{"id":2915,"date":"2020-08-23T11:00:00","date_gmt":"2020-08-23T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2915"},"modified":"2020-08-20T21:03:26","modified_gmt":"2020-08-20T12:03:26","slug":"student-t-distribution","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/student-t-distribution","title":{"rendered":"t\u5206\u5e03\uff08\u30b9\u30c1\u30e5\u30fc\u30c7\u30f3\u30c8\u306et\u5206\u5e03\uff09"},"content":{"rendered":"\n\n\n\n\n<p>\u3000t\u5206\u5e03\uff08\u30b9\u30c1\u30e5\u30fc\u30c7\u30f3\u30c8\u306et\u5206\u5e03\u3001(Student&#8217;s)t distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002t\u5206\u5e03\u306f\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u6a19\u672c\u304b\u3089\u306e\u63a8\u5b9a\u91cf\uff08\u6a19\u672c\u5e73\u5747\u3084\u56de\u5e30\u4fc2\u6570\u306a\u3069\uff09\u306b\u898b\u3089\u308c\u308b\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u7d71\u8a08\u691c\u5b9a\u3084\u5b66\u6821\u306e\u8a66\u9a13\u306a\u3069\u6a19\u672c\u3092\u6271\u3046\u554f\u984c\u304c\u51fa\u305f\u5834\u5408\u307b\u307c\u307b\u307c\u5fc5\u9808\u306e\u77e5\u8b58\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001\u5fc5\u305a\u899a\u3048\u3066\u304a\u304d\u305f\u3044\u5206\u5e03\u306e\u3072\u3068\u3064\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u8a18\u4e8b\u3067\u306ft\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002t\u5206\u5e03\u3092\u5b66\u3076\u524d\u306b\u3001<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/normal-distribution\">\uff1c\u6b63\u898f\u5206\u5e03\uff1e<\/a>\u3084<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/chi-square-distribution\">\uff1c\u30ab\u30a4\uff12\u4e57\u5206\u5e03\uff1e<\/a>\u306b\u3064\u3044\u3066\u77e5\u3063\u3066\u304a\u304f\u3068\u30b9\u30e0\u30fc\u30ba\u306b\u5b66\u7fd2\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u601d\u3044\u307e\u3059\u3002<br>\u3000t\u5206\u5e03\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u306f\u81ea\u7531\u5ea6\u306e\u307f\u3067\u3042\u308a\u3001\u81ea\u7531\u5ea6\\(m\\)\u306et\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\)\u306f\u8a18\u53f7\\(X\\sim t(m)\\)\u3067\u8868\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/normal-distribution\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09\u03bc\uff1d\uff10-100x100.png\" alt=\"\u6b63\u898f\u5206\u5e03\uff08\u30ac\u30a6\u30b9\u5206\u5e03\uff09\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u6b63\u898f\u5206\u5e03\uff08\u30ac\u30a6\u30b9\u5206\u5e03\uff09<\/div>\n    <div class=\"blogcard_excerpt\">\u3000\u6b63\u898f\u5206\u5e03\uff08\u30ac\u30a6\u30b9\u5206\u5e03\u3001normal distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u6b63\u898f\u5206\u5e03\u306f\u6570\u5b66\u7684\u306b\u3082\u4fbf\u5229\u306a\u6027\u8cea\u3092\u305f\u304f\u3055\u3093\u6301\u3063\u3066\u3044\u308b\u305f\u3081\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\/chi-square-distribution\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/\u30ab\u30a42\u4e57\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09-100x100.png\" alt=\"\u30ab\u30a42\u4e57\u5206\u5e03\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u30ab\u30a42\u4e57\u5206\u5e03<\/div>\n    <div class=\"blogcard_excerpt\">\u3000\u30ab\u30a42\u4e57\u5206\u5e03\uff08chi square distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u591a\u304f\u306e\u66f8\u7c4d\u3067\u306f\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306e\u5e73\u65b9\u548c\u3067\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\u30ab\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\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=\"#t\"><span class=\"toc_number toc_depth_1\">1<\/span> t\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831<\/a><\/li><li><a href=\"#i\"><span class=\"toc_number toc_depth_1\">2<\/span> \u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3068\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570<\/a><\/li><li><a href=\"#t-2\"><span class=\"toc_number toc_depth_1\">3<\/span> t\u5206\u5e03\u306e\u5b9a\u7fa9<\/a><\/li><li><a href=\"#tF\"><span class=\"toc_number toc_depth_1\">4<\/span> t\u5206\u5e03\u3068F\u5206\u5e03<\/a><\/li><li><a href=\"#t-3\"><span class=\"toc_number toc_depth_1\">5<\/span> t\u5206\u5e03\u3068\u6a19\u6e96\u6b63\u898f\u5206\u5e03<\/a><\/li><li><a href=\"#t-4\"><span class=\"toc_number toc_depth_1\">6<\/span> \u6b63\u898f\u6bcd\u96c6\u56e3\u304b\u3089\u306e\u6a19\u672c\u3068t\u5206\u5e03<\/a><\/li><li><a href=\"#t-5\"><span class=\"toc_number toc_depth_1\">7<\/span> \u56de\u5e30\u4fc2\u6570\u306e\u6700\u5c0f\u4e8c\u4e57\u63a8\u5b9a\u91cf\u3068t\u5206\u5e03<\/a><\/li><\/ul><\/div>\n\n\n\n\n<h2><span id=\"t\">t\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: 406px;\">\n<tbody>\n<tr style=\"height: 36px;\">\n<td style=\"width: 24.2529%; height: 36px;\">\u30d1\u30e9\u30e1\u30fc\u30bf<\/td>\n<td style=\"width: 75.5747%; height: 36px;\">\\(m&gt;0\\)\uff08\u6574\u6570\u3001\u81ea\u7531\u5ea6\u3092\u8868\u3057\u307e\u3059\uff09<\/td>\n<\/tr>\n<tr style=\"height: 56px;\">\n<td style=\"width: 24.2529%; height: 56px;\">\u78ba\u7387\u5909\u6570\u306e\u7bc4\u56f2<\/td>\n<td style=\"width: 75.5747%; height: 56px;\">\\(-\\infty&lt; x &lt;\\infty \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 24.2529%; 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: 75.5747%; height: 36px;\">\\(\\displaystyle \\frac{ \\Gamma\\left( \\frac{m+1}{2} \\right) }{ (\\pi m)^{\\frac{1}{2}}\\Gamma\\left( \\frac{m}{2} \\right)\\left( 1+\\frac{x^{2}}{m} \\right)^{\\frac{m+1}{2}} } \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 24.2529%; height: 36px;\">\u751f\u5b58\u95a2\u6570<\/td>\n<td style=\"width: 75.5747%; height: 36px;\">\\(\\displaystyle \\frac{1}{2}I_{j}\\left( \\frac{1}{2}m,\\ \\frac{1}{2} \\right) \\)<br \/>\u3053\u3053\u3067\\(I_{z}(a,b)\\)\u306f\u4e0d\u5b8c\u5168\u30d9\u30fc\u30bf\u95a2\u6570\u6bd4\u3092\u8868\u3057\u307e\u3059\u3002<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 24.2529%; height: 36px;\">\u5371\u967a\u5ea6\u95a2\u6570<\/td>\n<td style=\"width: 75.5747%; height: 36px;\">\\(\\displaystyle \\frac{f(x)}{\\frac{1}{2}I_{j}\\left( \\frac{1}{2}m,\\ \\frac{1}{2} \\right)} \\)<br \/>\u305f\u3060\u3057\u3001\\(f(x)\\)\u306f\u81ea\u7531\u5ea6\\(m\\)\u306et\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u8868\u3057\u307e\u3059\u3002<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 24.2529%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mean-definition\">\u671f\u5f85\u5024<\/a><\/td>\n<td style=\"width: 75.5747%; height: 36px;\">\\( \\displaystyle 0 \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 24.2529%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/variance\">\u5206\u6563<\/a><\/td>\n<td style=\"width: 75.5747%; height: 36px;\">\\( \\displaystyle \\frac{m}{m-2},\\ \\ \\ m&gt;2 \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 24.2529%; height: 35px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/05231022\">\u30e2\u30fc\u30c9<\/a><\/td>\n<td style=\"width: 75.5747%; height: 35px;\">\\( \\displaystyle 0 \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 24.2529%; height: 35px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/05242107-2\">\u5e73\u5747\u504f\u5dee<\/a><\/td>\n<td style=\"width: 75.5747%; height: 35px;\">\\( \\displaystyle \\frac{ m^{\\frac{1}{2}}\\Gamma\\left( \\frac{m-1}{2} \\right) }{ \\pi^{\\frac{1}{2}}\\Gamma\\left( \\frac{m}{2} \\right) } \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 24.2529%; height: 35px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/skewness\">\u6b6a\u5ea6<\/a><\/td>\n<td style=\"width: 75.5747%; height: 35px;\">\\(0,\\ \\ \\ m&gt;3\\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 24.2529%; height: 29px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/kurtosis\">\u5c16\u5ea6<\/a><\/td>\n<td style=\"width: 75.5747%; height: 29px;\">\\( \\displaystyle \\frac{3(m-2)}{m-4}=3+\\frac{6}{m-4},\\ \\ \\ m&gt;4\\)<\/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\/cstutd-mv\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/t\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09-100x100.png\" alt=\"t\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\">t\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\u306ft\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\u304a\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h2><span id=\"i\">\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3068\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570<\/span><\/h2>\n\n\n\n<p>\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3068\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u81ea\u7531\u5ea6\u304c\u5927\u304d\u304f\u306a\u308b\u3068\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306b\u8fd1\u3065\u3044\u3066\u3044\u304f\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"984\" height=\"546\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/t\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09.png\" alt=\"\" class=\"wp-image-2999\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/t\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09.png 984w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/t\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09-300x166.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/t\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09-768x426.png 768w\" sizes=\"(max-width: 984px) 100vw, 984px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"986\" height=\"546\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/t\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09.png\" alt=\"\" class=\"wp-image-3000\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/t\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09.png 986w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/t\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09-300x166.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/t\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09-768x425.png 768w\" sizes=\"(max-width: 986px) 100vw, 986px\" \/><\/figure>\n\n\n\n<h2><span id=\"t-2\">t\u5206\u5e03\u306e\u5b9a\u7fa9<\/span><\/h2>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>t\u5206\u5e03\u306e\u5b9a\u7fa9<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\u81ea\u7531\u5ea6\\(m\\)\u306et\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\)\u306f\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(Z\\)\u3068\u3001\u81ea\u7531\u5ea6\\(m\\)\u306e\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(V\\sim \\chi^{2}(m)\\)\u3092\u7528\u3044\u3066<div style=\"overflow-x: auto;\">\\begin{align}X=\\frac{Z}{\\sqrt{\\frac{V}{m}}}\\sim t(m)\\end{align}<\/div>\u3068\u306a\u308a\u307e\u3059\u3002\n<\/div>\n\n\n\n<h2><span id=\"tF\">t\u5206\u5e03\u3068F\u5206\u5e03<\/span><\/h2>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>t\u5206\u5e03\u3068F\u5206\u5e03<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\u81ea\u7531\u5ea6\\(m\\)\u306et\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\)\u306f\u3001F\u5206\u5e03\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<div style=\"overflow-x: auto;\">\\begin{align}X^{2}\\sim F(1,\\ m)\\end{align}<\/div>\n<\/div>\n\n\n\n<p>\u3053\u306e\u95a2\u4fc2\u306ft\u5206\u5e03\u306e\u5b9a\u7fa9\u3068F\u5206\u5e03\u306e\u5b9a\u7fa9\u304b\u3089\u660e\u3089\u304b\u3067\u3059\u3002\u3042\u307e\u308a\u3053\u306e\u95a2\u4fc2\u306f\u5229\u7528\u3059\u308b\u304c\u306a\u3044\u3067\u3059\u304c\u3001\u77e5\u8b58\u3068\u3057\u3066\u899a\u3048\u3066\u304a\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n<h2><span id=\"t-3\">t\u5206\u5e03\u3068\u6a19\u6e96\u6b63\u898f\u5206\u5e03<\/span><\/h2>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>t\u5206\u5e03\u3068\u6a19\u6e96\u6b63\u898f\u5206\u5e03<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\u81ea\u7531\u5ea6\\(m\\)\u306et\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\)\u306b\u3064\u3044\u3066\u3001\u81ea\u7531\u5ea6\\(m\\)\u304c\u5341\u5206\u306b\u5927\u304d\u3044\u3068\u304d\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306b\u53ce\u675f\u3057\u307e\u3059\u3002<div style=\"overflow-x: auto;\">\\begin{align}X\\rightarrow^{d}_{m\\rightarrow\\infty}N(0,1)\\end{align}<\/div>\n<\/div>\n\n\n\n<h4>\u8a3c\u660e<\/h4>\n\n\n\n<p>\u4e2d\u5fc3\u6975\u9650\u5b9a\u7406\u3092\u7528\u3044\u308b\u65b9\u6cd5\u304c\u3042\u308a\u307e\u3059\u304c\u3001\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304b\u3089\u76f4\u63a5\u6c42\u3081\u308b\u65b9\u6cd5\u3092\u7d39\u4ecb\u3057\u307e\u3059\u3002\u307e\u305a<div style=\"overflow-x: auto;\">\\begin{align}\\left( 1+\\frac{x^{2}}{m} \\right)^{-\\frac{m+1}{2}} &amp;= \\left[ \\left( 1+\\frac{x^{2}}{m} \\right)^{\\frac{m}{x^{2}}}\\right]^{\\frac{x^{2}}{m}\\left( -\\frac{m+1}{2} \\right)}\\\\<br>&amp;\\rightarrow_{m\\rightarrow\\infty}e^{-\\frac{x^{2}}{2}}\\end{align}<\/div>\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u307e\u3059\u3002\u5f0f\u5909\u5f62\u306b\u306f\u81ea\u7136\u5bfe\u6570\\(e\\)\u306e\u5b9a\u7fa9\u3092\u4f7f\u7528\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u304b\u3089\u3001\u81ea\u7531\u5ea6\\(m\\)\u306et\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\\(f(x)\\)\u306f<div style=\"overflow-x: auto;\">\\begin{align}<br>f(x) &amp;= \\frac{ \\Gamma\\left( \\frac{m+1}{2} \\right) }{ (\\pi m)^{\\frac{1}{2}}\\Gamma\\left( \\frac{m}{2} \\right)\\left( 1+\\frac{x^{2}}{m} \\right)^{\\frac{m+1}{2}} } \\\\<br>&amp;\\rightarrow_{m\\rightarrow\\infty} \\frac{ \\Gamma\\left( \\frac{m+1}{2} \\right) }{ (\\pi m)^{\\frac{1}{2}}\\Gamma\\left( \\frac{m}{2} \\right) }e^{-\\frac{x^{2}}{2}} \\\\<br>&amp;= \\frac{ 1 }{ \\sqrt{2\\pi} }e^{-\\frac{x^{2}}{2}}<br>\\end{align}<\/div>\u3068\u5909\u5f62\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306b\u53ce\u675f\u3059\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"has-text-align-right\">\u25a1<\/p>\n\n\n\n<h2><span id=\"t-4\">\u6b63\u898f\u6bcd\u96c6\u56e3\u304b\u3089\u306e\u6a19\u672c\u3068t\u5206\u5e03<\/span><\/h2>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u6b63\u898f\u6bcd\u96c6\u56e3\u304b\u3089\u306e\u6a19\u672c\u3068t\u5206\u5e03<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\u6b63\u898f\u5206\u5e03\\(N(\\mu,\\ \\sigma^{2})\\)\u306b\u5f93\u3046\u6bcd\u96c6\u56e3\u304b\u3089\u306en\u500b\u306e\u7121\u4f5c\u70ba\u6a19\u672c\\(X_{1},\\cdots,X_{n}\\)\u306b\u3064\u3044\u3066\u3001\u671f\u5f85\u5024\u3068\u5206\u6563\u306e\u4e0d\u504f\u63a8\u5b9a\u91cf<div style=\"overflow-x: auto;\">\\begin{align}\\bar{X}=\\frac{1}{n}\\sum_{i=1}^{n}X_{i},\\ \\ \\ s^{2}=\\frac{1}{n-1}\\sum_{i=1}^{n}(X_{i}-\\bar{X})^{2}\\end{align}<\/div>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3068\u304d\\(\\bar{X}\\)\u3068\\(s^{2}\\)\u306f\u4e92\u3044\u306b\u72ec\u7acb\u3067\u3042\u308a\u3001<div style=\"overflow-x: auto;\">\\begin{align}\\frac{\\bar{X}-\\mu}{\\frac{s}{\\sqrt{n}}}\\sim t(n-1)\\end{align}<\/div>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\n<\/div>\n\n\n\n<h4>\u8a3c\u660e<\/h4>\n\n\n\n<p>\u3000\u3053\u3053\u3067\u306f\u7c21\u5358\u5316\u306e\u305f\u3081\u30b3\u30af\u30e9\u30f3\u306e\u5b9a\u7406\u3092\u4f7f\u3063\u3066\u8a3c\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u6a19\u672c\u3092\u307e\u3068\u3081\u305f\u30d9\u30af\u30c8\u30eb\u3092\\(X={}^{T}\\!(X_{1},\\cdots,X_{n})\\)\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\\(1\\)\u304c\\(n\\)\u500b\u4e26\u3093\u3060\u30d9\u30af\u30c8\u30eb\\(1_{n}\\)\u3092\u7528\u3044\u3066\u3001\u671f\u5f85\u5024\u304a\u3088\u3073\u5206\u6563\u306e\u4e0d\u504f\u63a8\u5b9a\u91cf\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\bar{X} &amp;= (\\ {}^{T}\\!1_{n}1_{n})^{-1}\\ {}^{T}\\!1_{n}X\\\\<br>(n-1)\\frac{s^2}{\\sigma^{2}}&amp;=\\frac{^{T}\\!X}{\\sigma}(I_{n}-1_{n}({}^{T}\\!1_{n}1_{n})^{-1}\\ {}^{T}\\!1_{n})\\frac{X}{\\sigma} <br>\\end{align}<\/div>\u3053\u3053\u3067\u3001\u8a18\u53f7\u304c\u9577\u304f\u306a\u308b\u306e\u3067\\(P_{1_{n}}=1_{n}({}^{T}\\!1_{n}1_{n})^{-1}\\ {}^{T}\\!1_{n}\\)\u3001\\(H=I_{n}-P_{1_{n}}\\)\u3068\u304a\u304d\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u30b3\u30af\u30e9\u30f3\u306e\u5b9a\u7406\u304b\u3089<div style=\"overflow-x: auto;\">\\begin{align}1_{n}H=0\\end{align}<\/div>\u304c\u6210\u308a\u7acb\u3064\u306e\u3067\u3001\\(\\bar{X}\\)\u3068\\(s^{2}\\)\u304c\u4e92\u3044\u306b\u72ec\u7acb\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u3055\u3089\u306b\u3001\u884c\u5217\\(H\\)\u306b\u3064\u3044\u3066\u3001\u3053\u306e\u884c\u5217\u306f\u30d9\u30ad\u7b49\u884c\u5217\u3067\u3042\u308a\\(\\mathrm{tr}H=n-1\\)\u304c\u6210\u7acb\u3057\u3066\u3044\u308b\u306e\u3067\u3001\\(\\mathrm{Var}[X\/\\sigma]=I_{n}\\)\u3088\u308a\u30b3\u30af\u30e9\u30f3\u306e\u5b9a\u7406\u304b\u3089<div style=\"overflow-x: auto;\">\\begin{align}(n-1)\\frac{s^{2}}{\\sigma^{2}}=\\frac{^{T}\\!X}{\\sigma}H\\frac{X}{\\sigma}\\sim \\chi^{2}(n-1)\\end{align}<\/div>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u304b\u3089\u4eca\u3001\u793a\u3057\u305f\u3044\u3053\u3068\u306f\u6c42\u3081\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"has-text-align-right\">\u25a1<\/p>\n\n\n\n<p>\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u304a\u3088\u3073\u30b3\u30af\u30e9\u30f3\u306e\u5b9a\u7406\u306b\u3064\u3044\u3066\u306f\u5225\u8a18\u4e8b\u3067\u7d39\u4ecb\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/chi-square-distribution\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/\u30ab\u30a42\u4e57\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09-100x100.png\" alt=\"\u30ab\u30a42\u4e57\u5206\u5e03\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u30ab\u30a42\u4e57\u5206\u5e03<\/div>\n    <div class=\"blogcard_excerpt\">\u3000\u30ab\u30a42\u4e57\u5206\u5e03\uff08chi square distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u591a\u304f\u306e\u66f8\u7c4d\u3067\u306f\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306e\u5e73\u65b9\u548c\u3067\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\u30ab\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h2><span id=\"t-5\">\u56de\u5e30\u4fc2\u6570\u306e\u6700\u5c0f\u4e8c\u4e57\u63a8\u5b9a\u91cf\u3068t\u5206\u5e03<\/span><\/h2>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u56de\u5e30\u4fc2\u6570\u306e\u6700\u5c0f\u4e8c\u4e57\u63a8\u5b9a\u91cf\u3068t\u5206\u5e03<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\u56de\u5e30\u30e2\u30c7\u30eb<div style=\"overflow-x: auto;\">\\begin{align}y_{i}=\\beta_{0}+\\beta_{1}x_{i1}+\\cdots+\\beta_{p}x_{ip}+\\varepsilon_{i}\\end{align}<\/div>\u3092\u8003\u3048\u307e\u3059\u3002\u3053\u3053\u3067\u8aa4\u5dee\u9805\\(\\varepsilon_{i}\\)\u306f\u4e92\u3044\u306b\u72ec\u7acb\u306b\u6b63\u898f\u5206\u5e03\\(N(0,\\sigma^{2})\\)\u306b\u5f93\u3046\u3082\u306e\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\\(\\beta\\)\u306e\u6700\u5c0f\u4e8c\u4e57\u63a8\u5b9a\u91cf\u3092\\(\\widehat{\\beta}\\)\u3001\\(y_{i}\\)\u306e\u63a8\u5b9a\u91cf\u3092\\(\\widehat{y}_{i}\\)\u3001\\(\\sigma^{2}\\)\u306e\u4e0d\u504f\u63a8\u5b9a\u91cf\u3092<div style=\"overflow-x: auto;\">\\begin{align}s^{2}=\\frac{1}{n-(p+1)}\\sum_{i=1}^{n}(y_{i}-\\widehat{y}_{i})\\end{align}<\/div>\u3068\u3057\u305f\u3068\u304d\u3001\u6b21\u306e\u3088\u3046\u306a\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\n<div style=\"overflow-x: auto;\">\\begin{align}\\frac{\\widehat{\\beta}_{i}-\\beta_{i}}{s}\\sim t(n-(p+1))\\end{align}<\/div>\n<\/div>\n\n\n\n<p>\u8a3c\u660e\u306f\u56de\u5e30\u5206\u6790\u3067\u8a73\u3057\u304f\u6271\u3044\u307e\u3059\u304c\u3001\u30b3\u30c1\u30e9\u306e\u8a3c\u660e\u3082\u30b3\u30af\u30e9\u30f3\u306e\u5b9a\u7406\u3092\u4f7f\u7528\u3057\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u3000t\u5206\u5e03\uff08\u30b9\u30c1\u30e5\u30fc\u30c7\u30f3\u30c8\u306et\u5206\u5e03\u3001(Student&#8217;s)t distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002t\u5206\u5e03\u306f\u6b63\u898f\u5206\u5e03 &#8230; <\/p>\n","protected":false},"author":1,"featured_media":2999,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[84,43],"tags":[124,71],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/t\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2915"}],"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=2915"}],"version-history":[{"count":32,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2915\/revisions"}],"predecessor-version":[{"id":3011,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2915\/revisions\/3011"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2999"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2915"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2915"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2915"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}