{"id":215641,"date":"2023-10-06T07:30:00","date_gmt":"2023-10-06T07:30:00","guid":{"rendered":"https:\/\/datascientest.com\/?p=215641"},"modified":"2023-11-09T13:41:39","modified_gmt":"2023-11-09T13:41:39","slug":"theoreme-central-limite-tout-savoir","status":"publish","type":"post","link":"https:\/\/datascientest.com\/theoreme-central-limite-tout-savoir","title":{"rendered":"Le th\u00e9or\u00e8me central limite : Comprendre et savoir l&rsquo;utiliser"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"215641\" class=\"elementor elementor-215641\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b26d478 e-flex e-con-boxed e-con\" data-id=\"b26d478\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-2553bb5 elementor-widget elementor-widget-heading\" data-id=\"2553bb5\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-heading-title{padding:0;margin:0;line-height:1}.elementor-widget-heading .elementor-heading-title[class*=elementor-size-]>a{color:inherit;font-size:inherit;line-height:inherit}.elementor-widget-heading .elementor-heading-title.elementor-size-small{font-size:15px}.elementor-widget-heading .elementor-heading-title.elementor-size-medium{font-size:19px}.elementor-widget-heading .elementor-heading-title.elementor-size-large{font-size:29px}.elementor-widget-heading .elementor-heading-title.elementor-size-xl{font-size:39px}.elementor-widget-heading .elementor-heading-title.elementor-size-xxl{font-size:59px}<\/style><h2 class=\"elementor-heading-title elementor-size-default\">R\u00e9affirmant l\u2019importance de la loi normale, le th\u00e9or\u00e8me central limite figure parmi les concepts incontournables des statistiques.<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ee2fd45 elementor-widget elementor-widget-text-editor\" data-id=\"ee2fd45\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-widget-text-editor.elementor-drop-cap-view-stacked .elementor-drop-cap{background-color:#69727d;color:#fff}.elementor-widget-text-editor.elementor-drop-cap-view-framed .elementor-drop-cap{color:#69727d;border:3px solid;background-color:transparent}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap{margin-top:8px}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap-letter{width:1em;height:1em}.elementor-widget-text-editor .elementor-drop-cap{float:left;text-align:center;line-height:1;font-size:50px}.elementor-widget-text-editor .elementor-drop-cap-letter{display:inline-block}<\/style>\t\t\t\t<p><span style=\"font-weight: 400;\">Mais alors de quoi s\u2019agit-il exactement ? Comment se traduit-il ? Et surtout, comment s\u2019applique-t-il ? C\u2019est ce que nous allons voir dans cet article.\u00a0<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-f81a0cb e-flex e-con-boxed e-con\" data-id=\"f81a0cb\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-b558b80 elementor-widget elementor-widget-heading\" data-id=\"b558b80\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Le th\u00e9or\u00e8me central limite et la convergence vers la loi normale<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-14e8cf2 elementor-widget elementor-widget-text-editor\" data-id=\"14e8cf2\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Le th\u00e9or\u00e8me central limite \u00e9tablit<\/span><b> la convergence d\u2019une suite de variables d\u00e9finies sur le m\u00eame espace vers une loi normale centr\u00e9e r\u00e9duite.<\/b><span style=\"font-weight: 400;\"> Autrement dit : plus on additionne de variables al\u00e9atoires ind\u00e9pendantes et identiquement distribu\u00e9es, plus la distribution de probabilit\u00e9 de la nouvelle variable sera proche d\u2019une distribution normale (\u00e9galement appel\u00e9 <a href=\"https:\/\/datascientest.com\/courbe-gaussienne\">courbe gaussienne<\/a> ou une courbe en cloche). Et <\/span><b>cela vaut pour tout type de distributions de probabilit\u00e9 <\/b><span style=\"font-weight: 400;\">d\u2019un \u00e9v\u00e9nement al\u00e9atoire, comme la distribution uniforme continue, la distribution triangulaire, la distribution exponentielle, etc.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">D\u00e8s lors que cet \u00e9v\u00e9nement est r\u00e9p\u00e9t\u00e9 suffisamment souvent, sa moyenne converge progressivement vers une distribution normale. M\u00eame si au d\u00e9but (apr\u00e8s avoir additionn\u00e9 les variables une ou deux fois), la distribution n\u2019avait rien de normal, elle le devient progressivement \u00e0 mesure que l\u2019addition des variables augmente.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">C\u2019est justement ce principe qui permet aux statisticiens d\u2019affirmer que <\/span><b>la loi normale est la loi des ph\u00e9nom\u00e8nes naturels.\u00a0<\/b><\/p><p><span style=\"font-weight: 400;\"><span style=\"text-decoration: underline;\"><strong>Un peu d\u2019histoire<\/strong><\/span> : Le th\u00e9or\u00e8me central limite a commenc\u00e9 \u00e0 voir le jour d\u00e8s 1718 gr\u00e2ce \u00e0 De Moivre qui a d\u00e9montr\u00e9 l\u2019importance de la loi normale. Mais c\u2019est v\u00e9ritablement avec les travaux de Pierre-Simon de Laplace en 1812 que ce th\u00e9or\u00e8me est v\u00e9ritablement appliqu\u00e9 pour des cas particuliers. Il faudra ensuite 1920 et l\u2019ouvrage de George P\u00f3lya (\u00ab\u00a0<i>Sur le th\u00e9or\u00e8me central du calcul probabiliste, parmi ceux ayant rapport \u00e0 la notion de limite, et le probl\u00e8me des moments\u00a0\u00bb<\/i>) pour lui donner le nom de Th\u00e9or\u00e8me central limite. <\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ac00841 elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"ac00841\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/formation-data-scientist\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">En apprendre plus sur la loi normale<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-88f16be e-flex e-con-boxed e-con\" data-id=\"88f16be\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-3bcdc02 elementor-widget elementor-widget-heading\" data-id=\"3bcdc02\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">L\u2019application math\u00e9matique du th\u00e9or\u00e8me de la limite centrale<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3d1136d elementor-widget elementor-widget-text-editor\" data-id=\"3d1136d\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<span style=\"font-weight: 400;\">Pour calculer le th\u00e9or\u00e8me central limite, il convient de consid\u00e9rer :\u00a0<\/span>\n<ul>\n \t<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\"><em><strong>X<sub>1<\/sub>, X<sub>2<\/sub>, &#8230;<\/strong><\/em> : comme une suite de variables al\u00e9atoires r\u00e9elles ind\u00e9pendantes et identiquement distribu\u00e9es.<\/span><\/li>\n \t<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\"><em><strong>\u03c3 \u2260 0<\/strong><\/em> : l\u2019esp\u00e9rance \u03bc et l&rsquo;\u00e9cart-type \u03c3 de la loi <i>D<\/i> ne sont pas infinis.\u00a0<\/span><\/li>\n \t<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\"><em><strong>S<sub>n<\/sub> = X<sub>1<\/sub>+X<sub>2<\/sub>+X<sub>3 <\/sub>&#8230;<\/strong><\/em>\u00a0: comme la somme de toutes les variables al\u00e9atoires.<\/span><\/li>\n \t<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\"><strong><i>n<\/i><\/strong><span style=\"font-weight: 400;\"><strong>\u03bc<\/strong> : correspond \u00e0 l&rsquo;esp\u00e9rance de <\/span><i><span style=\"font-weight: 400;\">S<\/span><\/i><i><sub>n<\/sub><\/i><\/span><\/li>\n \t<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\"><em><strong>\u03c3\u221an<\/strong><\/em>: correspond \u00e0 l\u2019\u00e9cart-type de <i>S<\/i><i><sub>n<\/sub><\/i><\/span><\/li>\n<\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-155690b elementor-widget elementor-widget-text-editor\" data-id=\"155690b\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Le th\u00e9or\u00e8me central limite stipule : <\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7a43518 elementor-widget elementor-widget-image\" data-id=\"7a43518\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-widget-image{text-align:center}.elementor-widget-image a{display:inline-block}.elementor-widget-image a img[src$=\".svg\"]{width:48px}.elementor-widget-image img{vertical-align:middle;display:inline-block}<\/style>\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"919\" height=\"299\" src=\"https:\/\/datascientest.com\/wp-content\/uploads\/2023\/09\/theoreme-central-limite.jpg\" class=\"attachment-full size-full wp-image-215653\" alt=\"\" srcset=\"https:\/\/datascientest.com\/wp-content\/uploads\/2023\/09\/theoreme-central-limite.jpg 919w, https:\/\/datascientest.com\/wp-content\/uploads\/2023\/09\/theoreme-central-limite-300x98.jpg 300w, https:\/\/datascientest.com\/wp-content\/uploads\/2023\/09\/theoreme-central-limite-768x250.jpg 768w\" sizes=\"(max-width: 919px) 100vw, 919px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-91fc4c1 elementor-widget elementor-widget-text-editor\" data-id=\"91fc4c1\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Ici, l&rsquo;esp\u00e9rance de <i>Z<sub>n<\/sub><\/i><span style=\"font-weight: 400;\">vaut 0 et son \u00e9cart-type vaut 1. Autrement dit, la <a href=\"https:\/\/www.techno-science.net\/glossaire-definition\/Variable.html\">variable<\/a> est <i>centr\u00e9e et r\u00e9duite<\/i>.<\/span><\/span><\/p><p><i><span style=\"font-weight: 400;\"><i>Z<sub>n<\/sub><\/i><\/span><\/i><span style=\"font-weight: 400;\">tend donc vers la loi normale centr\u00e9e r\u00e9duite lorsque <i><span style=\"font-weight: 400;\"><i>n<\/i><\/span><\/i><span style=\"font-weight: 400;\">tend vers l&rsquo;infini<\/span><i><span style=\"font-weight: 400;\">.<\/span><\/i><\/span><\/p><p><span style=\"font-weight: 400;\">Ainsi pour tout r\u00e9el <i>z<\/i>\u00a0:<\/span><\/p><p style=\"padding-left: 360px;\"><strong><em>lim <sub>n\u2192\u221e<\/sub> P (Zn \u2264 z) = \u03a6(z)<\/em><\/strong><\/p><p>o\u00f9 : \u03a6 est la fonction de r\u00e9partition de \u039d(0,1)<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-c1d7e23 e-flex e-con-boxed e-con\" data-id=\"c1d7e23\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-2b95ec6 elementor-widget elementor-widget-heading\" data-id=\"2b95ec6\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Exemple d\u2019application du th\u00e9or\u00e8me central limite<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-dc4e001 elementor-widget elementor-widget-text-editor\" data-id=\"dc4e001\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Pour vous aider \u00e0 mieux comprendre le th\u00e9or\u00e8me de la limite centrale, voici un exemple concret.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">Marine et Julien jouent aux d\u00e9s. \u00c0 chaque fois qu\u2019ils tombent sur pair, ils gagnent 1 \u20ac, et \u00e0 chaque fois qu\u2019ils tombent sur impair, ils perdent 1 \u20ac. Dans ce cas, les nombres pairs et impairs sont des variables al\u00e9atoires identiquement distribu\u00e9es, puisque Marine et Julien ont autant de chance de gagner que de perdre. Au cours des premi\u00e8res parties, il se peut que la balance penche davantage du c\u00f4t\u00e9 des nombres pairs (et donc des gains) ou des nombres impairs (et donc des pertes). Mais au fur et \u00e0 mesure qu\u2019ils jouent, leur probabilit\u00e9 de gain ou de perte est proche de 0.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">Ce th\u00e9or\u00e8me central limite se retrouve dans de nombreux cas de la vie courante. Par exemple, la taille et le poids des individus au sein d\u2019une population, la distribution des salaires au sein d\u2019une entreprise, \u2026 La moyenne de ces r\u00e9sultats\u00a0 finit toujours par ressembler \u00e0 une courbe en cloche \u00e0 mesure qu\u2019on \u00e9largit l\u2019\u00e9chantillon de la population,\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">Du fait de la grande vari\u00e9t\u00e9 de ces applications, le th\u00e9or\u00e8me central limite est un <\/span><b>outil statistique incontournable pour les data scientists,<\/b><span style=\"font-weight: 400;\"> puisqu\u2019il leur permet de mod\u00e9liser des <a href=\"https:\/\/datascientest.com\/machine-learning-tout-savoir\">mod\u00e8les de Machine Learning<\/a>. En plus des outils de science des donn\u00e9es, il convient donc de ma\u00eetriser les <a href=\"https:\/\/datascientest.com\/top-10-des-pre-requis-mathematiques-importants-en-data-science\">concepts math\u00e9matiques<\/a>. C\u2019est justement pour cette raison qu\u2019une formation s\u2019av\u00e8re indispensable. <a href=\"https:\/\/datascientest.com\/formation-data-scientist\">D\u00e9couvrez nos programmes<\/a>.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6258ced elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"6258ced\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/formation-data-scientist\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Ma\u00eetriser le th\u00e9or\u00e8me central limite<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-4cc7203 e-flex e-con-boxed e-con\" data-id=\"4cc7203\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-3c67b68 elementor-widget elementor-widget-heading\" data-id=\"3c67b68\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Ce qu\u2019il faut retenir<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-75a692d elementor-widget elementor-widget-text-editor\" data-id=\"75a692d\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Les variables al\u00e9atoires identiquement distribu\u00e9es suivent une loi normale \u00e0 mesure qu\u2019elles sont additionn\u00e9es.\u00a0<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">De par cette affirmation, le th\u00e9or\u00e8me central limite souligne l\u2019importance de la loi normale.\u00a0<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">D\u2019ailleurs, ce th\u00e9or\u00e8me s\u2019applique pour une grande vari\u00e9t\u00e9 d&rsquo;hypoth\u00e8ses ; depuis un jeu de hasard, jusqu\u2019\u00e0 la d\u00e9termination du salaire moyen ou la taille de la population.\u00a0<\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-aabeef6 elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"aabeef6\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"\/formations-data\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">D\u00e9couvrir nos formations<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-2d4f80f e-flex e-con-boxed e-con\" data-id=\"2d4f80f\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-44a9a62 elementor-widget elementor-widget-html\" data-id=\"44a9a62\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"FAQPage\",\r\n  \"mainEntity\": [{\r\n    \"@type\": \"Question\",\r\n    \"name\": \"Le th\u00e9or\u00e8me central limite et la convergence vers la loi normale\",\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"Le th\u00e9or\u00e8me central limite \u00e9tablit la convergence d\u2019une suite de variables d\u00e9finies sur le m\u00eame espace vers une loi normale centr\u00e9e r\u00e9duite. Autrement dit : plus on additionne de variables al\u00e9atoires ind\u00e9pendantes et identiquement distribu\u00e9es, plus la distribution de probabilit\u00e9 de la nouvelle variable sera proche d\u2019une distribution normale (\u00e9galement appel\u00e9 courbe gaussienne ou une courbe en cloche). Et cela vaut pour tout type de distributions de probabilit\u00e9 d\u2019un \u00e9v\u00e9nement al\u00e9atoire, comme la distribution uniforme continue, la distribution triangulaire, la distribution exponentielle, etc.\"\r\n    }\r\n  },{\r\n    \"@type\": \"Question\",\r\n    \"name\": \"Exemple d\u2019application du th\u00e9or\u00e8me central limite\",\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"Pour vous aider \u00e0 mieux comprendre le th\u00e9or\u00e8me de la limite centrale, voici un exemple concret. Marine et Julien jouent aux d\u00e9s. \u00c0 chaque fois qu\u2019ils tombent sur pair, ils gagnent 1 \u20ac, et \u00e0 chaque fois qu\u2019ils tombent sur impair, ils perdent 1 \u20ac. Dans ce cas, les nombres pairs et impairs sont des variables al\u00e9atoires identiquement distribu\u00e9es, puisque Marine et Julien ont autant de chance de gagner que de perdre. Au cours des premi\u00e8res parties, il se peut que la balance penche davantage du c\u00f4t\u00e9 des nombres pairs (et donc des gains) ou des nombres impairs (et donc des pertes). Mais au fur et \u00e0 mesure qu\u2019ils jouent, leur probabilit\u00e9 de gain ou de perte est proche de 0.\"\r\n    }\r\n  }]\r\n}\r\n<\/script>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>R\u00e9affirmant l\u2019importance de la loi normale, le th\u00e9or\u00e8me central limite figure parmi les concepts incontournables des statistiques. Mais alors de quoi s\u2019agit-il exactement ? Comment se traduit-il ? Et surtout, comment s\u2019applique-t-il ? C\u2019est ce que nous allons voir dans cet article.&nbsp; Le th\u00e9or\u00e8me central limite et la convergence vers la loi normale Le th\u00e9or\u00e8me [&hellip;]<\/p>\n","protected":false},"author":77,"featured_media":215643,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_theme","format":"standard","meta":{"content-type":"","inline_featured_image":false},"categories":[2497],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Le th\u00e9or\u00e8me central limite : Comprendre et savoir l&#039;utiliser<\/title>\n<meta name=\"description\" content=\"Le th\u00e9or\u00e8me central limite d\u00e9montre la convergence progressive d\u2019une suite de variables vers la loi normale.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/datascientest.com\/theoreme-central-limite-tout-savoir\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Le th\u00e9or\u00e8me central limite : Comprendre et savoir l&#039;utiliser\" \/>\n<meta property=\"og:url\" content=\"https:\/\/datascientest.com\/theoreme-central-limite-tout-savoir\" \/>\n<meta property=\"og:site_name\" content=\"Formation Data Science | DataScientest.com\" \/>\n<meta property=\"article:published_time\" content=\"2023-10-06T07:30:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-11-09T13:41:39+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/datascientest.com\/wp-content\/uploads\/2023\/09\/theoreme-central-limite.png\" \/>\n\t<meta property=\"og:image:width\" content=\"1200\" \/>\n\t<meta property=\"og:image:height\" content=\"600\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"author\" content=\"Raphael Kassel\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"Raphael Kassel\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/datascientest.com\/theoreme-central-limite-tout-savoir\",\"url\":\"https:\/\/datascientest.com\/theoreme-central-limite-tout-savoir\",\"name\":\"Le th\u00e9or\u00e8me central limite : Comprendre et savoir l'utiliser\",\"isPartOf\":{\"@id\":\"https:\/\/datascientest.com\/#website\"},\"datePublished\":\"2023-10-06T07:30:00+00:00\",\"dateModified\":\"2023-11-09T13:41:39+00:00\",\"author\":{\"@id\":\"https:\/\/datascientest.com\/#\/schema\/person\/6eca16f63daaafcbc96e15549b9d4ee1\"},\"description\":\"Le th\u00e9or\u00e8me central limite d\u00e9montre la convergence progressive d\u2019une suite de variables vers la loi normale.\",\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/datascientest.com\/theoreme-central-limite-tout-savoir\"]}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/datascientest.com\/#website\",\"url\":\"https:\/\/datascientest.com\/\",\"name\":\"Formation Data Science | DataScientest.com\",\"description\":\"Leader de la formation en Data Science\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/datascientest.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"fr-FR\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/datascientest.com\/#\/schema\/person\/6eca16f63daaafcbc96e15549b9d4ee1\",\"name\":\"Raphael Kassel\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\/\/datascientest.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/datascientest.com\/wp-content\/uploads\/2023\/11\/raph.jpeg\",\"contentUrl\":\"https:\/\/datascientest.com\/wp-content\/uploads\/2023\/11\/raph.jpeg\",\"caption\":\"Raphael Kassel\"},\"description\":\"Raphael Kassel est le Head of Data Programs chez DataScientest. Il supervise une \u00e9quipe Data, impl\u00e9mente de nouveaux parcours comme \\\"Chef de projet IA\\\" et \\\"Data Product Manager\\\", et contribue \u00e0 la modernisation des op\u00e9rations du groupe. En collaboration \u00e9troite avec l'\u00e9quipe de d\u00e9veloppement, il adapte l'accompagnement acad\u00e9mique aux \u00e9volutions de l'\u00e9quipe data et \u00e0 l'internationalisation.\",\"sameAs\":[\"https:\/\/www.linkedin.com\/in\/raphaelkassel\/\",\"Data Science\"],\"url\":\"https:\/\/datascientest.com\/author\/raphaelkassel\"}]}<\/script>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"Le th\u00e9or\u00e8me central limite : Comprendre et savoir l'utiliser","description":"Le th\u00e9or\u00e8me central limite d\u00e9montre la convergence progressive d\u2019une suite de variables vers la loi normale.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/datascientest.com\/theoreme-central-limite-tout-savoir","og_locale":"fr_FR","og_type":"article","og_title":"Le th\u00e9or\u00e8me central limite : Comprendre et savoir l'utiliser","og_url":"https:\/\/datascientest.com\/theoreme-central-limite-tout-savoir","og_site_name":"Formation Data Science | DataScientest.com","article_published_time":"2023-10-06T07:30:00+00:00","article_modified_time":"2023-11-09T13:41:39+00:00","og_image":[{"width":1200,"height":600,"url":"https:\/\/datascientest.com\/wp-content\/uploads\/2023\/09\/theoreme-central-limite.png","type":"image\/png"}],"author":"Raphael Kassel","twitter_card":"summary_large_image","twitter_misc":{"\u00c9crit par":"Raphael Kassel"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/datascientest.com\/theoreme-central-limite-tout-savoir","url":"https:\/\/datascientest.com\/theoreme-central-limite-tout-savoir","name":"Le th\u00e9or\u00e8me central limite : Comprendre et savoir l'utiliser","isPartOf":{"@id":"https:\/\/datascientest.com\/#website"},"datePublished":"2023-10-06T07:30:00+00:00","dateModified":"2023-11-09T13:41:39+00:00","author":{"@id":"https:\/\/datascientest.com\/#\/schema\/person\/6eca16f63daaafcbc96e15549b9d4ee1"},"description":"Le th\u00e9or\u00e8me central limite d\u00e9montre la convergence progressive d\u2019une suite de variables vers la loi normale.","inLanguage":"fr-FR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/datascientest.com\/theoreme-central-limite-tout-savoir"]}]},{"@type":"WebSite","@id":"https:\/\/datascientest.com\/#website","url":"https:\/\/datascientest.com\/","name":"Formation Data Science | DataScientest.com","description":"Leader de la formation en Data Science","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/datascientest.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"fr-FR"},{"@type":"Person","@id":"https:\/\/datascientest.com\/#\/schema\/person\/6eca16f63daaafcbc96e15549b9d4ee1","name":"Raphael Kassel","image":{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/datascientest.com\/#\/schema\/person\/image\/","url":"https:\/\/datascientest.com\/wp-content\/uploads\/2023\/11\/raph.jpeg","contentUrl":"https:\/\/datascientest.com\/wp-content\/uploads\/2023\/11\/raph.jpeg","caption":"Raphael Kassel"},"description":"Raphael Kassel est le Head of Data Programs chez DataScientest. 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