span id="MJXc-Node-11055" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 3.689em; bottom: -1.045em;"><span id="MJXc-Node-11056" class="mjx-mrow"><span id="MJXc-Node-11057" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11058" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11059" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-11060" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-11061" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11062" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11063" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.689em;"></span></span><span class="mjx-vsize" style="height: 2.412em; vertical-align: -1.045em;"></span></span><span id="MJXc-Node-11064" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>i</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>j</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>i</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-890">P\big( (X=i),(Y=j)\big)=P(X=i)P(Y=j)=\frac{1}{(n+1)^2}.</script>
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>36</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Couple géométrique<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1282"> [Signaler une erreur]</a><a id="exo36" href="javascript:ajouteexo(1282,36,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1282&#39;,&#39;imageenonce1282&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce1282"></a><div class="inner" id="innerenonce1282">Soient <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-891-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11065" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11066" class="mjx-mrow"><span id="MJXc-Node-11067" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-891">X</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-892-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11068" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11069" class="mjx-mrow"><span id="MJXc-Node-11070" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-892">Y</script> deux variables aléatoires à valeurs dans <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-893-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11071" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11072" class="mjx-mrow"><span id="MJXc-Node-11073" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11074" class="mjx-texatom"><span id="MJXc-Node-11075" class="mjx-mrow"><span id="MJXc-Node-11076" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">N</span></span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.615em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11077" class="mjx-mo" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.198em; padding-bottom: 0.328em;">∗</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow><mo>∗</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-893">\mtn^*</script>, telles que :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-894-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11078" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11079" class="mjx-mrow"><span id="MJXc-Node-11080" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11081" class="mjx-texatom"><span id="MJXc-Node-11082" class="mjx-mrow"><span id="MJXc-Node-11083" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.652em; padding-bottom: 0.652em;">(</span></span></span></span><span id="MJXc-Node-11084" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-11085" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11086" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-11087" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">i</span></span><span id="MJXc-Node-11088" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-11089" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">∩</span></span><span id="MJXc-Node-11090" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-11091" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11092" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-11093" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em;">j</span></span><span id="MJXc-Node-11094" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-11095" class="mjx-texatom"><span id="MJXc-Node-11096" class="mjx-mrow"><span id="MJXc-Node-11097" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.652em; padding-bottom: 0.652em;">)</span></span></span></span><span id="MJXc-Node-11098" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-11099" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.821em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.821em; top: -1.143em;"><span id="MJXc-Node-11100" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span><span class="mjx-denominator" style="width: 1.821em; bottom: -0.941em;"><span id="MJXc-Node-11101" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11102" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11103" class="mjx-texatom" style=""><span id="MJXc-Node-11104" class="mjx-mrow"><span id="MJXc-Node-11105" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">i</span></span><span id="MJXc-Node-11106" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-11107" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em;">j</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.291em; width: 1.821em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.941em;"></span></span><span id="MJXc-Node-11108" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>i</mi><mo stretchy="false">)</mo><mo>∩</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>j</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>=</mo><mfrac><mi>a</mi><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup></mfrac><mo>,</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-894">P\big((X=i)\cap(Y=j)\big)=\frac{a}{2^{i+j}},</script>
pour tous <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-895-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11109" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11110" class="mjx-mrow"><span id="MJXc-Node-11111" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">i</span></span><span id="MJXc-Node-11112" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-11113" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.458em;">j</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>,</mo><mi>j</mi></math></span></span><script type="math/tex" id="MathJax-Element-895">i,j</script> de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-896-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11114" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11115" class="mjx-mrow"><span id="MJXc-Node-11116" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11117" class="mjx-texatom"><span id="MJXc-Node-11118" class="mjx-mrow"><span id="MJXc-Node-11119" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">N</span></span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.615em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11120" class="mjx-mo" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.198em; padding-bottom: 0.328em;">∗</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow><mo>∗</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-896">\mtn^*</script>.
<ol class="enumeratechiffre">
<li> Calculer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-897-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11121" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11122" class="mjx-mrow"><span id="MJXc-Node-11123" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-897">a</script>.
</li><li> Déterminer les lois marginales de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-898-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11124" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11125" class="mjx-mrow"><span id="MJXc-Node-11126" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-898">X</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-899-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11127" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11128" class="mjx-mrow"><span id="MJXc-Node-11129" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-899">Y</script>.
</li><li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-900-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11130" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11131" class="mjx-mrow"><span id="MJXc-Node-11132" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-900">X</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-901-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11133" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11134" class="mjx-mrow"><span id="MJXc-Node-11135" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-901">Y</script> sont-elles indépendantes?
</li></ol>
</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1282&#39;,&#39;imageindication1282&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication1282"></a><div class="inner" id="innerindication1282" style="display:none;"><ol class="enumeratechiffre">
<li> Il faut que la somme sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-902-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11136" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11137" class="mjx-mrow"><span id="MJXc-Node-11138" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-902">i</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-903-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11139" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11140" class="mjx-mrow"><span id="MJXc-Node-11141" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></span></span><script type="math/tex" id="MathJax-Element-903">j</script> fasse 1.
</li><li> Utiliser la définition.
</li><li> Comparer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-904-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11142" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11143" class="mjx-mrow"><span id="MJXc-Node-11144" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11145" class="mjx-texatom"><span id="MJXc-Node-11146" class="mjx-mrow"><span id="MJXc-Node-11147" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-11148" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11149" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11150" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11151" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11152" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11153" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-11154" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11155" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11156" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11157" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span><span id="MJXc-Node-11158" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11159" class="mjx-texatom"><span id="MJXc-Node-11160" class="mjx-mrow"><span id="MJXc-Node-11161" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>i</mi><mo stretchy="false">)</mo><mo>∩</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>j</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></math></span></span><script type="math/tex" id="MathJax-Element-904">P\big((X=i)\cap(Y=j)\big)</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-905-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11162" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11163" class="mjx-mrow"><span id="MJXc-Node-11164" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11165" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11166" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11167" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11168" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11169" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11170" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-11171" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11172" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11173" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11174" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11175" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span><span id="MJXc-Node-11176" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>i</mi><mo stretchy="false">)</mo><mo>×</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>j</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-905">P(X=i)\times P(Y=j)</script>.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1282&#39;,&#39;imagecorrige1282&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige1282"></a><div class="inner" id="innercorrige1282" style="display:none;"><ol class="enumeratechiffre">
<li> Il est d'abord nécessaire que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-906-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11177" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11178" class="mjx-mrow"><span id="MJXc-Node-11179" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-11180" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-11181" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≥</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-906">a\geq 0</script>. On a ensuite :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-907-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;munder&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mfrac&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27FA;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27FA;&lt;/mo&gt;&lt;mspace width=&quot;thickmathspace&quot; /&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1.&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11182" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11183" class="mjx-mrow"><span id="MJXc-Node-11184" class="mjx-munderover"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-op" style="padding-left: 0.096em;"><span id="MJXc-Node-11185" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em;"><span id="MJXc-Node-11186" class="mjx-texatom" style=""><span id="MJXc-Node-11187" class="mjx-mrow"><span id="MJXc-Node-11188" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11189" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-11190" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span><span id="MJXc-Node-11191" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≥</span></span><span id="MJXc-Node-11192" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span id="MJXc-Node-11193" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 1.821em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.821em; top: -1.143em;"><span id="MJXc-Node-11194" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-denominator" style="width: 1.821em; bottom: -0.92em;"><span id="MJXc-Node-11195" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11196" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11197" class="mjx-texatom" style=""><span id="MJXc-Node-11198" class="mjx-mrow"><span id="MJXc-Node-11199" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11200" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-11201" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.821em;"></span></span><span class="mjx-vsize" style="height: 2.063em; vertical-align: -0.92em;"></span></span><span id="MJXc-Node-11202" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11203" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-11204" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-11205" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟺</span></span><span id="MJXc-Node-11206" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-11207" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-11208" class="mjx-munderover MJXc-space1"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.176em; padding-top: 0.141em; padding-left: 0.132em;"><span id="MJXc-Node-11215" class="mjx-texatom" style=""><span id="MJXc-Node-11216" class="mjx-mrow"><span id="MJXc-Node-11217" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-11218" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-op"><span id="MJXc-Node-11209" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.21em;"><span id="MJXc-Node-11210" class="mjx-texatom" style=""><span id="MJXc-Node-11211" class="mjx-mrow"><span id="MJXc-Node-11212" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11213" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11214" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span id="MJXc-Node-11219" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 0.979em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.979em; top: -1.368em;"><span id="MJXc-Node-11220" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.979em; bottom: -0.92em;"><span id="MJXc-Node-11221" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11222" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11223" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.979em;"></span></span><span class="mjx-vsize" style="height: 2.288em; vertical-align: -0.92em;"></span></span><span id="MJXc-Node-11224" class="mjx-munderover MJXc-space1"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.176em; padding-top: 0.141em; padding-left: 0.132em;"><span id="MJXc-Node-11231" class="mjx-texatom" style=""><span id="MJXc-Node-11232" class="mjx-mrow"><span id="MJXc-Node-11233" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-11234" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-op"><span id="MJXc-Node-11225" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.176em;"><span id="MJXc-Node-11226" class="mjx-texatom" style=""><span id="MJXc-Node-11227" class="mjx-mrow"><span id="MJXc-Node-11228" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span><span id="MJXc-Node-11229" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11230" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span id="MJXc-Node-11235" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 1.027em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.027em; top: -1.368em;"><span id="MJXc-Node-11236" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.027em; bottom: -0.92em;"><span id="MJXc-Node-11237" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11238" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11239" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.027em;"></span></span><span class="mjx-vsize" style="height: 2.288em; vertical-align: -0.92em;"></span></span><span id="MJXc-Node-11240" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11241" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-11242" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-11243" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⟺</span></span><span id="MJXc-Node-11244" class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span id="MJXc-Node-11245" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-11246" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11247" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><munder><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi><mo>≥</mo><mn>1</mn></mrow></munder><mfrac><mi>a</mi><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup></mfrac><mo>=</mo><mn>1</mn><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mi>a</mi><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mfrac><mn>1</mn><msup><mn>2</mn><mi>i</mi></msup></mfrac><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mfrac><mn>1</mn><msup><mn>2</mn><mi>j</mi></msup></mfrac><mo>=</mo><mn>1</mn><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mi>a</mi><mo>=</mo><mn>1.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-907">\sum_{i,j\geq 1}\frac{a}{2^{i+j}}=1\iff a\sum_{i=1}^{+\infty}\frac{1}{2^i}\sum_{j=1}^{+\infty}\frac 1{2^j}=1\iff a=1.</script>
</li><li> Pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-908-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11248" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11249" class="mjx-mrow"><span id="MJXc-Node-11250" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11251" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">∈</span></span><span id="MJXc-Node-11252" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-11253" class="mjx-texatom"><span id="MJXc-Node-11254" class="mjx-mrow"><span id="MJXc-Node-11255" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.47em; padding-bottom: 0.295em;">N</span></span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.615em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11256" class="mjx-mo" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.295em;">∗</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>∈</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow><mo>∗</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-908">i\in\mtn^*</script>, on a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-909-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11257" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11258" class="mjx-mrow"><span id="MJXc-Node-11259" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11260" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11261" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11262" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11263" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11264" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11265" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11266" class="mjx-munderover MJXc-space3"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.176em; padding-top: 0.141em; padding-left: 0.132em;"><span id="MJXc-Node-11273" class="mjx-texatom" style=""><span id="MJXc-Node-11274" class="mjx-mrow"><span id="MJXc-Node-11275" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-11276" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-op"><span id="MJXc-Node-11267" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.176em;"><span id="MJXc-Node-11268" class="mjx-texatom" style=""><span id="MJXc-Node-11269" class="mjx-mrow"><span id="MJXc-Node-11270" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span><span id="MJXc-Node-11271" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11272" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span id="MJXc-Node-11277" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11278" class="mjx-texatom"><span id="MJXc-Node-11279" class="mjx-mrow"><span id="MJXc-Node-11280" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-11281" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11282" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11283" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11284" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11285" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11286" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-11287" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11288" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11289" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11290" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span><span id="MJXc-Node-11291" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11292" class="mjx-texatom"><span id="MJXc-Node-11293" class="mjx-mrow"><span id="MJXc-Node-11294" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span><span id="MJXc-Node-11295" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11296" class="mjx-munderover MJXc-space3"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.176em; padding-top: 0.141em; padding-left: 0.132em;"><span id="MJXc-Node-11303" class="mjx-texatom" style=""><span id="MJXc-Node-11304" class="mjx-mrow"><span id="MJXc-Node-11305" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-11306" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-op"><span id="MJXc-Node-11297" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.176em;"><span id="MJXc-Node-11298" class="mjx-texatom" style=""><span id="MJXc-Node-11299" class="mjx-mrow"><span id="MJXc-Node-11300" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span><span id="MJXc-Node-11301" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11302" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span id="MJXc-Node-11307" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 1.821em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.821em; top: -1.368em;"><span id="MJXc-Node-11308" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.821em; bottom: -0.92em;"><span id="MJXc-Node-11309" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11310" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11311" class="mjx-texatom" style=""><span id="MJXc-Node-11312" class="mjx-mrow"><span id="MJXc-Node-11313" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11314" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-11315" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.821em;"></span></span><span class="mjx-vsize" style="height: 2.288em; vertical-align: -0.92em;"></span></span><span id="MJXc-Node-11316" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11317" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.979em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.979em; top: -1.368em;"><span id="MJXc-Node-11318" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.979em; bottom: -0.92em;"><span id="MJXc-Node-11319" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11320" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11321" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.979em;"></span></span><span class="mjx-vsize" style="height: 2.288em; vertical-align: -0.92em;"></span></span><span id="MJXc-Node-11322" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11323" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-11324" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-11325" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-11326" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.883em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.883em; top: -1.368em;"><span id="MJXc-Node-11327" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.883em; bottom: -0.924em;"><span id="MJXc-Node-11328" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11329" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11330" class="mjx-texatom" style=""><span id="MJXc-Node-11331" class="mjx-mrow"><span id="MJXc-Node-11332" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11333" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-11334" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.883em;"></span></span><span class="mjx-vsize" style="height: 2.291em; vertical-align: -0.924em;"></span></span><span id="MJXc-Node-11335" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>i</mi><mo stretchy="false">)</mo><mo>∩</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>j</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mfrac><mn>1</mn><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup></mfrac><mo>=</mo><mfrac><mn>1</mn><msup><mn>2</mn><mi>i</mi></msup></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mn>1</mn><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msup></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-909">P(X=i)=\sum_{j=1}^{+\infty}P\big((X=i)\cap(Y=j)\big)=\sum_{j=1}^{+\infty}\frac{1}{2^{i+j}}=\frac{1}{2^i}=\frac{1}{2}\frac{1}{2^{i-1}}.</script>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-910-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11336" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11337" class="mjx-mrow"><span id="MJXc-Node-11338" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-910">X</script> suit donc une loi géométrique de paramètre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-911-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11339" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11340" class="mjx-mrow"><span id="MJXc-Node-11341" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-11342" class="mjx-texatom"><span id="MJXc-Node-11343" class="mjx-mrow"><span id="MJXc-Node-11344" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-11345" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-911">1/2</script>. Par raison de symétrie, il en est de même pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-912-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11346" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11347" class="mjx-mrow"><span id="MJXc-Node-11348" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-912">Y</script>.
</li><li> On a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-913-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11349" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11350" class="mjx-mrow"><span id="MJXc-Node-11351" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11352" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11353" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11354" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11355" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11356" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11357" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-11358" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11359" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11360" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11361" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11362" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span><span id="MJXc-Node-11363" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11364" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11365" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.821em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.821em; top: -1.368em;"><span id="MJXc-Node-11366" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.821em; bottom: -0.92em;"><span id="MJXc-Node-11367" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11368" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11369" class="mjx-texatom" style=""><span id="MJXc-Node-11370" class="mjx-mrow"><span id="MJXc-Node-11371" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11372" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-11373" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.821em;"></span></span><span class="mjx-vsize" style="height: 2.288em; vertical-align: -0.92em;"></span></span><span id="MJXc-Node-11374" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11375" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11376" class="mjx-texatom"><span id="MJXc-Node-11377" class="mjx-mrow"><span id="MJXc-Node-11378" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-11379" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11380" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11381" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11382" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-11383" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11384" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-11385" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11386" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11387" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11388" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.47em;">j</span></span><span id="MJXc-Node-11389" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11390" class="mjx-texatom"><span id="MJXc-Node-11391" class="mjx-mrow"><span id="MJXc-Node-11392" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span><span id="MJXc-Node-11393" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>i</mi><mo stretchy="false">)</mo><mo>×</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup></mfrac><mo>=</mo><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>i</mi><mo stretchy="false">)</mo><mo>∩</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>j</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-913">P(X=i)\times P(Y=j)=\frac{1}{2^{i+j}}=P\big((X=i)\cap(Y=j)\big).</script>
Les variables aléatoires sont indépendantes.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>37</span> <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Guichet de poste<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1281"> [Signaler une erreur]</a><a id="exo37" href="javascript:ajouteexo(1281,37,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1281&#39;,&#39;imageenonce1281&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/bas.png" id="imageenonce1281"></a><div class="inner" id="innerenonce1281">Dans un bureau de poste, il y a deux guichets. Chacune des personnes arrivant à la poste choisit le premier guichet avec une probabilité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-914-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11394" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11395" class="mjx-mrow"><span id="MJXc-Node-11396" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-914">p</script>,
ou le deuxième guichet avec une probabilité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-915-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11397" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11398" class="mjx-mrow"><span id="MJXc-Node-11399" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em; padding-right: 0.014em;">q</span></span><span id="MJXc-Node-11400" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-11401" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-11402" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-11403" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>1</mn><mo>−</mo><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-915">q=1-p</script>. Les personnes effectuent leur choix de façon indépendante. En une heure, le nombre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-916-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11404" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11405" class="mjx-mrow"><span id="MJXc-Node-11406" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-916">X</script> de personnes arrivés à la poste suit une loi de Poisson <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-917-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi class=&quot;MJX-tex-caligraphic&quot; mathvariant=&quot;script&quot;&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11407" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11408" class="mjx-mrow"><span id="MJXc-Node-11409" class="mjx-texatom"><span id="MJXc-Node-11410" class="mjx-mrow"><span id="MJXc-Node-11411" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.393em; padding-bottom: 0.393em; padding-right: 0.037em;">P</span></span></span></span><span id="MJXc-Node-11412" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-11413" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">m</span></span><span id="MJXc-Node-11414" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-tex-caligraphic" mathvariant="script">P</mi></mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-917">\mathcal{P}(m)</script>.
On désigne par <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-918-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11415" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11416" class="mjx-mrow"><span id="MJXc-Node-11417" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-918">Y</script> le nombre de personnes ayant choisi le premier guichet.
<ol class="enumeratechiffre">
<li> Exprimer la probabilité conditionnelle de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-919-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11418" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11419" class="mjx-mrow"><span id="MJXc-Node-11420" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11421" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-11422" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-919">Y=k</script> sachant que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-920-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11423" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11424" class="mjx-mrow"><span id="MJXc-Node-11425" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11426" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-11427" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-920">X=n</script>.
</li><li> En déduire la loi conjointe du couple <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-921-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11428" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11429" class="mjx-mrow"><span id="MJXc-Node-11430" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-11431" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11432" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-11433" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11434" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-921">(X,Y)</script>.
</li><li> Déterminer la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-922-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11435" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11436" class="mjx-mrow"><span id="MJXc-Node-11437" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-922">Y</script>. On trouvera que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-923-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11438" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11439" class="mjx-mrow"><span id="MJXc-Node-11440" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.182em;">Y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-923">Y</script> suit une loi de Poisson de paramètre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-924-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11441" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11442" class="mjx-mrow"><span id="MJXc-Node-11443" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">m</span></span><span id="MJXc-Node-11444" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-924">mp</script>.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1281&#39;,&#39;imageindication1281&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imageindication1281"></a><div class="inner" id="innerindication1281" style="display:none;"><ol class="enumeratechiffre">
<li> Reconnaitre le schéma d'une loi binomiale. On séparera le cas <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-925-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11445" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11446" class="mjx-mrow"><span id="MJXc-Node-11447" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11448" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-11449" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>≤</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-925">k\leq n</script> du cas <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-926-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11450" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11451" class="mjx-mrow"><span id="MJXc-Node-11452" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11453" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-11454" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>&gt;</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-926">k>n</script>.
</li><li> Utiliser la formule définissant une probabilité conditionnelle. On fera la même séparation de cas.
</li><li> On rappelle que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-927-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;msup&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11455" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11456" class="mjx-mrow"><span id="MJXc-Node-11457" class="mjx-munderover"><span class="mjx-base"><span id="MJXc-Node-11458" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">∑</span></span></span><span class="mjx-stack" style="vertical-align: -0.31em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.323em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11464" class="mjx-texatom" style=""><span id="MJXc-Node-11465" class="mjx-mrow"><span id="MJXc-Node-11466" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-11467" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-11459" class="mjx-texatom" style=""><span id="MJXc-Node-11460" class="mjx-mrow"><span id="MJXc-Node-11461" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11462" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11463" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span></span></span><span id="MJXc-Node-11468" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-11469" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11470" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-11471" class="mjx-texatom"><span id="MJXc-Node-11472" class="mjx-mrow"><span id="MJXc-Node-11473" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-11474" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11475" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-11476" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11477" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-11478" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11479" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><msup><mi>λ</mi><mi>k</mi></msup><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>k</mi><mo>!</mo><mo>=</mo><msup><mi>e</mi><mi>λ</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-927">\sum_{k=0}^{+\infty}\lambda^k/k!=e^\lambda</script>. Il faut juste aller doucement, et faire un changement d'indice dans la somme.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1281&#39;,&#39;imagecorrige1281&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice2 - Bibm@th.net_files/droite.png" id="imagecorrige1281"></a><div class="inner" id="innercorrige1281" style="display:none;"><ol class="enumeratechiffre">
<li> Pour chaque personne, le choix du premier guichet se fait avec une probabilité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-928-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11480" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11481" class="mjx-mrow"><span id="MJXc-Node-11482" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-928">p</script>. Les choix sont indépendants les uns des autres, et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-929-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11483" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11484" class="mjx-mrow"><span id="MJXc-Node-11485" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11486" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11487" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11488" class="mjx-texatom"><span id="MJXc-Node-11489" class="mjx-mrow"><span id="MJXc-Node-11490" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-11491" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11492" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11493" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>X</mi><mo>=</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-929">Y=k|X=n</script> compte le nombre de "succès"
lorsqu'on réalise <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-930-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" style="font-size: 119%; position: relative;" role="presentation"><span id="MJXc-Node-11494" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11495" class="mjx-mrow"><span id="MJXc-Node-11496" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-930">n</script> fois l'épreuve. On reconnait le schéma théorique d'une variable aléatoire de loi binomiale. On a donc :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-931-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable columnalign=&quot;left left&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow&gt;&lt;mtext&gt;si&amp;#xA0;&lt;/mtext&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow&gt;&lt;mtext&gt;si&amp;#xA0;&lt;/mtext&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mtext&gt;.&lt;/mtext&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; symmetric=&quot;true&quot;&gt;&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11497" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11498" class="mjx-mrow"><span id="MJXc-Node-11499" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11500" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11501" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11502" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11503" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11504" class="mjx-texatom"><span id="MJXc-Node-11505" class="mjx-mrow"><span id="MJXc-Node-11506" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-11507" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11508" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11509" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11510" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11511" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11512" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-11513" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">{</span></span><span id="MJXc-Node-11514" class="mjx-mtable" style="vertical-align: -1.075em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-11515" class="mjx-mtr" style="height: 1.449em;"><span id="MJXc-Node-11516" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: left; width: 4.139em;"><span id="MJXc-Node-11517" class="mjx-mrow" style="margin-top: -0.125em;"><span id="MJXc-Node-11518" class="mjx-mrow"><span id="MJXc-Node-11519" class="mjx-TeXmathchoice"><span id="MJXc-Node-11520" class="mjx-texatom"><span id="MJXc-Node-11521" class="mjx-mrow"><span id="MJXc-Node-11522" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-11523" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.424em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.6em; top: -1.095em;"><span id="MJXc-Node-11524" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.6em; bottom: -0.524em;"><span id="MJXc-Node-11525" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="mjx-vsize" style="height: 1.144em; vertical-align: -0.37em;"></span></span><span id="MJXc-Node-11526" class="mjx-TeXmathchoice"><span id="MJXc-Node-11527" class="mjx-texatom"><span id="MJXc-Node-11528" class="mjx-mrow"><span id="MJXc-Node-11529" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-11530" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11531" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11532" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-11533" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.014em;"><span id="MJXc-Node-11534" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.096em; padding-right: 0.071em;"><span id="MJXc-Node-11535" class="mjx-texatom" style=""><span id="MJXc-Node-11536" class="mjx-mrow"><span id="MJXc-Node-11537" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11538" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-11539" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11540" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 5.21em;"><span id="MJXc-Node-11541" class="mjx-mrow" style="margin-top: -0.125em;"><span id="MJXc-Node-11542" class="mjx-texatom"><span id="MJXc-Node-11543" class="mjx-mrow"><span id="MJXc-Node-11544" class="mjx-mrow"><span id="MJXc-Node-11545" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">si&nbsp;</span></span><span id="MJXc-Node-11546" class="mjx-texatom"><span id="MJXc-Node-11547" class="mjx-mrow"><span id="MJXc-Node-11548" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-11549" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-11550" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11551" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-11552" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-11553" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-11554" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: left;"><span id="MJXc-Node-11555" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-11556" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11557" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-11558" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-11559" class="mjx-texatom"><span id="MJXc-Node-11560" class="mjx-mrow"><span id="MJXc-Node-11561" class="mjx-mrow"><span id="MJXc-Node-11562" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">si&nbsp;</span></span><span id="MJXc-Node-11563" class="mjx-texatom"><span id="MJXc-Node-11564" class="mjx-mrow"><span id="MJXc-Node-11565" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11566" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">&gt;</span></span><span id="MJXc-Node-11567" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-11568" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-11569" class="mjx-mo" style="width: 0.12em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>X</mi><mo>=</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mi>n</mi><mi>k</mi></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><msup><mi>p</mi><mi>k</mi></msup><msup><mi>q</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mrow><mtext>si&nbsp;</mtext><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mrow><mtext>si&nbsp;</mtext><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>&gt;</mo><mi>n</mi></mrow><mtext>.</mtext></mrow></mrow></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-931">P(Y=k|X=n)=\left\{
\begin{array}{ll}
\binom nk p^kq^{n-k}&\textrm{si $0\leq k\leq n$}\\
0&\textrm{si $k>n$.}
\end{array}\right.</script>
</li><li> On a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-932-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0 thickmathspace&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable columnalign=&quot;left left&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow&gt;&lt;mtext&gt;si&amp;#xA0;&lt;/mtext&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;sinon.&lt;/mtext&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; symmetric=&quot;true&quot;&gt;&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11570" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11571" class="mjx-mrow"><span id="MJXc-Node-11572" class="mjx-mtable" style="vertical-align: -1.962em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-11573" class="mjx-mtr" style="height: 1.225em;"><span id="MJXc-Node-11574" class="mjx-mtd" style="padding: 0px; text-align: right; width: 7.377em;"><span id="MJXc-Node-11575" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-11576" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11577" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11578" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11579" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11580" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11581" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-11582" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11583" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11584" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11585" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11586" class="mjx-mtd" style="padding: 0px 0.139em 0px 0px; width: 1.056em;"><span id="MJXc-Node-11587" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-11588" class="mjx-mi"></span><span id="MJXc-Node-11589" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11590" class="mjx-mtd" style="padding: 0px 0px 0px 0.139em; text-align: left; width: 12.655em;"><span id="MJXc-Node-11591" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-11592" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11593" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11594" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11595" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11596" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11597" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11598" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11599" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11600" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11601" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11602" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11603" class="mjx-texatom"><span id="MJXc-Node-11604" class="mjx-mrow"><span id="MJXc-Node-11605" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-11606" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11607" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11608" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11609" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-11610" class="mjx-mtr" style="height: 3.199em;"><span id="MJXc-Node-11611" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-11612" class="mjx-mrow" style="margin-top: 0.775em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11613" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-11614" class="mjx-mrow" style="margin-top: 0.775em;"><span id="MJXc-Node-11615" class="mjx-mi"></span><span id="MJXc-Node-11616" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11617" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-11618" class="mjx-mrow"><span id="MJXc-Node-11619" class="mjx-mrow"><span id="MJXc-Node-11620" class="mjx-mo"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.52em; padding-bottom: 1.52em;">{</span></span><span id="MJXc-Node-11621" class="mjx-mtable" style="vertical-align: -1.158em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-11622" class="mjx-mtr" style="height: 1.617em;"><span id="MJXc-Node-11623" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: left; width: 7.139em;"><span id="MJXc-Node-11624" class="mjx-mrow" style="margin-top: -0.059em;"><span id="MJXc-Node-11625" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11626" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11627" class="mjx-texatom" style=""><span id="MJXc-Node-11628" class="mjx-mrow"><span id="MJXc-Node-11629" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-11630" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span></span></span><span id="MJXc-Node-11631" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.087em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.538em; top: -1.331em;"><span id="MJXc-Node-11632" class="mjx-msubsup" style=""><span class="mjx-base"><span id="MJXc-Node-11633" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11634" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.538em; bottom: -0.672em;"><span id="MJXc-Node-11635" class="mjx-mrow" style=""><span id="MJXc-Node-11636" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11637" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.087em;"></span></span><span class="mjx-vsize" style="height: 1.417em; vertical-align: -0.475em;"></span></span><span id="MJXc-Node-11638" class="mjx-mrow"><span id="MJXc-Node-11639" class="mjx-TeXmathchoice"><span id="MJXc-Node-11640" class="mjx-texatom"><span id="MJXc-Node-11641" class="mjx-mrow"><span id="MJXc-Node-11642" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-11643" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.424em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.6em; top: -1.095em;"><span id="MJXc-Node-11644" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.6em; bottom: -0.524em;"><span id="MJXc-Node-11645" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="mjx-vsize" style="height: 1.144em; vertical-align: -0.37em;"></span></span><span id="MJXc-Node-11646" class="mjx-TeXmathchoice"><span id="MJXc-Node-11647" class="mjx-texatom"><span id="MJXc-Node-11648" class="mjx-mrow"><span id="MJXc-Node-11649" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-11650" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11651" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11652" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-11653" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.014em;"><span id="MJXc-Node-11654" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.096em; padding-right: 0.071em;"><span id="MJXc-Node-11655" class="mjx-texatom" style=""><span id="MJXc-Node-11656" class="mjx-mrow"><span id="MJXc-Node-11657" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11658" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-11659" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11660" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 3.377em;"><span id="MJXc-Node-11661" class="mjx-mrow" style="margin-top: -0.059em;"><span id="MJXc-Node-11662" class="mjx-texatom"><span id="MJXc-Node-11663" class="mjx-mrow"><span id="MJXc-Node-11664" class="mjx-mrow"><span id="MJXc-Node-11665" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">si&nbsp;</span></span><span id="MJXc-Node-11666" class="mjx-texatom"><span id="MJXc-Node-11667" class="mjx-mrow"><span id="MJXc-Node-11668" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11669" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.47em;">≤</span></span><span id="MJXc-Node-11670" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-11671" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-11672" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: left;"><span id="MJXc-Node-11673" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-11674" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11675" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-11676" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-11677" class="mjx-texatom"><span id="MJXc-Node-11678" class="mjx-mrow"><span id="MJXc-Node-11679" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">sinon.</span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-11680" class="mjx-mo" style="width: 0.12em;"></span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>k</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>n</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>n</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>X</mi><mo>=</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mrow><mo>{</mo><mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>m</mi></mrow></msup><mfrac><msup><mi>m</mi><mi>n</mi></msup><mrow><mi>n</mi><mo>!</mo></mrow></mfrac><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mi>n</mi><mi>k</mi></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><msup><mi>p</mi><mi>k</mi></msup><msup><mi>q</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mrow><mtext>si&nbsp;</mtext><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>sinon.</mtext></mrow></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-932">\begin{eqnarray*}
P(Y=k,X=n)&=&P(X=n)P(Y=k|X=n)\\
&=&\left\{\begin{array}{ll}
e^{-m}\frac{m^n}{n!}\binom nkp^kq^{n-k}&\textrm{si $k\leq n$}\\
0&\textrm{sinon.}\end{array}\right.
\end{eqnarray*}</script>
</li><li> Il faut réaliser la sommation! On a, tenant compte du fait que les premiers termes sont nuls :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-933-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 119%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0 thickmathspace&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-11681" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11682" class="mjx-mrow"><span id="MJXc-Node-11683" class="mjx-mtable" style="vertical-align: -9.009em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-11684" class="mjx-mtr" style="height: 3.13em;"><span id="MJXc-Node-11685" class="mjx-mtd" style="padding: 0px; text-align: right; width: 4.147em;"><span id="MJXc-Node-11686" class="mjx-mrow" style="margin-top: 0.705em;"><span id="MJXc-Node-11687" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11688" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11689" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11690" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11691" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11692" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11693" class="mjx-mtd" style="padding: 0px 0.139em 0px 0px; width: 1.056em;"><span id="MJXc-Node-11694" class="mjx-mrow" style="margin-top: 0.705em;"><span id="MJXc-Node-11695" class="mjx-mi"></span><span id="MJXc-Node-11696" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11697" class="mjx-mtd" style="padding: 0px 0px 0px 0.139em; text-align: left; width: 13.795em;"><span id="MJXc-Node-11698" class="mjx-mrow"><span id="MJXc-Node-11699" class="mjx-mi"></span><span id="MJXc-Node-11700" class="mjx-munderover MJXc-space1"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.176em; padding-top: 0.141em; padding-left: 0.132em;"><span id="MJXc-Node-11707" class="mjx-texatom" style=""><span id="MJXc-Node-11708" class="mjx-mrow"><span id="MJXc-Node-11709" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-11710" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-op"><span id="MJXc-Node-11701" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.072em;"><span id="MJXc-Node-11702" class="mjx-texatom" style=""><span id="MJXc-Node-11703" class="mjx-mrow"><span id="MJXc-Node-11704" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11705" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11706" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span></span></span><span id="MJXc-Node-11711" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-11712" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11713" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.182em;">Y</span></span><span id="MJXc-Node-11714" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11715" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11716" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-11717" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span><span id="MJXc-Node-11718" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11719" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11720" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-11721" class="mjx-mtr" style="height: 3.286em;"><span id="MJXc-Node-11722" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-11723" class="mjx-mrow" style="margin-top: 0.71em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11724" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-11725" class="mjx-mrow" style="margin-top: 0.71em;"><span id="MJXc-Node-11726" class="mjx-mi"></span><span id="MJXc-Node-11727" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11728" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-11729" class="mjx-mrow"><span id="MJXc-Node-11730" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11731" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11732" class="mjx-texatom" style=""><span id="MJXc-Node-11733" class="mjx-mrow"><span id="MJXc-Node-11734" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-11735" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span></span></span><span id="MJXc-Node-11736" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11737" class="mjx-mrow"><span id="MJXc-Node-11738" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-11739" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.703em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.703em; top: -1.178em;"><span id="MJXc-Node-11740" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-denominator" style="width: 0.703em; bottom: -0.905em;"><span id="MJXc-Node-11741" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.703em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.905em;"></span></span><span id="MJXc-Node-11742" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11743" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-11744" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.999em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.999em; top: -1.368em;"><span id="MJXc-Node-11745" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.999em; bottom: -0.769em;"><span id="MJXc-Node-11746" class="mjx-mrow"><span id="MJXc-Node-11747" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11748" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.999em;"></span></span><span class="mjx-vsize" style="height: 2.137em; vertical-align: -0.769em;"></span></span><span id="MJXc-Node-11749" class="mjx-munderover MJXc-space1"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.176em; padding-top: 0.141em; padding-left: 0.132em;"><span id="MJXc-Node-11756" class="mjx-texatom" style=""><span id="MJXc-Node-11757" class="mjx-mrow"><span id="MJXc-Node-11758" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-11759" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-op"><span id="MJXc-Node-11750" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.072em;"><span id="MJXc-Node-11751" class="mjx-texatom" style=""><span id="MJXc-Node-11752" class="mjx-mrow"><span id="MJXc-Node-11753" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11754" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11755" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span></span></span><span id="MJXc-Node-11760" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 3.599em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.599em; top: -1.542em;"><span id="MJXc-Node-11761" class="mjx-mrow"><span id="MJXc-Node-11762" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11763" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-11764" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span><span id="MJXc-Node-11765" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11766" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11767" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="mjx-denominator" style="width: 3.599em; bottom: -1.042em;"><span id="MJXc-Node-11768" class="mjx-mrow"><span id="MJXc-Node-11769" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11770" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11771" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-11772" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11773" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11774" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.599em;"></span></span><span class="mjx-vsize" style="height: 2.585em; vertical-align: -1.042em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-11775" class="mjx-mtr" style="height: 3.286em;"><span id="MJXc-Node-11776" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-11777" class="mjx-mrow" style="margin-top: 0.71em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11778" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-11779" class="mjx-mrow" style="margin-top: 0.71em;"><span id="MJXc-Node-11780" class="mjx-mi"></span><span id="MJXc-Node-11781" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11782" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-11783" class="mjx-mrow"><span id="MJXc-Node-11784" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11785" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11786" class="mjx-texatom" style=""><span id="MJXc-Node-11787" class="mjx-mrow"><span id="MJXc-Node-11788" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-11789" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span></span></span><span id="MJXc-Node-11790" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11791" class="mjx-mrow"><span id="MJXc-Node-11792" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-11793" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.703em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.703em; top: -1.178em;"><span id="MJXc-Node-11794" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-denominator" style="width: 0.703em; bottom: -0.905em;"><span id="MJXc-Node-11795" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.703em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.905em;"></span></span><span id="MJXc-Node-11796" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11797" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-11798" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.999em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.999em; top: -1.368em;"><span id="MJXc-Node-11799" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.999em; bottom: -0.769em;"><span id="MJXc-Node-11800" class="mjx-mrow"><span id="MJXc-Node-11801" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11802" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.999em;"></span></span><span class="mjx-vsize" style="height: 2.137em; vertical-align: -0.769em;"></span></span><span id="MJXc-Node-11803" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11804" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-11805" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span><span id="MJXc-Node-11806" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11807" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11808" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-11809" class="mjx-munderover MJXc-space1"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.176em; padding-top: 0.141em; padding-left: 0.132em;"><span id="MJXc-Node-11816" class="mjx-texatom" style=""><span id="MJXc-Node-11817" class="mjx-mrow"><span id="MJXc-Node-11818" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-11819" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-op"><span id="MJXc-Node-11810" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.072em;"><span id="MJXc-Node-11811" class="mjx-texatom" style=""><span id="MJXc-Node-11812" class="mjx-mrow"><span id="MJXc-Node-11813" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11814" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11815" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span></span></span><span id="MJXc-Node-11820" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 3.694em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.694em; top: -1.639em;"><span id="MJXc-Node-11821" class="mjx-mrow"><span id="MJXc-Node-11822" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11823" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-11824" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span><span id="MJXc-Node-11825" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11826" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11827" class="mjx-texatom" style=""><span id="MJXc-Node-11828" class="mjx-mrow"><span id="MJXc-Node-11829" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11830" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-11831" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 3.694em; bottom: -1.042em;"><span id="MJXc-Node-11832" class="mjx-mrow"><span id="MJXc-Node-11833" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11834" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11835" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-11836" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11837" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11838" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.694em;"></span></span><span class="mjx-vsize" style="height: 2.681em; vertical-align: -1.042em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-11839" class="mjx-mtr" style="height: 3.274em;"><span id="MJXc-Node-11840" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-11841" class="mjx-mrow" style="margin-top: 0.71em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11842" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-11843" class="mjx-mrow" style="margin-top: 0.71em;"><span id="MJXc-Node-11844" class="mjx-mi"></span><span id="MJXc-Node-11845" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11846" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-11847" class="mjx-mrow"><span id="MJXc-Node-11848" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11849" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11850" class="mjx-texatom" style=""><span id="MJXc-Node-11851" class="mjx-mrow"><span id="MJXc-Node-11852" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-11853" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span></span></span><span id="MJXc-Node-11854" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11855" class="mjx-mrow"><span id="MJXc-Node-11856" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-11857" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.703em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.703em; top: -1.178em;"><span id="MJXc-Node-11858" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-denominator" style="width: 0.703em; bottom: -0.905em;"><span id="MJXc-Node-11859" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.703em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.905em;"></span></span><span id="MJXc-Node-11860" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11861" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-11862" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.999em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.999em; top: -1.368em;"><span id="MJXc-Node-11863" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.999em; bottom: -0.769em;"><span id="MJXc-Node-11864" class="mjx-mrow"><span id="MJXc-Node-11865" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11866" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.999em;"></span></span><span class="mjx-vsize" style="height: 2.137em; vertical-align: -0.769em;"></span></span><span id="MJXc-Node-11867" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11868" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-11869" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span><span id="MJXc-Node-11870" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11871" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11872" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-11873" class="mjx-munderover MJXc-space1"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.176em; padding-top: 0.141em; padding-left: 0.132em;"><span id="MJXc-Node-11880" class="mjx-texatom" style=""><span id="MJXc-Node-11881" class="mjx-mrow"><span id="MJXc-Node-11882" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-11883" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span><span class="mjx-op"><span id="MJXc-Node-11874" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.082em;"><span id="MJXc-Node-11875" class="mjx-texatom" style=""><span id="MJXc-Node-11876" class="mjx-mrow"><span id="MJXc-Node-11877" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11878" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-11879" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span></span></span></span><span id="MJXc-Node-11884" class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 2.776em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.776em; top: -1.542em;"><span id="MJXc-Node-11885" class="mjx-mrow"><span id="MJXc-Node-11886" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11887" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-11888" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span><span id="MJXc-Node-11889" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11890" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11891" class="mjx-texatom" style=""><span id="MJXc-Node-11892" class="mjx-mrow"><span id="MJXc-Node-11893" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 2.776em; bottom: -1.042em;"><span id="MJXc-Node-11894" class="mjx-mrow"><span id="MJXc-Node-11895" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11896" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-11897" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-11898" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.776em;"></span></span><span class="mjx-vsize" style="height: 2.585em; vertical-align: -1.042em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-11899" class="mjx-mtr" style="height: 2.984em;"><span id="MJXc-Node-11900" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-11901" class="mjx-mrow" style="margin-top: 0.71em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11902" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-11903" class="mjx-mrow" style="margin-top: 0.71em;"><span id="MJXc-Node-11904" class="mjx-mi"></span><span id="MJXc-Node-11905" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11906" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-11907" class="mjx-mrow"><span id="MJXc-Node-11908" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11909" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11910" class="mjx-texatom" style=""><span id="MJXc-Node-11911" class="mjx-mrow"><span id="MJXc-Node-11912" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-11913" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span></span></span></span><span id="MJXc-Node-11914" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11915" class="mjx-mrow"><span id="MJXc-Node-11916" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-11917" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.703em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.703em; top: -1.178em;"><span id="MJXc-Node-11918" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-denominator" style="width: 0.703em; bottom: -0.905em;"><span id="MJXc-Node-11919" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.703em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.905em;"></span></span><span id="MJXc-Node-11920" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11921" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-11922" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.999em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.999em; top: -1.368em;"><span id="MJXc-Node-11923" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.999em; bottom: -0.769em;"><span id="MJXc-Node-11924" class="mjx-mrow"><span id="MJXc-Node-11925" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11926" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.999em;"></span></span><span class="mjx-vsize" style="height: 2.137em; vertical-align: -0.769em;"></span></span><span id="MJXc-Node-11927" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11928" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-11929" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span><span id="MJXc-Node-11930" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11931" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11932" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span id="MJXc-Node-11933" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11934" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11935" class="mjx-texatom" style=""><span id="MJXc-Node-11936" class="mjx-mrow"><span id="MJXc-Node-11937" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-11938" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-11939" class="mjx-mtr" style="height: 2.558em;"><span id="MJXc-Node-11940" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-11941" class="mjx-mrow" style="margin-top: 0.639em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11942" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-11943" class="mjx-mrow" style="margin-top: 0.639em;"><span id="MJXc-Node-11944" class="mjx-mi"></span><span id="MJXc-Node-11945" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-11946" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-11947" class="mjx-mrow"><span id="MJXc-Node-11948" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11949" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11950" class="mjx-texatom" style=""><span id="MJXc-Node-11951" class="mjx-mrow"><span id="MJXc-Node-11952" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-11953" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-11954" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span></span></span><span id="MJXc-Node-11955" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.763em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.763em; top: -1.639em;"><span id="MJXc-Node-11956" class="mjx-mrow"><span id="MJXc-Node-11957" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-11958" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-11959" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-11960" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-11961" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-11962" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span class="mjx-denominator" style="width: 2.763em; bottom: -0.769em;"><span id="MJXc-Node-11963" class="mjx-mrow"><span id="MJXc-Node-11964" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-11965" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.763em;"></span></span><span class="mjx-vsize" style="height: 2.408em; vertical-align: -0.769em;"></span></span><span id="MJXc-Node-11966" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi></mi><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>=</mo><mi>k</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>k</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>m</mi></mrow></msup><msup><mrow><mo>(</mo><mfrac><mi>p</mi><mi>q</mi></mfrac><mo>)</mo></mrow><mi>k</mi></msup><mfrac><mn>1</mn><mrow><mi>k</mi><mo>!</mo></mrow></mfrac><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>=</mo><mi>k</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mfrac><mrow><mo stretchy="false">(</mo><mi>m</mi><mi>q</mi><msup><mo stretchy="false">)</mo><mi>n</mi></msup></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>m</mi></mrow></msup><msup><mrow><mo>(</mo><mfrac><mi>p</mi><mi>q</mi></mfrac><mo>)</mo></mrow><mi>k</mi></msup><mfrac><mn>1</mn><mrow><mi>k</mi><mo>!</mo></mrow></mfrac><mo stretchy="false">(</mo><mi>m</mi><mi>q</mi><msup><mo stretchy="false">)</mo><mi>k</mi></msup><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>=</mo><mi>k</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mfrac><mrow><mo stretchy="false">(</mo><mi>m</mi><mi>q</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>m</mi></mrow></msup><msup><mrow><mo>(</mo><mfrac><mi>p</mi><mi>q</mi></mfrac><mo>)</mo></mrow><mi>k</mi></msup><mfrac><mn>1</mn><mrow><mi>k</mi><mo>!</mo></mrow></mfrac><mo stretchy="false">(</mo><mi>m</mi><mi>q</mi><msup><mo stretchy="false">)</mo><mi>k</mi></msup><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mfrac><mrow><mo stretchy="false">(</mo><mi>m</mi><mi>q</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>m</mi></mrow></msup><msup><mrow><mo>(</mo><mfrac><mi>p</mi><mi>q</mi></mfrac><mo>)</mo></mrow><mi>k</mi></msup><mfrac><mn>1</mn><mrow><mi>k</mi><mo>!</mo></mrow></mfrac><mo stretchy="false">(</mo><mi>m</mi><mi>q</mi><msup><mo stretchy="false">)</mo><mi>k</mi></msup><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>q</mi></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>m</mi><mi>p</mi></mrow></msup><mfrac><mrow><mo stretchy="false">(</mo><mi>m</mi><mi>p</mi><msup><mo stretchy="false">)</mo><mi>k</mi></msup></mrow><mrow><mi>k</mi><mo>!</mo></mrow></mfrac><mo>.</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-933">\begin{eqnarray*}
P(Y=k)&=&\sum_{n=k}^{+\infty}P(Y=k,X=n)\\
&=&e^{-m}\left(\frac{p}{q}\right)^k\frac{1}{k!}\sum_{n=k}^{+\infty}\frac{(mq)^n}{(n-k)!}\\
&=&e^{-m}\left(\frac{p}{q}\right)^k\frac{1}{k!}(mq)^k\sum_{n=k}^{+\infty}\frac{(mq)^{n-k}}{(n-k)!}\\
&=&e^{-m}\left(\frac{p}{q}\right)^k\frac{1}{k!}(mq)^k\sum_{n=0}^{+\infty}\frac{(mq)^{n}}{(n)!}\\
&=&e^{-m}\left(\frac{p}{q}\right)^k\frac{1}{k!}(mq)^ke^{mq}\\
&=&e^{-mp}\frac{(mp)^k}{k!}.
\end{eqnarray*}</script>
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