{"id":26810,"date":"2021-06-13T13:00:02","date_gmt":"2021-06-13T13:00:02","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=26810"},"modified":"2024-05-22T04:16:16","modified_gmt":"2024-05-22T04:16:16","slug":"variables-aleatoires-et-distributions-de-probabilites","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/fr\/variables-aleatoires-et-distributions-de-probabilites\/","title":{"rendered":"Variables Al\u00e9atoires et Distributions de Probabilit\u00e9s"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Variables Al\u00e9atoires et Distributions de Probabilit\u00e9s<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>R\u00e9sum\u00e9<\/strong><br \/><em>Ce cours fournit une immersion profonde dans les concepts de variables al\u00e9atoires et de distributions de probabilit\u00e9, piliers fondamentaux de la th\u00e9orie des probabilit\u00e9s et de l&#8217;analyse statistique. La d\u00e9finition d&#8217;une variable al\u00e9atoire comme un nombre d\u00e9pendant du r\u00e9sultat d&#8217;une exp\u00e9rience al\u00e9atoire est introduite. La fonction de distribution d&#8217;une variable al\u00e9atoire est abord\u00e9e, en soulignant son importance ainsi que ses propri\u00e9t\u00e9s essentielles. Enfin, la relation entre les variables al\u00e9atoires et les distributions de probabilit\u00e9 est analys\u00e9e, expliquant que deux variables peuvent avoir la m\u00eame distribution sans \u00eatre la m\u00eame variable al\u00e9atoire.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>OBJECTIFS D&#8217;APPRENTISSAGE:<\/strong><br \/>\n\u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de :\n<\/p>\n<ol>\n<li><strong>Comprendre<\/strong> le concept de variables al\u00e9atoires : Les \u00e9tudiants doivent \u00eatre capables de d\u00e9crire et d&#8217;expliquer ce que sont les variables al\u00e9atoires et comment elles sont d\u00e9finies math\u00e9matiquement.<\/li>\n<li><strong>Comprendre<\/strong> le concept de distributions de probabilit\u00e9 : Les \u00e9tudiants doivent pouvoir expliquer ce que sont les distributions de probabilit\u00e9 et comment elles sont repr\u00e9sent\u00e9es.<\/li>\n<li><strong>D\u00e9crire<\/strong> les propri\u00e9t\u00e9s des distributions de probabilit\u00e9 : Les \u00e9tudiants doivent \u00eatre capables de reconna\u00eetre et d&#8217;expliquer les propri\u00e9t\u00e9s cl\u00e9s des distributions de probabilit\u00e9.<\/li>\n<li><strong>Analyser<\/strong> la relation entre les variables al\u00e9atoires et les distributions de probabilit\u00e9 : Les \u00e9tudiants doivent pouvoir discuter de la mani\u00e8re dont les variables al\u00e9atoires et les distributions de probabilit\u00e9 sont interconnect\u00e9es, et comment deux variables peuvent avoir la m\u00eame distribution sans \u00eatre la m\u00eame variable al\u00e9atoire.<\/li>\n<li><strong>D\u00e9montrer<\/strong> et appliquer les propri\u00e9t\u00e9s des distributions de probabilit\u00e9 dans des situations pratiques : Les \u00e9tudiants doivent \u00eatre capables de d\u00e9montrer math\u00e9matiquement les propri\u00e9t\u00e9s des distributions de probabilit\u00e9 et d&#8217;appliquer ces propri\u00e9t\u00e9s dans des situations r\u00e9elles.<\/li>\n<li><strong>Comprendre<\/strong> le concept de fonctions de distribution : Les \u00e9tudiants doivent \u00eatre capables de d\u00e9crire ce qu&#8217;est une fonction de distribution et comment elle est utilis\u00e9e pour d\u00e9crire une variable al\u00e9atoire.<\/li>\n<\/ol>\n<p><center><br \/>\n<strong><u>TABLE DES MATI\u00c8RES<\/u>:<\/strong><br \/>\n<a href=\"#1\"><strong>Qu&#8217;est-ce qu&#8217;une variable al\u00e9atoire ?<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>Qu&#8217;est-ce qu&#8217;une distribution de probabilit\u00e9 ?<\/strong><\/a><br \/>\n<a href=\"#3\">Propri\u00e9t\u00e9s des distributions de probabilit\u00e9<\/a><br \/>\n<a href=\"#4\">Relation entre les variables al\u00e9atoires et les distributions de probabilit\u00e9s<\/a><br \/>\n<\/center><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/gIKn9t1hnrw\" title=\"Lecteur vid\u00e9o YouTube\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p style=\"text-align: justify; color: #000000;\">Un des concepts cl\u00e9s de la th\u00e9orie des probabilit\u00e9s et de l&#8217;analyse statistique est celui des variables al\u00e9atoires et des distributions de probabilit\u00e9. Bien que la th\u00e9orie que nous avons d\u00e9velopp\u00e9e jusqu&#8217;\u00e0 pr\u00e9sent soit en un certain sens \u00abcompl\u00e8te\u00bb, la v\u00e9rit\u00e9 est qu&#8217;elle est actuellement assez rudimentaire ; les variables al\u00e9atoires et les distributions de probabilit\u00e9 sont, pour ainsi dire, des concepts qui nous permettent de \u00ablisser notre capacit\u00e9 \u00e0 travailler avec les probabilit\u00e9s et \u00e0 effectuer des analyses statistiques\u00bb.<\/p>\n<p><a name=\"1\"><\/a><\/br><\/br><\/p>\n<h2>Qu&#8217;est-ce qu&#8217;une variable al\u00e9atoire ?<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><strong><span style=\"color: #ff0000;\">Pour nous familiariser avec le concept de variable al\u00e9atoire<\/span><\/strong>, il est utile de commencer par une approche intuitive : une variable al\u00e9atoire peut \u00eatre interpr\u00e9t\u00e9e comme \u00abun nombre d\u00e9pendant du r\u00e9sultat d&#8217;une exp\u00e9rience al\u00e9atoire\u00bb. Cependant, pour une compr\u00e9hension plus pr\u00e9cise, il est essentiel d&#8217;explorer \u00e9galement sa d\u00e9finition formelle. Voici cette d\u00e9finition :<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000080;\"><strong>D\u00e9finition :<\/strong><\/span> Une variable al\u00e9atoire sur un ensemble <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{X}<\/span><\/span> est une fonction <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f:\\Omega \\longmapsto \\mathcal{X}<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Le cas le plus courant est lorsque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{X}= \\mathbb{R},<\/span><\/span> et, sauf indication contraire, c&#8217;est ce que nous supposerons dor\u00e9navant ; c&#8217;est-\u00e0-dire que nous travaillerons avec des variables al\u00e9atoires \u00e0 valeurs r\u00e9elles. G\u00e9n\u00e9ralement, les variables al\u00e9atoires sont d\u00e9sign\u00e9es par des lettres majuscules, telles que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X,Y,Z, \\cdots,<\/span><\/span>, tandis que les constantes sont d\u00e9sign\u00e9es par des lettres minuscules. Pour simplifier, nous nous r\u00e9f\u00e9rerons simplement aux variables al\u00e9atoires en les appelant \u00abvariables\u00bb.<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0e0ff;\">\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000000;\"><strong>Exemple :<\/strong><\/span> Supposons que l&#8217;on lance un d\u00e9 \u00e0 6 faces deux fois. Alors nous aurons :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2d6} = \\{(\\omega_1, \\omega_2)\\;|\\; \\omega_1,\\omega_2 \\in \\{1,2,3,4,5,6\\}\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u00c0 partir de cela, nous pouvons d\u00e9finir les variables al\u00e9atoires suivantes :<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=<\/span><\/span> \u00abLe nombre de fois que le un sort\u00bb<\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y=<\/span><\/span> \u00abLa somme des r\u00e9sultats obtenus\u00bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">=\\omega_1 + \\omega_2<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Z=<\/span><\/span> \u00abLe r\u00e9sultat du deuxi\u00e8me lancer\u00bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">= \\omega_2<\/span><\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"2\"><\/a><\/br><\/br><\/p>\n<h2>Qu&#8217;est-ce qu&#8217;une distribution de probabilit\u00e9 ?<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000080;\"><strong>D\u00e9finition :<\/strong><\/span><strong><span style=\"color: #ff0000;\">Une fonction de distribution (ou \u00abFD\u00bb)<\/span><\/strong> d&#8217;une variable al\u00e9atoire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> est une fonction <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X: \\mathbb{R} \\longmapsto \\mathbb{R}<\/span><\/span> d\u00e9finie par la relation <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x) = P(\\{\\omega \\;|\\; X(\\omega)\\leq x\\}),<\/span><\/span> ou plus bri\u00e8vement : <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X\\leq x).<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">G\u00e9n\u00e9ralement, ce qui int\u00e9resse d&#8217;une variable al\u00e9atoire n&#8217;est pas tant son expression explicite dans un espace d&#8217;\u00e9chantillonnage <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span><\/span>, mais sa fonction de distribution. L&#8217;indice <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> dans <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X<\/span><\/span> peut \u00eatre omis si le contexte est clair et sans ambigu\u00eft\u00e9. Il est courant d&#8217;utiliser la notation <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim F<\/span><\/span> pour indiquer que la variable al\u00e9atoire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> a une fonction de distribution <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F.<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/br><\/br><\/p>\n<h3>Propri\u00e9t\u00e9s des distributions de probabilit\u00e9<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=gIKn9t1hnrw&amp;t=806s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> est une distribution de probabilit\u00e9<\/span><\/strong><\/a> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b<\/span><\/span> sont des nombres r\u00e9els quelconques, alors les propri\u00e9t\u00e9s suivantes seront v\u00e9rifi\u00e9es :<\/p>\n<p style=\"text-align: justify; color: #000000;\">(a) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b \\longrightarrow [P(a\\lt X \\leq b) = F(b) - F(a)]<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">(b) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b \\longrightarrow F(a) \\leq F(b),<\/span><\/span> c&#8217;est-\u00e0-dire \u00abF est croissante\u00bb.<\/p>\n<p style=\"text-align: justify; color: #000000;\">(c) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to +\\infty} F(x) = 1<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to -\\infty} F(x) = 0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">(d) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X=x)=\\lim_{t\\to x^+}F(t) - \\lim_{t\\to x^-}F(t)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">(e) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle F(x)=\\lim_{t\\to x^+}F(t)<\/span><\/span><\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td style=\"background-color: #ffe0e0;\"><span class=\"collapseomatic \" id=\"id69e4449ba695f\"  tabindex=\"0\" title=\"D\u00c9MONSTRATION\"    >D\u00c9MONSTRATION<\/span><div id=\"target-id69e4449ba695f\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify; color: #000000;\"><strong>(a)<\/strong> Soient <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/span> les \u00e9v\u00e9nements <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{X\\leq a\\}<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{X\\leq b\\}<\/span><\/span> respectivement, avec <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b.<\/span><\/span> Si tout cela se produit, alors il se v\u00e9rifiera que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B<\/span><\/span> et par cons\u00e9quent, il se produira que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{blue}{P(a\\lt X\\leq b)} = P(B\\setminus A) = P(B) - P(B\\cap A) = P(B)-P(A) =\\color{blue}{F(b) - F(a)}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\"><strong>(b)<\/strong> De la partie (a), nous avons : Comme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B\\setminus A)\\geq 0,<\/span><\/span> nous avons donc que :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(b) - F(a) \\geq 0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">ce qui est \u00e9quivalent \u00e0 dire<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(a) \\leq F(b)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\"><strong>(c)<\/strong> Ici, nous utiliserons le fait que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> est monotone croissante (prouv\u00e9 en (b)) et born\u00e9e avec une valeur maximale \u00e9gale \u00e0 \u00ab1\u00bb (car la distribution est d\u00e9finie en termes de probabilit\u00e9). Ce seul fait est suffisant pour dire que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty} F(x) = 1<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Une approche compl\u00e9mentaire \u00e0 cela nous permet de faire les calculs suivants avec le m\u00eame r\u00e9sultat.<\/p>\n<p style=\"text-align: justify; color: #000000;\">D\u00e9finissons l&#8217;ensemble <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A_n=\\{\\omega\\;|\\;X(\\omega)\\leq n\\}.<\/span><\/span> \u00c0 partir de cela, il est facile de v\u00e9rifier que, pour tout <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> il se v\u00e9rifiera <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A_{n}\\subseteq A_{n+1},<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\bigcup_{n\\lt +\\infty} A_n = \\Omega<\/span><\/span> et par cons\u00e9quent, en utilisant la propri\u00e9t\u00e9 de <a href=\"http:\/\/toposuranos.com\/material\/fr\/theoremes-utiles-pour-le-calcul-des-probabilites\/\" rel=\"noopener\" target=\"_blank\">continuit\u00e9<\/a>, nous aurons :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 1=P(\\Omega) = P\\left( \\bigcup_{n\\lt +\\infty} A_n \\right) = \\lim_{n\\to +\\infty} P(A_n) = \\lim_{n\\to +\\infty} P(\\{\\omega\\;|\\;X(\\omega)\\leq n\\}) = \\lim_{n\\to +\\infty} P(X\\leq n)=\\lim_{n\\to +\\infty}F(n)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">C&#8217;est-\u00e0-dire :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\lim_{x\\to +\\infty} F(x) = 1}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">En revanche, pour la limite o\u00f9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to -\\infty<\/span><\/span>, nous avons ce qui suit :<\/p>\n<p style=\"text-align: justify; color: #000000;\">D&#8217;abord, d\u00e9finissons l&#8217;ensemble <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B_n=\\{\\omega\\;|\\;-n\\lt X(\\omega)\\}.<\/span><\/span> \u00c0 partir de cela, nous v\u00e9rifions que :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{n \\to -\\infty}F(n) = \\lim_{n\\to -\\infty} P(X\\leq n) = \\lim_{n\\to \\infty} P(X\\leq -n)= 1 - \\lim_{n\\to \\infty} P(-n \\lt X) = 1 - \\lim_{n\\to \\infty}P(B_n)) = 1 - P(\\Omega) = 1-1=0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\"><strong>(d)<\/strong> Le raisonnement est similaire \u00e0 la partie (c). Nous commen\u00e7ons par d\u00e9finir l&#8217;ensemble<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle C_n = \\left\\{x - \\frac{1}{n} \\leq X \\leq x + \\frac{1}{n}\\right\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Et \u00e0 partir de cela, nous avons que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> C_{n+1}\\subseteq C_n<\/span><\/span><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcap_{n\\gt 0} C_n = \\{X=x\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Par cons\u00e9quent, en utilisant <a href=\"http:\/\/toposuranos.com\/material\/fr\/theoremes-utiles-pour-le-calcul-des-probabilites\/\" rel=\"noopener\" target=\"_blank\">un r\u00e9sultat de la propri\u00e9t\u00e9 de continuit\u00e9<\/a>, nous avons :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X=x)=P\\left(\\bigcap_{n\\gt 0} C_n \\right) = \\lim_{n\\to \\infty} P(C_n) = \\lim_{x+1\/n \\to x^+}F\\left(x+1\/n\\right) - \\lim_{x-1\/n \\to x^-}F\\left(x-1\/n\\right)= \\lim_{t \\to x^+}F\\left(t\\right) - \\lim_{t \\to x^-}F\\left(t\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\"><strong>(e)<\/strong> Ce dernier cas est obtenu \u00e0 partir du r\u00e9sultat pr\u00e9c\u00e9dent. En effet, comme nous l&#8217;avons d\u00e9j\u00e0 prouv\u00e9<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X=x)= \\lim_{t \\to x^+}F\\left(t\\right) - \\lim_{t \\to x^-}F\\left(t\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Nous pouvons \u00e9crire<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{t \\to x^+}F\\left(t\\right) = P(X=x) + \\lim_{t \\to x^-}F\\left(t\\right) = P(X=x) + \\lim_{t\\to x^-}P(X\\leq t)= P(X\\leq x) = F(x)<\/span><\/span><\/p>\n<\/div><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"4\"><\/a><\/br><\/br><\/p>\n<h3>Relation entre les variables al\u00e9atoires et les distributions de probabilit\u00e9s<\/h3>\n<p style=\"text-align: justify; color: #000000;\">On dit que deux variables <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y<\/span><\/span> ont la m\u00eame distribution de probabilit\u00e9 si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall A\\subseteq \\mathbb{R})(P(X\\in A) = P(Y\\in A)).<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Deux variables <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y<\/span><\/span> d\u00e9finies sur le m\u00eame espace d&#8217;\u00e9chantillonnage <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span><\/span> peuvent avoir la m\u00eame distribution mais ne pas \u00eatre pour autant la m\u00eame variable al\u00e9atoire. Par exemple, si nous consid\u00e9rons l&#8217;exp\u00e9rience de lancer une pi\u00e8ce \u00e9quilibr\u00e9e \u00e0 deux faces et que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=1<\/span><\/span> correspond \u00e0 face et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=0<\/span><\/span> correspond \u00e0 pile, nous pouvons d\u00e9finir la variable al\u00e9atoire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y=1-X<\/span><\/span> et nous aurons que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X=1) = P(Y=1)=0.5,<\/span><\/span> et que les deux ont la m\u00eame distribution, mais si nous calculons la probabilit\u00e9 que les deux aient la m\u00eame valeur, nous aurons <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X=Y)=0<\/span><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Variables Al\u00e9atoires et Distributions de Probabilit\u00e9s R\u00e9sum\u00e9Ce cours fournit une immersion profonde dans les concepts de variables al\u00e9atoires et de distributions de probabilit\u00e9, piliers fondamentaux de la th\u00e9orie des probabilit\u00e9s et de l&#8217;analyse statistique. La d\u00e9finition d&#8217;une variable al\u00e9atoire comme un nombre d\u00e9pendant du r\u00e9sultat d&#8217;une exp\u00e9rience al\u00e9atoire est introduite. La fonction de distribution d&#8217;une [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":26787,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":9,"footnotes":""},"categories":[569,682],"tags":[],"class_list":["post-26810","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","category-probabilites-et-statistiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Variables Al\u00e9atoires et Distributions de Probabilit\u00e9s - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"D\u00e9couvrez les concepts essentiels des variables al\u00e9atoires et des distributions de probabilit\u00e9 dans ce cours d\u00e9taill\u00e9.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" 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