{"id":5007,"date":"2016-10-10T09:44:12","date_gmt":"2016-10-10T08:44:12","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=5007"},"modified":"2024-02-11T21:23:15","modified_gmt":"2024-02-11T20:23:15","slug":"processus-de-markov","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/markov-process\/","title":{"rendered":"Markov Process 101"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"5007\" class=\"elementor elementor-5007\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ed79885 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ed79885\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-7f0aff2\" data-id=\"7f0aff2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5b2126f elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"5b2126f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/2020\/04\/03\/theories-et-algorithmes\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Th\u00e9ories<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-86f93cf\" data-id=\"86f93cf\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f0cfc19 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"f0cfc19\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Page d'accueil<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-6762493\" data-id=\"6762493\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fbe60d6 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"fbe60d6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/fr.wikipedia.org\/wiki\/Processus_stochastique\" target=\"_blank\" rel=\"noopener\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Wiki<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1ba09a0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1ba09a0\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a4f1e49\" data-id=\"a4f1e49\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8e18ffa elementor-widget elementor-widget-toggle\" data-id=\"8e18ffa\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1491\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1491\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">I. G\u00e9n\u00e9ralit\u00e9 sur processus de Markov<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1491\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1491\"><ul>\n<li><span class=\"nowrap\">Marche al\u00e9atoire<\/span><\/li>\n<li><span class=\"nowrap\">Martingale<\/span><\/li>\n<li><span class=\"nowrap\">Mouvement brownien<\/span><\/li>\n<\/ul><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1492\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-1492\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">II. Ordre 1 temps discret<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1492\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-1492\"><ul>\n<li><a href=\"https:\/\/complex-systems-ai.com\/processus-de-markov\/chaines-de-markov-en-temps-discret\/\">Chaines de Markov en temps discret<\/a><\/li>\n<li><a href=\"https:\/\/complex-systems-ai.com\/processus-de-markov\/criteres-de-recurrence-et-transience\/\">R\u00e9currence et transcience<\/a><\/li>\n<li><a href=\"https:\/\/complex-systems-ai.com\/processus-de-markov\/loi-invariante-et-comportement-asymptotique\/\">Loi invariante et comportement asymptotique<\/a><\/li>\n<li><a href=\"https:\/\/complex-systems-ai.com\/processus-de-markov\/probabilite-datteinte-dun-etat\/\">Temps d&rsquo;atteinte d&rsquo;un \u00e9tat<\/a><\/li>\n<li><a href=\"https:\/\/complex-systems-ai.com\/processus-de-markov\/probabilite-dabsorption-dun-etat\/\">Probabilit\u00e9 d\u2019absorption d\u2019un \u00e9tat<\/a><\/li>\n<\/ul><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1493\" class=\"elementor-tab-title\" data-tab=\"3\" role=\"button\" aria-controls=\"elementor-tab-content-1493\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">III. Ordre 1 temps continu<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1493\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"3\" role=\"region\" aria-labelledby=\"elementor-tab-title-1493\"><ul>\n<li><a href=\"https:\/\/complex-systems-ai.com\/processus-de-markov\/chaines-de-markov-en-temps-continu\/\">Chaines de Markov en temps continu<\/a><\/li>\n<li><a href=\"https:\/\/complex-systems-ai.com\/processus-de-markov\/regime-permanent\/\">R\u00e9gime permanent<\/a><\/li>\n<li><a href=\"https:\/\/complex-systems-ai.com\/processus-de-markov\/processus-de-poisson\/\">Processus de Poisson<\/a><\/li>\n<li><a href=\"https:\/\/complex-systems-ai.com\/processus-de-markov\/les-files-dattente\/\">Files d&rsquo;attente (g\u00e9n\u00e9ralisation)<\/a><\/li>\n<li><a href=\"https:\/\/complex-systems-ai.com\/processus-de-markov\/la-file-m-m-1\/\">File M\/M\/1<\/a><\/li>\n<\/ul><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1494\" class=\"elementor-tab-title\" data-tab=\"4\" role=\"button\" aria-controls=\"elementor-tab-content-1494\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">IV. Automate Stochastique et Hidden Markov Model<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1494\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"4\" role=\"region\" aria-labelledby=\"elementor-tab-title-1494\"><ul>\n<li><a href=\"https:\/\/docs.google.com\/presentation\/d\/12GF5VqE0K_CvFjl33HfuYvLHa4UlxzmVdt1sjFt7sdQ\/edit?usp=sharing\" target=\"_blank\" rel=\"noopener\">Rappel Stochastique et Automate<\/a><\/li>\n<li><a href=\"https:\/\/docs.google.com\/presentation\/d\/1UFM2o_8EGH_92JzNHBVtNY4jL5BemeI3qi1ekLa3F84\/edit?usp=sharing\" target=\"_blank\" rel=\"noopener\">Hidden Markov Model<\/a><\/li>\n<li><a href=\"https:\/\/docs.google.com\/presentation\/d\/1UFM2o_8EGH_92JzNHBVtNY4jL5BemeI3qi1ekLa3F84\/edit?usp=sharing\" target=\"_blank\" rel=\"noopener\">Viterbi<\/a><\/li>\n<li><a href=\"https:\/\/docs.google.com\/presentation\/d\/1UFM2o_8EGH_92JzNHBVtNY4jL5BemeI3qi1ekLa3F84\/edit?usp=sharing\" target=\"_blank\" rel=\"noopener\">Backward-Forward<\/a><\/li>\n<li><a href=\"https:\/\/docs.google.com\/presentation\/d\/1UFM2o_8EGH_92JzNHBVtNY4jL5BemeI3qi1ekLa3F84\/edit?usp=sharing\" target=\"_blank\" rel=\"noopener\">Baum-Welch<\/a><\/li>\n<li><a href=\"https:\/\/docs.google.com\/presentation\/d\/1x21qOkexEy8nGAWTahRAIPLwQmZ3End3PKDaOXRvIEQ\/edit?usp=sharing\" target=\"_blank\" rel=\"noopener\">Probabiliste finite automaton<\/a><\/li>\n<li><a href=\"https:\/\/docs.google.com\/presentation\/d\/1x21qOkexEy8nGAWTahRAIPLwQmZ3End3PKDaOXRvIEQ\/edit?usp=sharing\" target=\"_blank\" rel=\"noopener\">Transformation HMM vers PFA<\/a><\/li>\n<li><a href=\"https:\/\/docs.google.com\/presentation\/d\/1x21qOkexEy8nGAWTahRAIPLwQmZ3End3PKDaOXRvIEQ\/edit?usp=sharing\" target=\"_blank\" rel=\"noopener\">Algorithme de Merge &amp; Fold<\/a><\/li>\n<li><a href=\"https:\/\/docs.google.com\/presentation\/d\/1x21qOkexEy8nGAWTahRAIPLwQmZ3End3PKDaOXRvIEQ\/edit?usp=sharing\" target=\"_blank\" rel=\"noopener\">Merge &amp; Fold avec mot interdit<\/a><\/li>\n<li><a href=\"https:\/\/docs.google.com\/presentation\/d\/1x21qOkexEy8nGAWTahRAIPLwQmZ3End3PKDaOXRvIEQ\/edit?usp=sharing\" target=\"_blank\" rel=\"noopener\">Inf\u00e9rence grammaticale<\/a><\/li>\n<\/ul><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-52c15713 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"52c15713\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5c8bfe98\" data-id=\"5c8bfe98\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2eac3cba elementor-widget elementor-widget-text-editor\" data-id=\"2eac3cba\" data-element_type=\"widget\" data-e-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\t\t\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Contenus<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/#Processus-de-Markov\" >Processus de Markov<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/#Variables-aleatoires-et-probabilite\" >Variables al\u00e9atoires et probabilit\u00e9<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Processus-de-Markov\"><\/span>Processus de Markov<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Un processus de Markov repr\u00e9sente tous processus ayant des arguments d&rsquo;exp\u00e9rience al\u00e9atoire.<\/p>\n\n<p>Une exp\u00e9rience al\u00e9atoire, not\u00e9e E est une exp\u00e9rience dont l&rsquo;issue est soumise au hasard. On note\u00a0\u03a9 l&rsquo;ensemble de tous les r\u00e9sultats possibles \u00e0 cette exp\u00e9rience, et est appel\u00e9 univers, espace des possibles ou encore espace d&rsquo;\u00e9tats. Un r\u00e9sultat de E est un \u00e9l\u00e9ment de\u00a0\u03a9 not\u00e9\u00a0\u03c9.<\/p>\n\n<p>Par exemple dans le jeu pile ou face, l&rsquo;univers de l&rsquo;exp\u00e9rience \u00ab\u00a0lancer une pi\u00e8ce\u00a0\u00bb est\u00a0\u03a9={P, F}. Pour l&rsquo;exp\u00e9rience \u00ab\u00a0lancer deux pi\u00e8ces l&rsquo;une apr\u00e8s l&rsquo;autre\u00a0\u00bb, l&rsquo;univers est\u00a0\u03a9={PP, PF, FP, FF}.<\/p>\n\n<p>Un \u00e9v\u00e9nement al\u00e9atoire A li\u00e9 \u00e0 l&rsquo;exp\u00e9rience E est un sous-ensemble de\u00a0\u03a9 dont on peut dire au vu de l&rsquo;exp\u00e9rience s&rsquo;il est r\u00e9alis\u00e9 ou non. Sur l&rsquo;exemple pr\u00e9c\u00e9dent l&rsquo;\u00e9v\u00e9nement al\u00e9atoire \u00ab\u00a0obtenir pile\u00a0\u00bb dans pile ou face peut facilement \u00eatre observer en lan\u00e7ant une pi\u00e8ce. Un \u00e9v\u00e9nement al\u00e9atoire est un ensemble et poss\u00e8de donc les principales propri\u00e9t\u00e9s de la th\u00e9orie des ensembles.<\/p>\n\n<p>Op\u00e9rations \u00e9l\u00e9mentaires sur les parties d&rsquo;un ensemble :<\/p>\n\n<ul class=\"wp-block-list\">\n<li>Intersection : l&rsquo;intersection des ensembles A et B not\u00e9e A \u2229B est l&rsquo;ensemble<br \/>des points appartenant \u00e0 la fois \u00e0 A et \u00e0 B.<\/li>\n<li>R\u00e9union : la r\u00e9union de deux ensembles A et B not\u00e9e A\u222aB est l&rsquo;ensemble des<br \/>points appartenant \u00e0 au moins l&rsquo;un des deux ensembles.<\/li>\n<li>Ensemble vide : l&rsquo;ensemble vide, not\u00e9 \u00d8, est l\u2019ensemble ne contenant aucun<br \/>\u00e9l\u00e9ment.<\/li>\n<li>Ensembles disjoints : les ensembles A et B sont dits disjoints si A \u2229B = \u00d8.<\/li>\n<li>Compl\u00e9mentaire : le compl\u00e9mentaire de l\u2019ensemble A \u2282 \u2126 dans \u2126, not\u00e9 A<sup>c<\/sup> ou \u2126\\ A, est l\u2019ensemble des \u00e9l\u00e9ments n\u2019appartenant pas \u00e0 A. Les ensembles A<br \/>et\u00a0A<sup>c<\/sup>\u00a0sont disjoints.<\/li>\n<\/ul>\n\n<p>Op\u00e9rations ensemblistes :<\/p>\n\n<ul class=\"wp-block-list\">\n<li>Non : la r\u00e9alisation de l&rsquo;\u00e9v\u00e9nement contraire \u00e0 A est repr\u00e9sent\u00e9 par A<sup>c<\/sup>: le<br \/>r\u00e9sultat de l\u2019exp\u00e9rience n\u2019appartient pas \u00e0 A.<\/li>\n<li>Et : l&rsquo;\u00e9v\u00e9nement \u00ab A et B sont r\u00e9alis\u00e9s \u00bb est repr\u00e9sent\u00e9 par A\u2229B; le r\u00e9sultat de<br \/>l\u2019exp\u00e9rience se trouve \u00e0 la fois dans A et dans B.<\/li>\n<li>Ou : l&rsquo;\u00e9v\u00e9nement \u00ab A ou B sont r\u00e9alis\u00e9s \u00bb est repr\u00e9sent\u00e9 par A\u222aB; le r\u00e9sultat<br \/>de l&rsquo;exp\u00e9rience se trouve soit dans A soit dans B soit dans les deux.<\/li>\n<li>Implication : le fait que la r\u00e9alisation de l&rsquo;\u00e9v\u00e9nement A entra\u00eene la r\u00e9alisation<br \/>de B se traduit par A \u2282 B.<\/li>\n<li>Incompatibilit\u00e9 : si A\u2229B = \u00d8, A et B sont dits incompatibles. Un r\u00e9sultat de<br \/>l\u2019exp\u00e9rience ne peut \u00eatre \u00e0 la fois dans A et dans B.<\/li>\n<\/ul>\n\n<p>A chaque \u00e9v\u00e9nement, nous cherchons \u00e0 associ\u00e9 une mesure (que nous ne d\u00e9finirons pas dans ce cours) compris entre 0 et 1 et repr\u00e9sentant la probabilit\u00e9 que l&rsquo;\u00e9v\u00e9nement soit r\u00e9alis\u00e9. Pour une exp\u00e9rience A, cette mesure est not\u00e9e P(A).<\/p>\n\n<p>Formellement, soit E une exp\u00e9rience al\u00e9atoire d\u2019univers \u2126. On appelle mesure de probabilit\u00e9 sur \u2126 (ou plus simplement probabilit\u00e9) une application P qui associe \u00e0 tout \u00e9v\u00e9nement al\u00e9atoire A un nombre r\u00e9el P(A) telle que<br \/>(i) Pour tout A tel que P(A) existe, on a 0 \u2264 P(A) \u2264 1.<br \/>(ii) P(\u00d8) = 0 et P(\u2126) = 1.<br \/>(iii) A\u2229B = ; implique que P(A \u222aB) = P(A)+P(B).<\/p>\n\n<p>La probabilit\u00e9 d&rsquo;un \u00e9v\u00e9nement peut se comprendre ainsi P(A)=nombre de cas r\u00e9alisables\/nombre de cas possible. Le \u00ab\u00a0nombre de cas\u00a0\u00bb est le cardinal d&rsquo;un \u00e9v\u00e9nement\/univers.<\/p>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Variables-aleatoires-et-probabilite\"><\/span>Variables al\u00e9atoires et probabilit\u00e9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p>Une variable al\u00e9atoire est une fonction dont la valeur d\u00e9pend de l&rsquo;issue d&rsquo;une<br \/>exp\u00e9rience al\u00e9atoire E d\u2019univers \u2126. On dit qu&rsquo;une variable al\u00e9atoire X est discr\u00e8te si elle prend un nombre de valeurs fini ou d\u00e9nombrables. L&rsquo;ensemble des issues \u03c9 sur lesquelles X prend une valeur fix\u00e9e x forme l\u2019\u00e9v\u00e8nement {\u03c9 : X(\u03c9) = x} que l\u2019on note [X = x]. La probabilit\u00e9 de cet \u00e9v\u00e9nement est not\u00e9e P(X = x).<\/p>\n\n<p>La fonction p<sub>X<\/sub> : x \u2192 P(X = x) est appel\u00e9e la loi de la variable al\u00e9atoire X. Si {x<sub>1<\/sub>,x<sub>2<\/sub>,&#8230;} est l\u2019ensemble des valeurs possibles pour X, on a :<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" class=\"alignnone wp-image-6559\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/08\/proba1.png\" alt=\"probabilit\u00e9 processus de Markov\" width=\"181\" height=\"44\" title=\"\"><\/figure>\n<\/div>\n\n<p>Soit S<sub>2<\/sub>\u00a0le nombre de piles obtenus lors du lancer de deux pi\u00e8ces. L&rsquo;ensemble des valeurs possibles pour S<sub>2<\/sub>\u00a0est {0,1,2}. Si l\u2019on munit l\u2019univers \u2126 associ\u00e9 \u00e0 cette exp\u00e9rience al\u00e9atoire de la probabilit\u00e9 uniforme P, il vient les solutions suivantes :<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" class=\"alignnone wp-image-6560\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/08\/proba2.png\" alt=\"probabilit\u00e9\" width=\"268\" height=\"153\" title=\"\"><\/figure>\n<\/div>\n\n<p>Lorsqu&rsquo;elle existe (l&rsquo;esp\u00e9rance est toujours d\u00e9finie si X prend un nombre fini de valeurs, ou bien si X est \u00e0 valeurs positives.), on appelle esp\u00e9rance ou moyenne d\u2019une variable al\u00e9atoire discr\u00e8te X la quantit\u00e9 not\u00e9e E(X) d\u00e9finie par :<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" class=\"alignnone wp-image-6561\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/08\/proba3.png\" alt=\"probabilit\u00e9 esp\u00e9rance\" width=\"243\" height=\"44\" title=\"\"><\/figure>\n<\/div>\n\n<p>Lorsqu&rsquo;elle existe, on appelle variance d\u2019une variable al\u00e9atoire discr\u00e8te X la<br \/>quantit\u00e9 not\u00e9e Var(X) d\u00e9finie par :<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6562\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/08\/proba4.png\" alt=\"probabilit\u00e9 variance\" width=\"313\" height=\"65\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/08\/proba4.png 313w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/08\/proba4-300x62.png 300w\" sizes=\"(max-width: 313px) 100vw, 313px\" \/><\/figure>\n<\/div>\n\n<p>L&rsquo;id\u00e9e de base du conditionnement est la suivante : une information suppl\u00e9mentaire concernant l&rsquo;exp\u00e9rience modifie la vraisemblance que l&rsquo;on accorde \u00e0 l&rsquo;\u00e9v\u00e9nement \u00e9tudi\u00e9.<\/p>\n\n<p>Par exemple, pour un lancer de deux d\u00e9s (un rouge et un bleu), la probabilit\u00e9 de<br \/>l\u2019\u00e9v\u00e9nement \u00ab la somme est sup\u00e9rieure ou \u00e9gale \u00e0 10 \u00bb vaut 1\/6 sans information<br \/>suppl\u00e9mentaire. En revanche si l&rsquo;on sait que le r\u00e9sultat du d\u00e9 rouge est 6, elle est<br \/>\u00e9gale \u00e0 1\/2 tandis qu&rsquo;elle est \u00e9gale \u00e0 0 si le r\u00e9sultat du d\u00e9 rouge est 2.<\/p>\n\n<p>Soient P une mesure de probabilit\u00e9 sur \u2126 et B un \u00e9v\u00e8nement tel que P(B) &gt; 0. La<br \/>probabilit\u00e9 conditionnelle de A sachant B est le r\u00e9el P(A|B) d\u00e9fini par :<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6563\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/08\/proba5.png\" alt=\"probabilit\u00e9 conditionnelle\" width=\"124\" height=\"45\" title=\"\"><\/figure>\n<\/div>\n\n<p>Les \u00e9v\u00e9nements A et B sont dits ind\u00e9pendants si :<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6565\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba6.png\" alt=\"probabilit\u00e9 ind\u00e9pendante\" width=\"174\" height=\"41\" title=\"\"><\/figure>\n<\/div>\n\n<p>On peut \u00e9tendre l&rsquo;ind\u00e9pendants \u00e0 n \u00e9v\u00e9nements.\u00a0Soient A<sub>1<\/sub>, A<sub>2<\/sub>,&#8230;, A<sub>n<\/sub>\u00a0des \u00e9v\u00e9nements. Ils sont dits ind\u00e9pendants (dans leur ensemble) si pour tout k \u2208 {1,&#8230;,n} et pour tout ensemble d&rsquo;entiers distincts {i<sub>1<\/sub>,&#8230;,i<sub>k<\/sub>\u00a0} \u2282 {1,&#8230;n}, on a :<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6566\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba7.png\" alt=\"probabilit\u00e9 ind\u00e9pendante\" width=\"268\" height=\"33\" title=\"\"><\/figure>\n<\/div>\n\n<p>Des variables al\u00e9atoires peuvent \u00eatre ind\u00e9pendantes deux \u00e0 deux, sans \u00eatre ind\u00e9pendantes dans leur ensemble :<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6567\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba8.png\" alt=\"probabilit\u00e9 ind\u00e9pendante\" width=\"432\" height=\"33\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba8.png 432w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba8-300x23.png 300w\" sizes=\"(max-width: 432px) 100vw, 432px\" \/><\/figure>\n<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Theories Home page Wiki I. General information on Markov process Random walk Martingale Brownian motion II. Order 1 discrete time Markov chains in time \u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-5007","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/5007","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/comments?post=5007"}],"version-history":[{"count":30,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/5007\/revisions"}],"predecessor-version":[{"id":20470,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/5007\/revisions\/20470"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/media?parent=5007"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}