{"id":2007,"date":"2020-07-19T11:00:00","date_gmt":"2020-07-19T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2007"},"modified":"2020-07-19T20:39:52","modified_gmt":"2020-07-19T11:39:52","slug":"multinomial-distribution","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/multinomial-distribution","title":{"rendered":"\u591a\u9805\u5206\u5e03"},"content":{"rendered":"\n\n\n\n\n<p>\u591a\u9805\u5206\u5e03\uff08multinomial distribution\uff09\u306f\u96e2\u6563\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\\(1\\)\u56de\u306e\u8a66\u884c\u3067\\(k\\)\u901a\u308a\u306e\u30d1\u30bf\u30fc\u30f3\\(A_{1},\\cdots,A_{k}\\)\u304c\u5f97\u3089\u308c\u308b\u3068\u304d\u3001\u305d\u308c\u305e\u308c\u306e\u30d1\u30bf\u30fc\u30f3\u304c\u5f97\u3089\u308c\u308b\u78ba\u7387\u3092<br><div style=\"overflow-x: auto;\">\\begin{align}<br>p_{i}=\\mathrm{P}(A_{i}),\\ \\ \\ i=1,\\cdots,k<br>\\end{align}<\/div><br>\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u8a66\u884c\u3092\\(n\\)\u56de\u884c\u3046\u3068\u304d\u3001\u4e8b\u8c61\\(A_{i}\\)\u304c\u5f97\u3089\u308c\u308b\u56de\u6570\u3092\\(X_{i}\\)\u3068\u3059\u308b\u3068\u304d\u3001\u3053\u306e\\(X_{i}\\)\u306f\u591a\u9805\u5206\u5e03\u306b\u5f93\u3046\u3068\u3044\u3044\u307e\u3059\u3002\u3053\u306e\u8a18\u4e8b\u3067\u306f\u3001\u591a\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u3068\u4e3b\u306a\u6027\u8cea\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<div id=\"toc_container\" class=\"toc_light_blue no_bullets\"><p class=\"toc_title\">\u76ee\u6b21<\/p><ul class=\"toc_list\"><li><a href=\"#i\"><span class=\"toc_number toc_depth_1\">1<\/span> \u591a\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831<\/a><\/li><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_1\">2<\/span> \u591a\u9805\u5206\u5e03\u3068\uff12\u9805\u5206\u5e03<\/a><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_1\">3<\/span> \u30d1\u30e9\u30e1\u30fc\u30bf\u63a8\u5b9a<\/a><\/li><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_1\">4<\/span> \u591a\u9805\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u5206\u5e03<\/a><\/li><\/ul><\/div>\n<h2><span id=\"i\">\u591a\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831<\/span><\/h2>\n\n\n\n<p>\u203b\u3000\u8868\u306f\u6a2a\u306b\u30b9\u30af\u30ed\u30fc\u30eb\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-table\">\n<table style=\"width: 100%; height: 415px;\">\n<tbody>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/probability-function\">\u78ba\u7387\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(n!\\displaystyle\\prod_{i=1}^{k}\\left( \\frac{p_{i}^{x_{i}}}{x_{i}!} \\right)\\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mgf\">\u7a4d\u7387\u6bcd\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\displaystyle\\left( \\sum_{i=1}^{k}p_{i}\\exp[t_{i}]\\right)^{n}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 17.5862%;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cf\">\u7279\u6027\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%;\">\\(\\displaystyle\\left( \\sum_{i=1}^{k}p_{i}\\exp[it_{i}]\\right)^{n}\\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/pgf\">\u78ba\u7387\u6bcd\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\displaystyle\\left( \\sum_{i=1}^{k}p_{i}t_{i}\\right)^{n}\\)<\/td>\n<\/tr>\n<tr style=\"height: 56px;\">\n<td style=\"width: 17.5862%; height: 56px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cgf\">\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 56px;\">\\(n\\log\\displaystyle\\left( \\sum_{i=1}^{k}p_{i}\\exp[t_{i}]\\right)\\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 38px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mean-definition\">\u671f\u5f85\u5024<\/a><\/td>\n<td style=\"width: 82.2414%; height: 38px;\">\\( \\displaystyle np_{i} \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/variance\">\u5206\u6563<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\displaystyle np_{i}(1-p_{i}) \\)<\/td>\n<\/tr>\n<tr style=\"height: 100px;\">\n<td style=\"width: 17.5862%; height: 100px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/covariance\">\u5171\u5206\u6563<\/a><\/td>\n<td style=\"width: 82.2414%; height: 100px;\">\n<p>\\(\\displaystyle np_{i}p_{j},\\ \\ \\ \\ i\\neq j\\)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/figure>\n\n\n\n<h3>\u8a3c\u660e\u4e00\u89a7<\/h3>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/dmuld-mv\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/05\/PAK39_toumeinadice_TP_V-100x100.jpg\" alt=\"\u591a\u9805\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u591a\u9805\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011<\/div>\n    <div class=\"blogcard_excerpt\">\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u3053\u306e\u8a18\u4e8b\u3067\u306f\u591a\u9805\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u3092\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u6c42\u3081\u65b9\u304c\u5206\u304b\u3089\u306a\u3044\u65b9\u306f\u662f\u975e\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/dmuld-mc\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/05\/PAK39_toumeinadice_TP_V-100x100.jpg\" alt=\"\u591a\u9805\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u591a\u9805\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9<\/div>\n    <div class=\"blogcard_excerpt\">\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u3053\u306e\u8a18\u4e8b\u3067\u306f\u591a\u9805\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u3092\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9\u304c\u5206\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h2><span id=\"i-2\">\u591a\u9805\u5206\u5e03\u3068\uff12\u9805\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u3000\u591a\u9805\u5206\u5e03\u306e\u5b9a\u7fa9\u304b\u3089\u660e\u3089\u304b\u3067\u3059\u304c\u3001\\(k=2,\\ \\ p_{1}=p,\\ \\ \\ p_{2}=1-p\\)\u3068\u3059\u308b\u3068\u3001\u591a\u9805\u5206\u5e03\u306f\uff12\u9805\u5206\u5e03\u3068\u4e00\u81f4\u3057\u307e\u3059\u3002\uff12\u9805\u5206\u5e03\u306b\u3064\u3044\u3066\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/binomial-distribution\">\uff1c\uff12\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\uff1e<\/a>\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/binomial-distribution\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\uff12\u9805\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570-n10\uff09-100x100.png\" alt=\"\uff12\u9805\u5206\u5e03\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\uff12\u9805\u5206\u5e03<\/div>\n    <div class=\"blogcard_excerpt\">\uff12\u9805\u5206\u5e03\uff08binomial distribution\uff09\u306f\u96e2\u6563\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3001\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u8907\u6570\u56de\u884c\u3063\u305f\u969b\u306b\u5f97\u3089\u308c\u308b\u30c7\u30fc\u30bf\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u6210\u529f\u78ba\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h2><span id=\"i-3\">\u30d1\u30e9\u30e1\u30fc\u30bf\u63a8\u5b9a<\/span><\/h2>\n\n\n\n<p>\u78ba\u7387\\(p_{i}\\)\u306e\u6700\u5c24\u63a8\u5b9a\u91cf\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\widehat{p}_{i} &amp;= \\frac{X_{i}}{n}<br>\\end{align}<\/div><\/p>\n\n\n\n<h2><span id=\"i-4\">\u591a\u9805\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u591a\u9805\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u5206\u5e03\u3092\u56f3\u30fb\u8868\u3067\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002\u5404\u5206\u5e03\u306e\u8a73\u3057\u3044\u60c5\u5831\u306f\u8868\u306e\u4e2d\u306e\u30ea\u30f3\u30af\u304b\u3089\u304a\u9858\u3044\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"738\" height=\"514\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7.png\" alt=\"\" class=\"wp-image-2055\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7.png 738w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7-300x209.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7-320x223.png 320w\" sizes=\"(max-width: 738px) 100vw, 738px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/bernoulli-distribution\">\u30d9\u30eb\u30cc\u30fc\u30a4\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\uff11\u56de\u884c\u3046\u3068\u304d\u306e\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/categorical-distribution\">\u30ab\u30c6\u30b4\u30ea\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">1\u56de\u306e\u8a66\u884c\u3067\\(k\\)\u901a\u308a\u306e\u30d1\u30bf\u30fc\u30f3\u306e\u4e2d\u304b\u3089\u3069\u308c\u304b\u304c\u5f97\u3089\u308c\u308b\u53ef\u80fd\u6027\u304c\u3042\u308b\u8a66\u884c\u3092\u8868\u3059\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/bind\">\uff12\u9805\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\\(n\\)\u56de\u306e\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3067\\(x\\)\u56de\u6210\u529f\u3059\u308b\u3068\u304d\u306e\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/dgeod\">\u5e7e\u4f55\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u8907\u6570\u56de\u884c\u3046\u4e0a\u3067\u3001\u521d\u3081\u3066\u6210\u529f\u3059\u308b\u307e\u3067\u306e<br>\u8a66\u884c\u56de\u6570\u3092\u8868\u3059\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/dhgeod\">\u8d85\u5e7e\u4f55\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u8907\u6570\u56de\u884c\u3063\u3066\u3044\u304f\u4e0a\u3067\u3001\u305d\u306e\u90fd\u5ea6\u6210\u529f\u78ba\u7387\u304c<br>\u5909\u5316\u3059\u308b\u5206\u5e03\uff08\u6709\u9650\u500b\u306e\u30a2\u30bf\u30ea\u304f\u3058\u306a\u3069\uff09<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/nbiob\">\u8ca0\u306e\uff12\u9805\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u8907\u6570\u56de\u884c\u3046\u4e0a\u3067\u3001\\(k\\)\u56de\u6210\u529f\u3059\u308b\u307e\u3067\u306e<br>\u5931\u6557\u3059\u308b\u56de\u6570\u3092\u8868\u3059\u5206\u5e03\uff08\\(k=1\\)\u306e\u5834\u5408\u3001\u5e7e\u4f55\u5206\u5e03\u306b\u306a\u308a\u307e\u3059\uff09<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/dmuld\">\u591a\u9805\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u6210\u529f\u30fb\u5931\u6557\u306e\uff12\u7a2e\u985e\u3060\u3051\u3067\u306a\u304f\u3001\u8a66\u884c\u306e\u7d50\u679c\u304c<br>\u8907\u6570\u500b\u3042\u308b\u5834\u5408\u306e\u5206\u5e03\u3092\u8868\u3057\u307e\u3059<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/dpoid\">\u30dd\u30a2\u30bd\u30f3\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u6210\u529f\u78ba\u7387\u304c\u6975\u7aef\u306b\u5c0f\u3055\u304f\uff08\u3064\u307e\u308a\u6ec5\u591a\u306b\u8d77\u3053\u3089\u306a\u3044\uff09\u3001<br>\u8a66\u884c\u56de\u6570\u304c\u6975\u7aef\u306b\u5927\u304d\u3044\u5834\u5408\u306e\u5206\u5e03<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u591a\u9805\u5206\u5e03\uff08multinomial distribution\uff09\u306f\u96e2\u6563\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\\(1\\)\u56de\u306e\u8a66\u884c\u3067\\(k\\)\u901a\u308a\u306e\u30d1\u30bf\u30fc\u30f3\\(A_{1 &#8230; 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