{"id":514,"date":"2013-05-26T20:38:11","date_gmt":"2013-05-26T20:38:11","guid":{"rendered":"http:\/\/frederic-junier.org\/?p=514"},"modified":"2013-06-01T19:55:18","modified_gmt":"2013-06-01T19:55:18","slug":"theoreme-de-moivre-laplace","status":"publish","type":"post","link":"https:\/\/frederic-junier.org\/wp\/?p=514","title":{"rendered":"Th\u00e9or\u00e8me de Moivre-Laplace"},"content":{"rendered":"<p> La correction du dernier DS me laisse penser que certains ont oubli\u00e9 le th\u00e9or\u00e8me de Moivre-Laplace, r\u00e9sultat fondamental de notre programme de probabilit\u00e9. Petit rappel : <\/p>\n<blockquote><p>\nSoit <script type=\"math\/tex\">\\displaystyle p \\in [0;1]<\/script>. On suppose que pour tout entier naturel  <em>n<\/em> non nul, la variable al\u00e9atoire <script type=\"math\/tex\">\\displaystyle X_{n}<\/script> suit la loi binomiale <script type=\"math\/tex\">\\displaystyle \\mathcal{B}\\left(n \\, ; \\, p\\right)<\/script>.<br \/>\nSoit <script type=\"math\/tex\">\\displaystyle Z_{n}=\\frac{X_{n}-np}{\\sqrt{np(1-p)}}<\/script>, la variable centr\u00e9e et r\u00e9duite associ\u00e9e \u00e0 <script type=\"math\/tex\"> \\displaystyle X_{n}<\/script>.<br \/>\nPour tous r\u00e9els <em>a<\/em> et <em>b<\/em> on a :<br \/>\n<script type=\"math\/tex; mode=display\">\\lim_{n \\to +\\infty} \\text{P}\\left( a \\leqslant Z_{n} \\leqslant b \\right)=\\int_{a}^{b} \\frac{1}{\\sqrt{2\\pi}} \\text{e}^{-\\frac{t^2}{2}} \\, dt<\/script>\n<\/p><\/blockquote>\n<p> Les formules ci-dessus ont \u00e9t\u00e9 obtenues avec en utilisant <script type=\"math\/tex\"> \\LaTeX <\/script>  et <a href=\"http:\/\/www.mathjax.org\/\" target=\"_blank\">MathJax<\/a> <\/p>\n<p>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>La correction du dernier DS me laisse penser que certains ont oubli\u00e9 le th\u00e9or\u00e8me de Moivre-Laplace, r\u00e9sultat fondamental de notre programme de probabilit\u00e9. Petit rappel : Soit . On suppose que pour tout entier naturel n non nul, la variable al\u00e9atoire suit la loi binomiale . Soit , la variable centr\u00e9e et r\u00e9duite associ\u00e9e \u00e0 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-514","post","type-post","status-publish","format-standard","hentry","category-blog732"],"_links":{"self":[{"href":"https:\/\/frederic-junier.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/514","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/frederic-junier.org\/wp\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/frederic-junier.org\/wp\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/frederic-junier.org\/wp\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/frederic-junier.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=514"}],"version-history":[{"count":19,"href":"https:\/\/frederic-junier.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/514\/revisions"}],"predecessor-version":[{"id":560,"href":"https:\/\/frederic-junier.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/514\/revisions\/560"}],"wp:attachment":[{"href":"https:\/\/frederic-junier.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=514"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/frederic-junier.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=514"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/frederic-junier.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=514"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}