{"id":2442,"date":"2020-08-08T11:00:00","date_gmt":"2020-08-08T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2442"},"modified":"2020-08-11T08:21:19","modified_gmt":"2020-08-10T23:21:19","slug":"gamma-distribution","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/gamma-distribution","title":{"rendered":"\u30ac\u30f3\u30de\u5206\u5e03"},"content":{"rendered":"\n\n\n\n\n<p>\u3000\u30ac\u30f3\u30de\u5206\u5e03\uff08gamma distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u30ac\u30f3\u30de\u5206\u5e03\u306f\u300c\u3042\u308b\u671f\u9593\\(f\\)\u306e\u9593\u306b\\(1\\)\u56de\u8d77\u3053\u308b\u4e8b\u8c61\u304c\u3001\\(n\\)\u56de\u8d77\u3053\u308b\u307e\u3067\u306b\u304b\u304b\u308b\u6642\u9593\u300d\u3092\u8868\u3057\u307e\u3059\u3002\u3082\u3046\u5c11\u3057\u5177\u4f53\u7684\u306a\u4f8b\u3092\u6319\u3052\u308b\u3068\u3001\u300c100\u5e74\u306b1\u56de\u306e\u5272\u5408\u3067\u3001\u5730\u9707\u304c\u767a\u751f\u3059\u308b\u3068\u304d\u3001\u5730\u9707\u304c2\u56de\u8d77\u3053\u308b\u307e\u3067\u306b\u304b\u304b\u308b\u671f\u9593\u306f\u3069\u306e\u304f\u3089\u3044\u306b\u306a\u308b\u304b\u300d\u3092\u8003\u3048\u305f\u3044\u3068\u304d\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<br>\u3000\u30ac\u30f3\u30de\u5206\u5e03\u306e\u5b9a\u7fa9\u304b\u3089\u3001\u6307\u6570\u5206\u5e03\u3068\u5f37\u304f\u3064\u306a\u304c\u308a\u304c\u3042\u308b\u3053\u3068\u3082\u5206\u304b\u308a\u307e\u3059\u3002\u6307\u6570\u5206\u5e03\u306f\u30ac\u30f3\u30de\u5206\u5e03\u306b\u304a\u3044\u3066\u300c\u3042\u308b\u671f\u9593\\(\\lambda\\)\u306e\u9593\u306b\\(1\\)\u56de\u8d77\u3053\u308b\u4e8b\u8c61\u304c\u3001\\(1\\)\u56de\u8d77\u3053\u308b\u307e\u3067\u306b\u304b\u304b\u308b\u6642\u9593\u300d\u3092\u8868\u3059\u3053\u3068\u304c\u3059\u3050\u306b\u308f\u304b\u308a\u307e\u3059\u3002<br>\u203b\u3053\u306e\u3053\u3068\u306f\u6307\u6570\u5206\u5e03\u306e\u8a18\u4e8b\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/exponential-distribution\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6307\u6570\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09-100x100.png\" alt=\"\u6307\u6570\u5206\u5e03\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u6307\u6570\u5206\u5e03<\/div>\n    <div class=\"blogcard_excerpt\">\u3000\u6307\u6570\u5206\u5e03\uff08exponential distribution\uff09\u306f\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u6307\u6570\u5206\u5e03\u306f\u3042\u308b\u4e8b\u8c61\u304c\u8d77\u3053\u3063\u3066\u304b\u3089\u3001\u6b21\u306b\u4e8b\u8c61\u304c\u767a\u751f\u3059\u308b\u307e\u3067\u306e\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<p>\u3000\u30ac\u30f3\u30de\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\)\u306f\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\alpha,\\beta\\)\u3092\u300c\u3042\u308b\u671f\u9593\\(\\beta\\)\u306e\u9593\u306b\\(1\\)\u56de\u8d77\u3053\u308b\u4e8b\u8c61\u306b\u5bfe\u3057\u3066\u3001\\(\\alpha\\)\u56de\u8d77\u3053\u308b\u307e\u3067\u306e\u6642\u9593\u300d\u3067\u304a\u304f\u3053\u3068\u3067\u3001\u78ba\u7387\u5909\u6570\\(X\\)\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\alpha,\\beta\\)\u306e\u30ac\u30f3\u30de\u5206\u5e03\u306b\u5f93\u3046\u3068\u3044\u3044\u3001\u8a18\u53f7\\(X\\sim Gam(\\alpha,\\beta)\\)\u3067\u8868\u3057\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> \u30ac\u30f3\u30de\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831<\/a><ul><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_2\">1.1<\/span> \u78ba\u7387\u5bc6\u5ea6\u95a2\u6570<\/a><\/li><\/ul><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_1\">2<\/span> \u30ac\u30f3\u30de\u5206\u5e03\u3092\u8003\u3048\u308b\u4e0a\u3067\u306e\u57fa\u790e\u77e5\u8b58<\/a><ul><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_2\">2.1<\/span> \u30d9\u30fc\u30bf\u95a2\u6570\u3068\u30ac\u30f3\u30de\u95a2\u6570<\/a><\/li><li><a href=\"#i-5\"><span class=\"toc_number toc_depth_2\">2.2<\/span> \u30d9\u30fc\u30bf\u95a2\u6570\u306e\u57fa\u672c\u6027\u8cea<\/a><\/li><li><a href=\"#i-6\"><span class=\"toc_number toc_depth_2\">2.3<\/span> \u30ac\u30f3\u30de\u95a2\u6570\u306e\u57fa\u672c\u6027\u8cea<\/a><\/li><\/ul><\/li><li><a href=\"#i-7\"><span class=\"toc_number toc_depth_1\">3<\/span> \u30ac\u30f3\u30de\u5206\u5e03\u3068\u6307\u6570\u5206\u5e03<\/a><\/li><li><a href=\"#i-8\"><span class=\"toc_number toc_depth_1\">4<\/span> \u30ac\u30f3\u30de\u5206\u5e03\u3068\u30a2\u30fc\u30e9\u30f3\u5206\u5e03<\/a><\/li><li><a href=\"#i-9\"><span class=\"toc_number toc_depth_1\">5<\/span> \u30ac\u30f3\u30de\u5206\u5e03\u3068\u30ab\u30a4\uff12\u4e57\u5206\u5e03<\/a><\/li><li><a href=\"#i-10\"><span class=\"toc_number toc_depth_1\">6<\/span> \u30ac\u30f3\u30de\u5206\u5e03\u306e\u518d\u751f\u6027<\/a><\/li><li><a href=\"#i-11\"><span class=\"toc_number toc_depth_1\">7<\/span> \u30ac\u30f3\u30de\u5206\u5e03\u3068\u30d9\u30fc\u30bf\u5206\u5e03<\/a><\/li><li><a href=\"#i-12\"><span class=\"toc_number toc_depth_1\">8<\/span> \u6a19\u6e96\u30ac\u30f3\u30de\u5206\u5e03<\/a><\/li><\/ul><\/div>\n<h2><span id=\"i\">\u30ac\u30f3\u30de\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<figure class=\"wp-block-table\">\n<table style=\"width: 100%; height: 600px;\">\n<tbody>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\">\u30d1\u30e9\u30e1\u30fc\u30bf<\/td>\n<td style=\"width: 77.8578%; height: 36px;\">\\(0 &lt; \\alpha,\\ \\ \\ 0 &lt; \\beta\\)<\/td>\n<\/tr>\n<tr style=\"height: 56px;\">\n<td style=\"width: 21.9698%; height: 56px;\">\u78ba\u7387\u5909\u6570\u306e\u7bc4\u56f2<\/td>\n<td style=\"width: 77.8578%; height: 56px;\">\\(0\\leq x \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/probability-function\">\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570<\/a><\/td>\n<td style=\"width: 77.8578%; height: 36px;\">\\(\\displaystyle \\frac{x^{\\alpha-1}\\exp\\left[ -\\frac{x}{\\beta} \\right]}{\\beta^{\\alpha}\\Gamma(\\alpha)} \\)<br \/>\u305f\u3060\u3057\u3001\\(\\Gamma(\\alpha)\\)\u306f\u30ac\u30f3\u30de\u95a2\u6570\u3067\u3059\u3002<br \/>\\(\\displaystyle \\Gamma(\\alpha) = \\int_{0}^{\\infty}t^{\\alpha-1}e^{-t}dt\\)<\/td>\n<\/tr>\n<tr style=\"height: 56px;\">\n<td style=\"width: 21.9698%; height: 56px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cumulative-distribution-function\">\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570<\/a><\/td>\n<td style=\"width: 77.8578%; height: 56px;\">\u2460\\(\\displaystyle \\alpha \\)\u304c\u6574\u6570\u306e\u3068\u304d<br \/>\\(\\displaystyle 1-\\exp\\left[ -\\frac{x}{\\beta} \\right]\\sum_{i=1}^{\\alpha-1}\\frac{ \\left( \\frac{x}{\\beta} \\right)^{i} }{ i! } \\)\n<p>\u00a0<\/p>\n<p>\u2461\\(\\beta=1\\)\u306e\u3068\u304d<br \/>\\(\\displaystyle\\frac{\\Gamma_{x}(\\alpha)}{\\Gamma(\\alpha)}\\)<br \/>\u3053\u3053\u3067\u3001\\(\\Gamma_{x}(\\alpha)\\)\u306f\u4e0d\u5b8c\u5168\u30ac\u30f3\u30de\u95a2\u6570\u3067\u3059\u3002<br \/>\\(\\displaystyle \\Gamma_{x}(\\alpha)=\\int_{0}^{x}t^{\\alpha-1}e^{-t}dt \\)<\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mgf\">\u7a4d\u7387\u6bcd\u95a2\u6570<\/a><\/td>\n<td style=\"width: 77.8578%; height: 36px;\">\\(\\displaystyle (1-\\beta t)^{-\\alpha},\\ \\ \\ \\ t&lt;\\frac{1}{\\beta} \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cf\">\u7279\u6027\u95a2\u6570<\/a><\/td>\n<td style=\"width: 77.8578%; height: 36px;\">\\(\\displaystyle (1-i\\beta t)^{-\\alpha}\\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\"><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: 77.8578%; height: 36px;\">\\(\\displaystyle -\\alpha\\log(1-\\beta t)\\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cgf\">\\(r\\)\u6b21\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8<\/a><\/td>\n<td style=\"width: 77.8578%; height: 36px;\">\\(\\displaystyle (r-1)!\\alpha\\beta^{r}\\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/moment\">\\(r\\)\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8<\/a><br \/>\uff08\u539f\u70b9\u307e\u308f\u308a\uff09<\/td>\n<td style=\"width: 77.8578%; height: 36px;\">\\(\\displaystyle \\frac{ \\beta^{r}\\Gamma(\\alpha+r) }{ \\Gamma(\\alpha) } \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mean-definition\">\u671f\u5f85\u5024<\/a><\/td>\n<td style=\"width: 77.8578%; height: 36px;\">\\( \\displaystyle \\alpha\\beta \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/variance\">\u5206\u6563<\/a><\/td>\n<td style=\"width: 77.8578%; height: 36px;\">\\( \\displaystyle \\alpha\\beta^{2} \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 35px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/05231022\">\u30e2\u30fc\u30c9<\/a><\/td>\n<td style=\"width: 77.8578%; height: 35px;\">\\( \\displaystyle (\\alpha-1)\\beta,\\ \\ \\alpha &gt; 1\\)<\/td>\n<\/tr>\n<tr style=\"height: 100px;\">\n<td style=\"width: 21.9698%; height: 100px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/skewness\">\u6b6a\u5ea6<\/a><\/td>\n<td style=\"width: 77.8578%; height: 100px;\">\n<p>\\(\\displaystyle \\frac{2}{\\sqrt{\\alpha}}\\)<\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 21.9698%; height: 29px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/kurtosis\">\u5c16\u5ea6<\/a><\/td>\n<td style=\"width: 77.8578%; height: 29px;\">\\( \\displaystyle3+\\frac{6}{\\alpha} \\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/figure>\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\/cgamd-mv\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30ac\u30f3\u30de\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u03b14\uff09-100x100.png\" alt=\"\u30ac\u30f3\u30de\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\">\u30ac\u30f3\u30de\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\u30ac\u30f3\u30de\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\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\/cgamd-mc\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30ac\u30f3\u30de\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u03b21\uff09-100x100.png\" alt=\"\u30ac\u30f3\u30de\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3011\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u30ac\u30f3\u30de\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3011<\/div>\n    <div class=\"blogcard_excerpt\">\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2605\u2606\u2606 \u3000\u3053\u306e\u8a18\u4e8b\u3067\u306f\u30ac\u30f3\u30de\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\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h3><span id=\"i-2\">\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570<\/span><\/h3>\n\n\n\n<p>\u69d8\u3005\u306a\\(\\alpha,\\beta\\)\u306b\u5bfe\u5fdc\u3059\u308b\u30ac\u30f3\u30de\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\\(\\beta=1\\)\u3067\u56fa\u5b9a\u3057\u305f\u5834\u5408\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"977\" height=\"705\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30ac\u30f3\u30de\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u03b21\uff09.png\" alt=\"\" class=\"wp-image-2453\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30ac\u30f3\u30de\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u03b21\uff09.png 977w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30ac\u30f3\u30de\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u03b21\uff09-300x216.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30ac\u30f3\u30de\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u03b21\uff09-768x554.png 768w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30ac\u30f3\u30de\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u03b21\uff09-320x231.png 320w\" sizes=\"(max-width: 977px) 100vw, 977px\" \/><\/figure>\n\n\n\n<p>\\(\\alpha=4\\)\u3067\u56fa\u5b9a\u3057\u305f\u5834\u5408\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"977\" height=\"705\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30ac\u30f3\u30de\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u03b14\uff09.png\" alt=\"\" class=\"wp-image-2452\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30ac\u30f3\u30de\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u03b14\uff09.png 977w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30ac\u30f3\u30de\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u03b14\uff09-300x216.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30ac\u30f3\u30de\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u03b14\uff09-768x554.png 768w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30ac\u30f3\u30de\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u03b14\uff09-320x231.png 320w\" sizes=\"(max-width: 977px) 100vw, 977px\" \/><\/figure>\n\n\n\n<h2><span id=\"i-3\">\u30ac\u30f3\u30de\u5206\u5e03\u3092\u8003\u3048\u308b\u4e0a\u3067\u306e\u57fa\u790e\u77e5\u8b58<\/span><\/h2>\n\n\n\n<p>(1)\u30d9\u30fc\u30bf\u95a2\u6570<br><div style=\"overflow-x: auto;\">\\begin{align}<br>B(\\alpha,\\beta) = \\int_{0}^{1}u^{\\alpha-1}(1-u)^{\\beta-1}du,\\ \\ \\ 0\\leq u\\leq 1<br>\\end{align}<\/div><\/p>\n\n\n\n<p>(2)\u30ac\u30f3\u30de\u95a2\u6570<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\Gamma(a) = \\int_{0}^{\\infty}\\exp[-u]u^{a-1}du,\\ \\ \\ a&gt;0<br>\\end{align}<\/div><\/p>\n\n\n\n<p>(3)\u30c7\u30a3\u30ac\u30f3\u30de\u95a2\u6570\uff08\u30d7\u30b5\u30a4\u95a2\u6570\uff09<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\psi(a) = \\frac{d}{da}\\log\\Gamma(a) = \\frac{\\displaystyle\\frac{d\\Gamma(a)}{da}}{\\Gamma(a)}<br>\\end{align}<\/div><\/p>\n\n\n\n<h3><span id=\"i-4\">\u30d9\u30fc\u30bf\u95a2\u6570\u3068\u30ac\u30f3\u30de\u95a2\u6570<\/span><\/h3>\n\n\n\n<p>\u30d9\u30fc\u30bf\u95a2\u6570\u3068\u30ac\u30f3\u30de\u95a2\u6570\u306b\u306f\u6b21\u306e\u3088\u3046\u306a\u95a2\u4fc2\u304c\u3042\u308a\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>B(\\alpha,\\beta)=\\frac{\\Gamma(\\alpha)\\Gamma(\\beta)}{\\Gamma(\\alpha+\\beta)} = B(\\beta,\\alpha)<br>\\end{align}<\/div><\/p>\n\n\n\n<h3><span id=\"i-5\">\u30d9\u30fc\u30bf\u95a2\u6570\u306e\u57fa\u672c\u6027\u8cea<\/span><\/h3>\n\n\n\n<p>\u30d9\u30fc\u30bf\u95a2\u6570\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u6027\u8cea\u3092\u6301\u3063\u3066\u3044\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>B(\\alpha+1,\\beta)=\\frac{\\alpha}{\\alpha+\\beta}B(\\alpha,\\beta)<br>\\end{align}<\/div><br>\u7279\u306b\\(\\alpha,\\beta\\)\u304c\u6574\u6570\u306e\u3068\u304d\u3001\u6b21\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>B(\\alpha,\\beta) &amp;= \\frac{(\\alpha-1)!(\\beta-1)!}{(\\alpha+\\beta-1)!} \\\\<br>B(1,1) &amp;= 1\\\\<br>B\\left( \\frac{1}{2},\\frac{1}{2} \\right) &amp;= \\pi<br>\\end{align}<\/div><\/p>\n\n\n\n<h3><span id=\"i-6\">\u30ac\u30f3\u30de\u95a2\u6570\u306e\u57fa\u672c\u6027\u8cea<\/span><\/h3>\n\n\n\n<p>\u30ac\u30f3\u30de\u95a2\u6570\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u6027\u8cea\u3092\u6301\u3063\u3066\u3044\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\Gamma(a)=(a-1)\\Gamma(a-1)<br>\\end{align}<\/div><br>\u7279\u306b\\(a\\)\u304c\u6574\u6570\u306e\u3068\u304d\u3001\u6b21\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\Gamma(a) &amp;= (a-1)! \\\\<br>\\Gamma(0) &amp;= 1 \\\\<br>\\Gamma(1) &amp;= 1 \\\\<br>\\Gamma\\left( \\frac{1}{2} \\right) &amp;= \\sqrt{\\pi} \\\\<br>\\Gamma\\left( a+\\frac{1}{2} \\right) &amp;= \\frac{(2a)!\\sqrt{\\pi}}{a!2^{2a}}<br>\\end{align}<\/div><\/p>\n\n\n\n<h2><span id=\"i-7\">\u30ac\u30f3\u30de\u5206\u5e03\u3068\u6307\u6570\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u3000\u672c\u8a18\u4e8b\u306e\u5192\u982d\u3067\u3082\u66f8\u3044\u3066\u3044\u307e\u3059\u304c\u3001\u30ac\u30f3\u30de\u5206\u5e03\u3068\u6307\u6570\u5206\u5e03\u306f\u5f37\u3044\u95a2\u9023\u6027\u304c\u3042\u308a\u307e\u3059\u3002\u30ac\u30f3\u30de\u5206\u5e03\\(Gamma(\\alpha,\\beta)\\)\u306b\u3064\u3044\u3066\u3001\\(\\alpha=1\\)\u3068\u3057\u305f\u3068\u304d\u30ac\u30f3\u30de\u5206\u5e03\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\beta\\)\u306e\u6307\u6570\u5206\u5e03\\(Exp(\\beta)\\)\u3068\u4e00\u81f4\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u9006\u306b\u3001\u78ba\u7387\u5909\u6570\\(X_{1},\\cdots,X_{n}\\)\u304c\u4e92\u3044\u306b\u72ec\u7acb\u306b\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\lambda\\)\u306e\u6307\u6570\u5206\u5e03\\(Exp(\\lambda)\\)\u306b\u5f93\u3046\u3068\u304d\u3001\u3053\u306e\u78ba\u7387\u5909\u6570\u306e\u548c\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\\(n,\\lambda\\)\u306e\u30ac\u30f3\u30de\u5206\u5e03<br><div style=\"overflow-x: auto;\">\\begin{align}<br>X_{1}+\\cdots+X_{n} \\sim Gamma(n,\\lambda)<br>\\end{align}<\/div><br>\u306b\u5f93\u3044\u307e\u3059\u3002<\/p>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/exponential-distribution\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6307\u6570\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09-100x100.png\" alt=\"\u6307\u6570\u5206\u5e03\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u6307\u6570\u5206\u5e03<\/div>\n    <div class=\"blogcard_excerpt\">\u3000\u6307\u6570\u5206\u5e03\uff08exponential distribution\uff09\u306f\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u6307\u6570\u5206\u5e03\u306f\u3042\u308b\u4e8b\u8c61\u304c\u8d77\u3053\u3063\u3066\u304b\u3089\u3001\u6b21\u306b\u4e8b\u8c61\u304c\u767a\u751f\u3059\u308b\u307e\u3067\u306e\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2><span id=\"i-8\">\u30ac\u30f3\u30de\u5206\u5e03\u3068\u30a2\u30fc\u30e9\u30f3\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u3000\\(\\alpha\\)\u304c\u6574\u6570\u306e\u3068\u304d\u3001\u30ac\u30f3\u30de\u5206\u5e03\u306f\u30a2\u30fc\u30e9\u30f3\u5206\u5e03\u3068\u4e00\u81f4\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2><span id=\"i-9\">\u30ac\u30f3\u30de\u5206\u5e03\u3068\u30ab\u30a4\uff12\u4e57\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u3000\u78ba\u7387\u5909\u6570\\(X\\)\u304c\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\alpha,\\beta\\)\u306e\u30ac\u30f3\u30de\u5206\u5e03\\(X\\sim Gam(\\alpha,\\beta)\\)\u306b\u5f93\u3046\u3068\u304d\u3001\\(2X\/\\beta\\)\u306f\u81ea\u7531\u5ea6\\(2\\alpha\\)\u306e\u30ab\u30a4\uff12\u4e57\u5206\u5e03<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\frac{2X}{\\beta}\\sim \\chi^{2}(2\\alpha)<br>\\end{align}<\/div><br>\u306b\u5f93\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2><span id=\"i-10\">\u30ac\u30f3\u30de\u5206\u5e03\u306e\u518d\u751f\u6027<\/span><\/h2>\n\n\n\n<p>\u78ba\u7387\u5909\u6570\\(X_{i}\\)\uff08\\(i=1,\\cdots,n\\)\uff09\u304c\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\alpha_{i},\\beta\\)\u306e\u30ac\u30f3\u30de\u5206\u5e03<br><div style=\"overflow-x: auto;\">\\begin{align}<br>X_{i} \\sim Gam(\\alpha_{i},\\beta)<br>\\end{align}<\/div><br>\u306b\u5f93\u3046\u3068\u304d\u3001\u3053\u306e\u78ba\u7387\u5909\u6570\u306e\u548c\u3082\u30ac\u30f3\u30de\u5206\u5e03\u306b\u5f93\u3044<br><div style=\"overflow-x: auto;\">\\begin{align}<br>X_{1}+\\cdots+X_{n} \\sim Gam\\left( \\sum_{i=1}^{n}\\alpha_{i},\\beta \\right)<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u8a3c\u660e\u306f\u3068\u3066\u3082\u7c21\u5358\u3067\u3001\u78ba\u7387\u5909\u6570\\(X_{i}\\)\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u304c\\((1-\\beta t)^{-\\alpha_{i}}\\)\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3001\u78ba\u7387\u5909\u6570\u306e\u548c\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>M_{X_{1}+\\cdots+X_{n}}(t) &amp;= \\mathrm{E}[e^{t(X_{1}+\\cdots+X_{n})}]\\\\<br>&amp;= \\prod_{i=1}^{n}\\mathrm{E}[e^{tX_{i}}] \\\\<br>&amp;= \\prod_{i=1}^{n} M_{X_{i}}(t) \\\\<br>&amp;= \\prod_{i=1}^{n} (1-\\beta t)^{-\\alpha_{i}} \\\\<br>&amp;= (1-\\beta t)^{-\\alpha_{1}-\\cdots-\\alpha_{n}}<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u306e\u6027\u8cea\u304b\u3089\u30ac\u30f3\u30de\u5206\u5e03\u306e\u518d\u751f\u6027\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mgf\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/9fc17caf-9cad-4670-b857-eb43ec911ab9-100x100.png\" alt=\"\u7a4d\u7387\u6bcd\u95a2\u6570\u3068\u306f\u3010\u5b9a\u7fa9\u3068\u6027\u8cea\u3092\u89e3\u8aac\u3011\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u7a4d\u7387\u6bcd\u95a2\u6570\u3068\u306f\u3010\u5b9a\u7fa9\u3068\u6027\u8cea\u3092\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 \u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u95a2\u6570\u306b\u7a4d\u7387\u6bcd\u95a2\u6570\u3068\u3044\u3046\u3082\u306e\u304c\u3042\u308a\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u7528\u3044\u308c\u3070\u671f\u5f85\u5024\u3084\u5206\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2><span id=\"i-11\">\u30ac\u30f3\u30de\u5206\u5e03\u3068\u30d9\u30fc\u30bf\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u78ba\u7387\u5909\u6570\\(X_{1},X_{2}\\)\u304c\u305d\u308c\u305e\u308c\u72ec\u7acb\u306b\u30ac\u30f3\u30de\u5206\u5e03\\(X_{1}\\sim Gam(\\alpha_{1},\\beta),\\ X_{2}\\sim Gam(\\alpha_{2},\\beta)\\)\u306b\u5f93\u3046\u3068\u304d<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\frac{X_{1}}{X_{1}+X_{2}}\\sim Beta(\\alpha_{1},\\alpha_{2})<br>\\end{align}<\/div><br>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u203b\u30d9\u30fc\u30bf\u5206\u5e03\u306b\u3064\u3044\u3066\u306f\u4e0b\u306e\u30ea\u30f3\u30af\u304b\u3089\u3069\u3046\u305e<\/p>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/beta-distribution\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30fc\u30bf\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09\uff11-100x100.png\" alt=\"\u30d9\u30fc\u30bf\u5206\u5e03\uff08\u7b2c\uff11\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\uff09\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u30d9\u30fc\u30bf\u5206\u5e03\uff08\u7b2c\uff11\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\uff09<\/div>\n    <div class=\"blogcard_excerpt\">\u3000\u30d9\u30fc\u30bf\u5206\u5e03\uff08beta distribution\uff09\u306f\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u30d9\u30fc\u30bf\u5206\u5e03\u306b\u306f\u7b2c\uff11\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\u3068\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\u304c\u3042\u308a\u3001\u5358\u306b\u30d9\u30fc\u30bf\u5206\u5e03\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h2><span id=\"i-12\">\u6a19\u6e96\u30ac\u30f3\u30de\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u3000\u30ac\u30f3\u30de\u5206\u5e03\u306b\u304a\u3044\u3066\\(\\beta=1\\)\u306e\u3068\u304d\u3001\u30ac\u30f3\u30de\u5206\u5e03\u306f\u6a19\u6e96\u30d9\u30fc\u30bf\u5206\u5e03\u3068\u547c\u3070\u308c\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u3000\u30ac\u30f3\u30de\u5206\u5e03\uff08gamma distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u30ac\u30f3\u30de\u5206\u5e03\u306f\u300c\u3042\u308b\u671f\u9593\\(f\\)\u306e\u9593\u306b\\(1\\)\u56de\u8d77\u3053\u308b\u4e8b\u8c61\u304c &#8230; <\/p>\n","protected":false},"author":1,"featured_media":2452,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[84],"tags":[114,71],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30ac\u30f3\u30de\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u03b14\uff09.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2442"}],"collection":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/comments?post=2442"}],"version-history":[{"count":18,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2442\/revisions"}],"predecessor-version":[{"id":2842,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2442\/revisions\/2842"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2452"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2442"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2442"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2442"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}