{"id":27281,"date":"2021-10-08T13:00:28","date_gmt":"2021-10-08T13:00:28","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27281"},"modified":"2024-06-20T11:48:54","modified_gmt":"2024-06-20T11:48:54","slug":"distributions-continues-de-probabilite","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/distributions-continues-de-probabilite\/","title":{"rendered":"Distributions continues de probabilit\u00e9"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Distributions continues de probabilit\u00e9<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>R\u00e9sum\u00e9<\/strong><br \/><em>Ici, nous examinerons en profondeur le concept de distributions continues de probabilit\u00e9, en soulignant les caract\u00e9ristiques et les utilisations des cinq plus connues : la distribution exponentielle, la distribution uniforme rectangulaire, la distribution normale (Gaussienne), la distribution de Weibull et la distribution Gamma. Les formules math\u00e9matiques d\u00e9finissant chacune de ces distributions sont fournies, et les implications et les applications pratiques de ces derni\u00e8res sont examin\u00e9es, telles que l&#8217;\u00e9valuation de l&#8217;\u00e9mission de particules dans des \u00e9chantillons radioactifs ou le calcul de la position d&#8217;une balle dans un rail avec des limites. En outre, il est d\u00e9taill\u00e9 comment ces distributions peuvent \u00eatre modifi\u00e9es et adapt\u00e9es par l&#8217;application de param\u00e8tres sp\u00e9cifiques.<\/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 cette le\u00e7on, l&#8217;\u00e9tudiant sera capable de :\n<\/p>\n<ol>\n<li><strong>Comprendre<\/strong> ce que sont les distributions continues de probabilit\u00e9.<\/li>\n<li><strong>Appliquer<\/strong> les distributions continues de probabilit\u00e9 les plus connues : exponentielle, uniforme rectangulaire, exponentielle, normale (Gaussienne), Weibull et Gamma.<\/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 que les distributions continues de probabilit\u00e9 ?<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>Les 5 distributions continues de probabilit\u00e9 les plus connues<\/strong><\/a><br \/>\n<a href=\"#3\">Distribution exponentielle<\/a><br \/>\n<a href=\"#4\">Distribution uniforme rectangulaire<\/a><br \/>\n<a href=\"#5\">Distribution normale (Gaussienne)<\/a><br \/>\n<a href=\"#6\">Distribution de Weibull<\/a><br \/>\n<a href=\"#7\">Distribution Gamma<\/a><br \/>\n<a href=\"#8\"><strong>Exercices<\/strong><\/a><br \/>\n<\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/REOTUa7K8uQ\" title=\"YouTube video player\" 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;\">Lorsque nous avons examin\u00e9 les <a href=\"http:\/\/toposuranos.com\/material\/es\/conoce-el-espacio-muestral-de-la-teoria-de-las-probabilidades\/\" target=\"_blank\" rel=\"noopener\">espaces d&#8217;\u00e9chantillonnage<\/a>, nous avons vu qu&#8217;ils peuvent \u00eatre de deux types : discrets et continus. Nous avons \u00e9galement examin\u00e9 ce qui constitue une <a href=\"http:\/\/toposuranos.com\/material\/es\/distribuciones-discretas-de-probabilidad-y-ejemplos\/\" target=\"_blank\" rel=\"noopener\">distribution de probabilit\u00e9 discr\u00e8te.<\/a> Il est maintenant temps de se pencher sur les distributions continues de probabilit\u00e9.<\/p>\n<p>&nbsp;<\/p>\n<p><a name=\"1\"><\/a><\/br><\/br><\/p>\n<h2>Qu&#8217;est-ce que les distributions continues de probabilit\u00e9 ?<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=86s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Nous dirons qu&#8217;une variable al\u00e9atoire<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> a une distribution continue de probabilit\u00e9 s&#8217;il existe une fonction <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f_X : \\mathbb{R} \\longrightarrow \\mathbb{R}^+,<\/span><\/span> que nous appellerons <strong>densit\u00e9 de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X,<\/span><\/span><\/strong> telle que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\forall A \\subseteq \\mathbb{R}<\/span><\/span> l&#8217;\u00e9galit\u00e9 suivante soit v\u00e9rifi\u00e9e<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X\\in A) = \\displaystyle \\int_A f_X(x)dx<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">En particulier, si nous prenons <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A=]a,b]<\/span><\/span> nous aurons<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(a\\lt X \\leq b) = \\displaystyle \\int_a^b f_X(x)dx<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">et si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=-\\infty<\/span><\/span><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x) = P( X \\leq x) = \\displaystyle \\int_{-\\infty}^x f_X(t)dt<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Et de plus, \u00e0 partir de la propri\u00e9t\u00e9 (c) des <a href=\"http:\/\/toposuranos.com\/material\/es\/variables-aleatorias-y-distribuciones-de-probabilidades\/\" rel=\"noopener\" target=\"_blank\">distributions de probabilit\u00e9<\/a> il en r\u00e9sulte que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int_{-\\infty}^{+\\infty} f_X(t)dt = 1<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">En appliquant le th\u00e9or\u00e8me fondamental du calcul \u00e0 cette derni\u00e8re expression, nous obtenons que pour une distribution continue, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x),<\/span><\/span> est continue pour tous les <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,<\/span><\/span> et sa d\u00e9riv\u00e9e est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f_X(x)<\/span><\/span> pour toutes les valeurs <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> o\u00f9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f_X(x)<\/span><\/span> est continue. De la continuit\u00e9 de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x)<\/span><\/span> et de la propri\u00e9t\u00e9 (d) (<a href=\"http:\/\/toposuranos.com\/material\/es\/variables-aleatorias-y-distribuciones-de-probabilidades\/\" rel=\"noopener\" target=\"_blank\">voir ici<\/a>) il en r\u00e9sulte que :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x=X)=0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Et donc<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x\\leq X)= P(x\\lt X)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est une fonction quelconque qui satisfait <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f\\geq 0<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int_{-\\infty}^{+\\infty}f(x)dx = 1,<\/span><\/span> alors on dit que c&#8217;est une densit\u00e9.<\/p>\n<p><a name=\"2\"><\/a><\/br><\/br><\/p>\n<h2>Les 5 distributions continues de probabilit\u00e9 les plus connues<\/h2>\n<p><a name=\"3\"><\/a><\/br><\/br><\/p>\n<h3>Distribution exponentielle<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=714s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Une fonction de distribution exponentielle<\/span><\/strong><\/a> avec le param\u00e8tre <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha \\gt 0 <\/span><\/span> est une fonction de distribution de la forme suivante :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(t) = \\left\\{\\begin{array}{lll}\n\n1 - e^{-t\/\\alpha} &amp; ; &amp; t\\geq 0 \\\\ \\\\\n\n0 &amp; ; &amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">En cons\u00e9quence, sa fonction de densit\u00e9 est de la forme<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f(t) = \\left\\{\\begin{array}{lll}\n\n\\frac{1}{\\alpha}e^{-t\/\\alpha} &amp; ; &amp; t\\geq 0 \\\\ \\\\\n\n0 &amp; ; &amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Si une variable al\u00e9atoire a une distribution exponentielle avec le param\u00e8tre <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span>, nous \u00e9crivons <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim Ex(\\alpha).<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Dans le contexte de la distribution de Poisson, si nous avons un \u00e9chantillon radioactif qui \u00e9met une particule avec un taux moyen d&#8217;\u00e9mission <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c,<\/span><\/span> alors le moment <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T<\/span><\/span> o\u00f9 la premi\u00e8re particule est \u00e9mise a une distribution exponentielle avec le param\u00e8tre <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1\/c.<\/span><\/span> En d&#8217;autres termes <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T\\sim Ex(1\/c),<\/span><\/span> et par cons\u00e9quent :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(T\\geq t)= e^{-ct}<\/span><\/span><\/p>\n<p><a name=\"4\"><\/a><\/br><\/br><\/p>\n<h3>Distribution uniforme rectangulaire<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=930s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Une distribution uniforme rectangulaire<\/span><\/strong><\/a> sur un intervalle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> est d\u00e9finie par la fonction de densit\u00e9 suivante<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\left\\{\\begin{array}{lll}\n\n\\displaystyle\\frac{1}{b-a} &amp; ; &amp; x\\in[a,b] \\\\ \\\\\n\n0 &amp; ; &amp; E.O.C.\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Si nous laissons tomber une petite balle sur un rail avec des limites aux extr\u00e9mit\u00e9s de l&#8217;intervalle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b],<\/span><\/span> et qu&#8217;elle rebondit \u00e9lastiquement en frappant les bords, alors la variable al\u00e9atoire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> associ\u00e9e \u00e0 la position d&#8217;arr\u00eat de la balle en raison du frottement a une distribution uniforme rectangulaire et s&#8217;\u00e9crit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim Un(a,b)<\/span>.<\/span><\/p>\n<p><a name=\"5\"><\/a><\/br><\/br><\/p>\n<h3>Distribution normale (Gaussienne)<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=1109s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Parmi les distributions continues<\/span><\/strong><\/a> de probabilit\u00e9, la distribution normale est l&#8217;une des plus populaires en pratique.<\/p>\n<h4>Distribution normale standard<\/h4>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=1150s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">La densit\u00e9 normale standard<\/span><\/strong><\/a> est d\u00e9finie par la fonction suivante<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\phi_{0,1}(x) = \\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Par sa d\u00e9finition, il est clair que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi\\gt 0.<\/span><\/span> Par cons\u00e9quent, il peut \u00eatre v\u00e9rifi\u00e9 que c&#8217;est une densit\u00e9 de probabilit\u00e9 simplement en v\u00e9rifiant que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int_{-\\infty}^{+\\infty}\\phi_{0,1}(x)dx<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Cette derni\u00e8re \u00e9galit\u00e9 peut \u00eatre d\u00e9montr\u00e9e en calculant la valeur de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I^2<\/span><\/span> lorsque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I =\\int_{-\\infty}^{+\\infty}\\phi(x)dx=1.<\/span><\/span> En effet, nous avons :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nI^2 &amp; = \\displaystyle \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2} dx \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2}dx \\\\ \\\\\n\n&amp; = \\displaystyle \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2} dx \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-y^2\/2} dy \\\\ \\\\\n\n&amp; = \\displaystyle \\frac{1}{{2\\pi}} \\int_{-\\infty}^{+\\infty} \\int_{-\\infty}^{+\\infty} e^{-\\frac{x^2 + y^2}{2}} dxdy \\\\ \\\\\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Mais il se trouve que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle \\int_{-\\infty}^{+\\infty} \\int_{-\\infty}^{+\\infty} e^{-\\frac{x^2 + y^2}{2}} dxdy = \\int_{0}^{2\\pi} \\int_{0}^{+\\infty} e^{-r^2\/2} rdr d\\theta = 2\\pi <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Par cons\u00e9quent, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I^2 = 1,<\/span><\/span> donc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I=\\int_{-\\infty}^{+\\infty}\\phi_{0,1}(x)dx = 1. <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u00c0 partir de la densit\u00e9 normale standard, la distribution normale standard est d\u00e9finie <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(x) = \\int_{-\\infty}^x\\phi_{0,1}(t)dt.<\/span><\/span> Si une variable al\u00e9atoire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> a une distribution normale standard, nous \u00e9crivons <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim N(0,1).<\/span><\/span> La distribution <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(x)<\/span><\/span> ne peut pas \u00eatre calcul\u00e9e explicitement, cependant, il existe des tables qui permettent d&#8217;obtenir rapidement des valeurs approximatives.<\/p>\n<h4>Distribution normale avec param\u00e8tres <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma<\/span><\/span><\/h4>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=1875s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u00c0 partir de la densit\u00e9 de la distribution normale standard<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_{0,1}<\/span><\/span> il est possible de construire la densit\u00e9 pour la distribution normale avec les param\u00e8tres <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma,<\/span><\/span> o\u00f9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu\\in\\mathbb{R}<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma\\gt 0 <\/span><\/span> sont respectivement la moyenne et l&#8217;\u00e9cart-type. La densit\u00e9 de la distribution normale avec ces param\u00e8tres est \u00e9crite de la mani\u00e8re suivante :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\phi_{\\mu,\\sigma}(x) = \\frac{1}{\\sigma}\\phi_{0,1}\\left(\\frac{x-\\mu}{\\sigma} \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Ainsi, la distribution normale avec param\u00e8tres <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{\\\u043c\u0443,\\\u03c3}(x)<\/span><\/span>, est de la forme suivante :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\Phi_{\\mu,\\sigma}(x) = \\int_{-\\infty}^x\\frac{1}{\\sigma}\\phi_{0,1}\\left(\\frac{t-\\mu}{\\sigma} \\right)dt = \\frac{1}{\\sqrt{2\\pi\\sigma}}\\int_{-\\infty}^x e^{-\\frac{(t-\\mu)^2}{2\\sigma^2}}dt<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Si la variable al\u00e9atoire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> a une distribution normale avec les param\u00e8tres <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu, \\sigma,<\/span><\/span> nous \u00e9crivons <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim N(\\mu, \\sigma).<\/span><\/span><\/p>\n<p><a name=\"6\"><\/a><\/br><\/br><\/p>\n<h3>Distribution de Weibull<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=2230s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">La distribution de Weibull<\/span><\/strong><\/a> avec les param\u00e8tres <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta \\gt 0<\/span><\/span> a une fonction de distribution de la forme suivante<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(t) = \\left\\{\\begin{array}{llr}\n\n\\left(1 - e^{-t\/\\alpha} \\right)^\\beta &amp;;&amp; t\\geq 0 \\\\ \\\\\n\n0 &amp;;&amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Si une variable al\u00e9atoire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> a une distribution de Weibull avec les param\u00e8tres <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta<\/span><\/span> nous \u00e9crivons <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim We(\\alpha,\\beta).<\/span><\/span> La distribution de Weibull est une g\u00e9n\u00e9ralisation de la distribution exponentielle, remarquez que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">We(\\alpha,1) = Ex(\\alpha).<\/span><\/span><\/p>\n<p><a name=\"7\"><\/a><\/br><\/br><\/p>\n<h3>Distribution Gamma<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=2311s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">La distribution Gamma<\/span><\/strong><\/a> avec les param\u00e8tres <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta,\\alpha<\/span><\/span> a une fonction de densit\u00e9 de la forme suivante<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(t) = \\left\\{\\begin{array}{llr}\n\n\\displaystyle \\frac{1}{\\alpha \\Gamma(\\beta)}\\left(\\frac{t}{\\alpha} \\right)^{\\beta-1}e^{-t\/\\alpha} &amp;;&amp; t\\geq 0 \\\\ \\\\\n\n0 &amp;;&amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">O\u00f9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma(s) = \\displaystyle \\int_0^{+\\infty}u^{s-1}e^{-u}du <\/span><\/span> est ce qu&#8217;on appelle la \u00abfonction Gamma\u00bb.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Une des propri\u00e9t\u00e9s les plus remarquables de la fonction Gamma est qu&#8217;elle permet de g\u00e9n\u00e9raliser les factorielles des nombres naturels aux r\u00e9els (et m\u00eame aux complexes). Il n&#8217;est pas difficile de v\u00e9rifier que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma(s+1) = s\\Gamma(s)<\/span><\/span> en int\u00e9grant par parties. De plus, comme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma(1)=1<\/span><\/span> il en r\u00e9sulte que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall n\\in\\mathbb{N}\\right)\\left(\\Gamma(n) = (n-1)! \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Si une variable al\u00e9atoire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> a une distribution Gamma avec les param\u00e8tres <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta, \\alpha<\/span><\/span> nous \u00e9crivons <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim Ga(\\alpha,\\beta).<\/span><\/span> La distribution Gamma est une autre g\u00e9n\u00e9ralisation de la distribution exponentielle, remarquez que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Ga(\\alpha,1) = Ex(\\alpha).<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Dans un processus de Poisson avec une fr\u00e9quence <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> (comme une d\u00e9sint\u00e9gration radioactive), si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T<\/span><\/span> est la variable al\u00e9atoire repr\u00e9sentant le moment o\u00f9 le m-\u00e8me \u00e9v\u00e9nement se produit ; alors, \u00e9tant donn\u00e9 un <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t\\geq 0<\/span><\/span> et un nombre <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">N<\/span><\/span> d&#8217;\u00e9v\u00e9nements qui se produisent dans l&#8217;intervalle de temps <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[0,t]<\/span><\/span> nous aurons <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t\\lt T \\leftrightarrow N\\lt m<\/span><\/span> et, comme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">N\\sim Po(ct),<\/span><\/span> il s&#8217;ensuit :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1-F_T(t) = P(T\\gt t) = \\displaystyle \\sum_{k=0}^{m-1}Po(k; ct)=e^{-ct}\\sum_{k=0}^{m-1}\\frac{(ct)^k}{k!}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Et donc, en d\u00e9rivant cela, nous d\u00e9couvrirons que la fonction de densit\u00e9 est<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f(t) = ce^{-ct}\\frac{(ct)^{m-1}}{(m-1)!}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Et donc, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T\\sim Ga(1\/c, m).<\/span><\/span><\/p>\n<p><a name=\"8\"><\/a><\/br><\/br><\/p>\n<h2>Exercices<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>Trouver la constante <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> telle que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f(x) = \\frac{c}{x^2+1}<\/span><\/span> soit une densit\u00e9 de probabilit\u00e9 et calculer la fonction de distribution correspondante (distribution de Cauchy)<\/li>\n<li>\u00c0 partir de la fonction de densit\u00e9 de la distribution <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Un(a.b),<\/span><\/span> d\u00e9terminer la fonction de distribution correspondante.<\/li>\n<li>D\u00e9montrer que la fonction <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{\\mu,\\sigma}(x)<\/span><\/span> est une fonction de distribution de probabilit\u00e9.<\/li>\n<\/ol>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/kdxgrB1h98g\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Distributions continues de probabilit\u00e9 R\u00e9sum\u00e9Ici, nous examinerons en profondeur le concept de distributions continues de probabilit\u00e9, en soulignant les caract\u00e9ristiques et les utilisations des cinq plus connues : la distribution exponentielle, la distribution uniforme rectangulaire, la distribution normale (Gaussienne), la distribution de Weibull et la distribution Gamma. Les formules math\u00e9matiques d\u00e9finissant chacune de ces distributions [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":26864,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":3,"footnotes":""},"categories":[569,682],"tags":[],"class_list":["post-27281","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>Distributions continues de probabilit\u00e9 - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"D\u00e9couvrez les distributions continues de probabilit\u00e9 les plus importantes: exponentielle, uniforme, normale, Weibull et Gamma.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/fr\/distributions-continues-de-probabilite\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Distributions continues de probabilit\u00e9\" \/>\n<meta property=\"og:description\" content=\"D\u00e9couvrez les distributions continues de probabilit\u00e9 les plus importantes: exponentielle, uniforme, normale, Weibull et Gamma.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/fr\/distributions-continues-de-probabilite\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-08T13:00:28+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-06-20T11:48:54+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/10\/distribucionescontinuas.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Distributions continues de probabilit\u00e9\" \/>\n<meta name=\"twitter:description\" content=\"D\u00e9couvrez les distributions continues de probabilit\u00e9 les plus importantes: exponentielle, uniforme, normale, Weibull et Gamma.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/10\/distribucionescontinuas.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/toposuranos.com\/material\/fr\/distributions-continues-de-probabilite\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/toposuranos.com\/material\/fr\/distributions-continues-de-probabilite\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Distributions continues de probabilit\u00e9\",\"datePublished\":\"2021-10-08T13:00:28+00:00\",\"dateModified\":\"2024-06-20T11:48:54+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/toposuranos.com\/material\/fr\/distributions-continues-de-probabilite\/\"},\"wordCount\":1858,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"https:\/\/toposuranos.com\/material\/fr\/distributions-continues-de-probabilite\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/10\/distribucionescontinuas.jpg\",\"articleSection\":[\"Math\u00e9matiques\",\"Probabilit\u00e9s et Statistiques\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/toposuranos.com\/material\/fr\/distributions-continues-de-probabilite\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/toposuranos.com\/material\/fr\/distributions-continues-de-probabilite\/\",\"url\":\"https:\/\/toposuranos.com\/material\/fr\/distributions-continues-de-probabilite\/\",\"name\":\"Distributions continues de probabilit\u00e9 - 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