{"id":28044,"date":"2024-08-18T07:18:06","date_gmt":"2024-08-18T07:18:06","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28044"},"modified":"2024-08-18T07:18:06","modified_gmt":"2024-08-18T07:18:06","slug":"comment-calculer-la-distribution-normale-en-utilisant-la-table","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/comment-calculer-la-distribution-normale-en-utilisant-la-table\/","title":{"rendered":"Comment calculer la distribution normale en utilisant la table ?"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n  <center><\/p>\n<h1>Comment calculer la distribution normale en utilisant la table ?<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\">\n      <strong>R\u00e9sum\u00e9<\/strong><br \/>\n      <em>Dans ce cours, nous aborderons le th\u00e8me de la distribution normale, l&#8217;une des distributions de probabilit\u00e9 continues les plus courantes. Nous analyserons comment une variable al\u00e9atoire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> avec des 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> peut suivre une distribution normale et comment cette distribution peut \u00eatre standardis\u00e9e par une substitution dans l&#8217;int\u00e9grale. Cependant, nous remarquerons que, malgr\u00e9 la simplification gr\u00e2ce \u00e0 la standardisation, le calcul analytique de l&#8217;int\u00e9grale reste un d\u00e9fi, car il n&#8217;existe pas d&#8217;expressions alg\u00e9briques permettant d&#8217;exprimer son r\u00e9sultat. Pour surmonter ce probl\u00e8me, nous pr\u00e9senterons une table de la distribution normale standard construite dans Excel, qui offre des approximations num\u00e9riques des valeurs de la fonction. Nous expliquerons en d\u00e9tail comment utiliser la table, y compris l&#8217;interpr\u00e9tation des probabilit\u00e9s cumul\u00e9es comme des aires sous la courbe. Enfin, le cours se termine par une s\u00e9rie d&#8217;exercices pratiques centr\u00e9s sur le calcul des probabilit\u00e9s et des estimations en utilisant la table de la distribution normale et le concept de distribution normale standard.<\/em>\n    <\/p>\n<p>  <\/center><br \/>\n  <\/p>\n<p style=\"text-align:center;\">\n    <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>Montrer<\/strong> la relation entre la distribution normale standard et la distribution normale avec des 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> \u00e9tablis, par une substitution int\u00e9grale.<\/li>\n<li><strong>R\u00e9soudre<\/strong> des probl\u00e8mes pratiques en utilisant la table de distribution normale standard.<\/li>\n<li><strong>Construire<\/strong> et utiliser une table de distribution normale standard en utilisant Excel.<\/li>\n<\/ol>\n<p>  <center><br \/>\n    <strong><u>TABLE DES MATI\u00c8RES<\/u> :<\/strong><br \/>\n    <a href=\"#1\">Le probl\u00e8me et la solution de la distribution normale<\/a><br \/>\n    <a href=\"#2\">Comment utiliser cette table ?<\/a><br \/>\n    <a href=\"#3\">Exercices<\/a><br \/>\n  <\/center><br \/>\n  <center><br \/>\n    <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/GdoFK696k3s\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><br \/>\n  <\/center>\n<\/div>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Le probl\u00e8me et la solution de la distribution normale<\/h2>\n<p style=\"text-align:justify;\">L&#8217;une des <a href=\"http:\/\/toposuranos.com\/material\/es\/distribuciones-continuas-de-probabilidad\/\" rel=\"noopener\" target=\"_blank\">distributions continues<\/a> de probabilit\u00e9 les plus utilis\u00e9es est la distribution normale. Une variable al\u00e9atoire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> suit une distribution normale avec des param\u00e8tres <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu,\\sigma<\/span><\/span> si<\/p>\n<p class=\"eq\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X\\leq x) =\\Phi_{\\mu,\\sigma}(x) = \\int_{-\\infty}^x \\frac{1}{\\sqrt{2\\pi}\\sigma} e^{-\\frac{1}{2} \\left(\\frac{t-\\mu}{\\sigma}\\right)^2}dt <\/span><\/span><\/p>\n<p>et aussi <a href=\"https:\/\/www.youtube.com\/watch?v=kdxgrB1h98g&amp;t=1188s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">nous avons vu<\/span><\/strong><\/a> que cette distribution peut \u00eatre \u00abstandardis\u00e9e\u00bb si nous faisons dans l&#8217;int\u00e9grale une substitution du type <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle z= \\frac{t-\\mu}{\\sigma},<\/span><\/span> obtenant ainsi :<\/p>\n<p class=\"eq\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\Phi_{\\mu,\\sigma}(x) = \\int_{-\\infty}^{\\frac{x-\\mu}{\\sigma}} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{t^2}{2} }dt = \\Phi_{0,1}\\left(\\frac{x-\\mu}{\\sigma} \\right) <\/span><\/span><\/p>\n<p style=\"text-align:justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=GdoFK696k3s&amp;t=270s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>Bien que ce processus de standardisation<\/strong><\/span><\/a> simplifie l&#8217;int\u00e9grale, il n&#8217;est toujours pas suffisant pour nous permettre de calculer une seule probabilit\u00e9 pour cette distribution. En fait, \u00e9valuer cette int\u00e9grale de mani\u00e8re analytique n&#8217;est pas quelque chose que nous pouvons faire ; en r\u00e9alit\u00e9, il n&#8217;existe pas d&#8217;expressions alg\u00e9briques permettant d&#8217;exprimer le r\u00e9sultat de ces int\u00e9grales, ce qui nous met en difficult\u00e9. Cependant, si nous disposons d&#8217;approximations num\u00e9riques pour certaines valeurs de la fonction de distribution normale standard, nous pouvons commencer \u00e0 faire certaines estimations. Ces valeurs sont r\u00e9sum\u00e9es dans cette <strong>table de distribution normale standard<\/strong> que j&#8217;ai construite dans Excel pour vous &gt;:D<\/p>\n<p><center><br \/>\n  <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/tabla-distribucion-normalb.png\" alt=\"\" width=\"784\" height=\"774\" class=\"aligncenter size-full wp-image-26819 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/tabla-distribucion-normalb.png\" alt=\"\" width=\"784\" height=\"774\" class=\"aligncenter size-full wp-image-26819 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/tabla-distribucion-normalb.png 784w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/tabla-distribucion-normalb-300x296.png 300w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/tabla-distribucion-normalb-768x758.png 768w\" sizes=\"(max-width: 784px) 100vw, 784px\" \/><\/noscript><br \/>\n  <em>Vous pouvez t\u00e9l\u00e9charger cette table au format Excel en suivant ce <a href=\"https:\/\/www.dropbox.com\/s\/9yff75b70wd75xn\/tabla%20-%20distribucion%20normal%20estandar.xlsx?dl=0\" rel=\"noopener\" target=\"_blank\">lien.<\/a> Je vous recommande de consulter ce fichier car il vous apprendra \u00e0 cr\u00e9er votre propre table si vous ne trouvez pas celle dont vous avez besoin \ud83d\ude42<\/em><br \/>\n<\/center><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Comment utiliser cette table ?<\/h2>\n<p style=\"text-align:justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=GdoFK696k3s&amp;t=622s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>Pour utiliser cette table sans difficult\u00e9<\/strong><\/span><\/a>, vous devez vous rappeler que, en plus de vous indiquer une probabilit\u00e9 cumulative, elle repr\u00e9sente une aire sous la courbe (de la fonction de densit\u00e9). Cela se voit dans l&#8217;illustration suivante :<\/p>\n<p><center><br \/>\n  <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/comousarlatablab.png\" alt=\"\" width=\"784\" height=\"774\" class=\"aligncenter size-full wp-image-26820 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/comousarlatablab.png\" alt=\"\" width=\"784\" height=\"774\" class=\"aligncenter size-full wp-image-26820 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/comousarlatablab.png 784w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/comousarlatablab-300x296.png 300w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/comousarlatablab-768x758.png 768w\" sizes=\"(max-width: 784px) 100vw, 784px\" \/><\/noscript><br \/>\n<\/center><\/p>\n<p style=\"text-align:justify;\">Apr\u00e8s avoir compris cela, l&#8217;\u00e9tape suivante consiste \u00e0 expliquer l&#8217;utilisation de la table elle-m\u00eame. Ce que nous voyons ici, c&#8217;est que la valeur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">z<\/span><\/span> \u00e0 laquelle nous allons \u00e9valuer la fonction <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(z)<\/span><\/span> appara\u00eet divis\u00e9e en deux parties : dans la colonne, vous trouverez son extension jusqu&#8217;au premier chiffre d\u00e9cimal, et dans la ligne, vous trouverez le deuxi\u00e8me chiffre d\u00e9cimal ; par exemple : le nombre <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">z=1,72<\/span><\/span> peut \u00eatre trouv\u00e9 comme s&#8217;il \u00e9tait form\u00e9 par des \u00abcoordonn\u00e9es\u00bb : l&#8217;axe vertical est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1,7,<\/span><\/span> et l&#8217;axe horizontal <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0,02.<\/span><\/span> Enfin, l&#8217;approximation num\u00e9rique de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(1,72)<\/span><\/span> appara\u00eetra dans le bloc de coordonn\u00e9es \u00ab1,7\u00bb et \u00ab0,02\u00bb, qui montre une valeur de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0,957284.<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Exercices<\/h2>\n<ol>\n<li>Estimez la valeur de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(2,93)<\/span><\/span><\/li>\n<li>Estimez la valeur de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{\\mu=12,\\sigma=3}(11,5)<\/span><\/span><\/li>\n<li>Une variable al\u00e9atoire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> suit une distribution normale <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">N(\\mu=37,\\sigma=9).<\/span><\/span> Calculez <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(35 \\lt X \\lt 43).<\/span><\/span><\/li>\n<li>La quantit\u00e9 de tomates qui pourrissent chaque jour sur l&#8217;\u00e9tal d&#8217;un marchand a une moyenne <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu=50<\/span><\/span> avec un \u00e9cart-type <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma=15.<\/span><\/span> Si le marchand apporte des tomates 3 fois par semaine, estimez le nombre de jours o\u00f9 plus de 60 tomates pourriront pendant le mois.<\/li>\n<li>En supposant que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X \\sim N(\\mu,\\sigma)<\/span><\/span>, calculez les probabilit\u00e9s suivantes :\n<ol>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(\\mu - \\sigma \\leq X \\leq \\mu + \\sigma)<\/span><\/span><\/li>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X \\leq \\mu - \\sigma \\vee \\mu + \\sigma \\leq X)<\/span><\/span><\/li>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X \\leq \\mu - n\\sigma \\vee \\mu + n\\sigma \\leq X)<\/span><\/span><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><center><br \/>\n  <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/6tEAj-_dd_c\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><br \/>\n<\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Comment calculer la distribution normale en utilisant la table ? R\u00e9sum\u00e9 Dans ce cours, nous aborderons le th\u00e8me de la distribution normale, l&#8217;une des distributions de probabilit\u00e9 continues les plus courantes. Nous analyserons comment une variable al\u00e9atoire avec des param\u00e8tres et peut suivre une distribution normale et comment cette distribution peut \u00eatre standardis\u00e9e par une [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":26821,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":4,"footnotes":""},"categories":[569,682],"tags":[],"class_list":["post-28044","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Comment calculer la distribution normale en utilisant la table ? - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Apprenez \u00e0 utiliser la table standardis\u00e9e de la distribution normale pour calculer des 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