{"id":2483,"date":"2020-08-11T11:00:00","date_gmt":"2020-08-11T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2483"},"modified":"2020-07-29T20:29:23","modified_gmt":"2020-07-29T11:29:23","slug":"normal-distribution","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/normal-distribution","title":{"rendered":"\u6b63\u898f\u5206\u5e03\uff08\u30ac\u30a6\u30b9\u5206\u5e03\uff09"},"content":{"rendered":"\n\n\n\n\n<p>\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\u3001\u7d71\u8a08\u7684\u63a8\u6e2c\u3092\u884c\u3046\u969b\u306b\u3001\u3053\u306e\u5206\u5e03\u3092\u4eee\u5b9a\u3059\u308b\u3053\u3068\u304c\u591a\u304f\u3001\u78ba\u7387\u5206\u5e03\u306e\u4e2d\u3067\u6700\u3082\u4f7f\u7528\u3055\u308c\u308b\u5206\u5e03\u306b\u306a\u308a\u307e\u3059\u3002<br>\u3000\u6b63\u898f\u5206\u5e03\u306f\u3001\u5206\u5e03\u306e\u5f62\u306f\u5e73\u5747\u3067\u5de6\u53f3\u5bfe\u79f0\u306b\u306a\u3063\u3066\u304a\u308a\u3001\u5e73\u5747\u306b\u8fd1\u3051\u308c\u3070\u8fd1\u3044\u307b\u3069\u89b3\u6e2c\u3055\u308c\u3084\u3059\u3044\u3082\u306e\u3068\u306a\u308a\u307e\u3059\u3002\u5177\u4f53\u7684\u306b\u306f\u30c6\u30b9\u30c8\u306e\u70b9\u6570\u306e\u5206\u5e03\u3084\u3001\u5b9f\u9a13\u306b\u304a\u3051\u308b\u6e2c\u5b9a\u306e\u8aa4\u5dee\u3001\u56de\u5e30\u5206\u6790\u306b\u304a\u3051\u308b\u8aa4\u5dee\u306a\u3069\u591a\u304f\u306e\u30c7\u30fc\u30bf\u306f\u3001\u307b\u307c\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002<br>\u3000\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\)\u306f\u3001\u671f\u5f85\u5024\\(\\mu\\)\u5206\u6563\\(\\sigma^{2}\\)\u3092\u7528\u3044\u3066\u8a18\u53f7\\(X\\sim N(\\mu,\\sigma^{2})\\)\u3067\u8868\u3055\u308c\u307e\u3059\u3002\u7279\u306b\\(\\mu=0,\\sigma^{2}=1\\)\u306e\u3068\u304d\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u3068\u547c\u3070\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u203b\u306a\u304a\u3001\u3053\u306e\u8a18\u4e8b\u3067\u306f\\(1\\)\u5909\u91cf\u306e\u6b63\u898f\u5206\u5e03\u306b\u3064\u3044\u3066\u6271\u3044\u307e\u3059\u3002\u591a\u5909\u91cf\u6b63\u898f\u5206\u5e03\u3001\u884c\u5217\u5909\u91cf\u6b63\u898f\u5206\u5e03\u306b\u3064\u3044\u3066\u306f\u4ee5\u4e0b\u306e\u30ea\u30f3\u30af\u304b\u3089\u304a\u9858\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p><\/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> \u6b63\u898f\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\u3068\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570<\/a><\/li><\/ul><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_1\">2<\/span> \u6b63\u898f\u5206\u5e03\u306e\u6027\u8cea<\/a><ul><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_2\">2.1<\/span> \u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306e\u898b\u5206\u3051\u65b9<\/a><\/li><li><a href=\"#i-5\"><span class=\"toc_number toc_depth_2\">2.2<\/span> \u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\u7dda\u5f62\u95a2\u6570<\/a><\/li><li><a href=\"#i-6\"><span class=\"toc_number toc_depth_2\">2.3<\/span> \u6b63\u898f\u5206\u5e03\u306e\u518d\u751f\u6027<\/a><\/li><li><a href=\"#i-7\"><span class=\"toc_number toc_depth_2\">2.4<\/span> \u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306e\u30a8\u30eb\u30df\u30fc\u30c8\u591a\u9805\u5f0f\u306b\u3088\u308b\u5c55\u958b<\/a><\/li><li><a href=\"#i-8\"><span class=\"toc_number toc_depth_2\">2.5<\/span> \u6b63\u898f\u5206\u5e03\u306e\u5e73\u5747\u504f\u5dee\u30e2\u30fc\u30e1\u30f3\u30c8<\/a><\/li><\/ul><\/li><li><a href=\"#i-9\"><span class=\"toc_number toc_depth_1\">3<\/span> \u6b63\u898f\u5206\u5e03\u3068\u4ed6\u306e\u5206\u5e03\u3068\u306e\u95a2\u4fc2<\/a><ul><li><a href=\"#i-10\"><span class=\"toc_number toc_depth_2\">3.1<\/span> \u30ab\u30a4\uff12\u4e57\u5206\u5e03<\/a><\/li><li><a href=\"#i-11\"><span class=\"toc_number toc_depth_2\">3.2<\/span> \u975e\u5fc3\u30ab\u30a4\uff12\u4e57\u5206\u5e03<\/a><\/li><li><a href=\"#i-12\"><span class=\"toc_number toc_depth_2\">3.3<\/span> \u5bfe\u6570\u6b63\u898f\u5206\u5e03<\/a><\/li><li><a href=\"#i-13\"><span class=\"toc_number toc_depth_2\">3.4<\/span> \u6b63\u898f\u5206\u5e03\u3068\u30e9\u30d7\u30e9\u30b9\u5206\u5e03<\/a><\/li><li><a href=\"#i-14\"><span class=\"toc_number toc_depth_2\">3.5<\/span> \u975e\u5bfe\u79f0\u6b63\u898f\u5206\u5e03<\/a><\/li><li><a href=\"#i-15\"><span class=\"toc_number toc_depth_2\">3.6<\/span> \u30d0\u30fc\u30f3\u30d0\u30a6\u30e0-\u30b5\u30f3\u30c0\u30fc\u30b9\u5206\u5e03<\/a><\/li><li><a href=\"#i-16\"><span class=\"toc_number toc_depth_2\">3.7<\/span> \u9006\u6b63\u898f\u5206\u5e03<\/a><\/li><\/ul><\/li><li><a href=\"#i-17\"><span class=\"toc_number toc_depth_1\">4<\/span> \u6b63\u898f\u5206\u5e03\u306b\u53ce\u675f\u3059\u308b\u78ba\u7387\u5206\u5e03<\/a><\/li><li><a href=\"#i-18\"><span class=\"toc_number toc_depth_1\">5<\/span> \u6b63\u898f\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u78ba\u7387\u5206\u5e03<\/a><\/li><\/ul><\/div>\n<h2><span id=\"i\">\u6b63\u898f\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: 615px;\">\n<tbody>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 36px;\">\u30d1\u30e9\u30e1\u30fc\u30bf<\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\mu,\\ \\ \\ 0 &lt; \\sigma\\)<\/td>\n<\/tr>\n<tr style=\"height: 56px;\">\n<td style=\"width: 17.5862%; height: 56px;\">\u78ba\u7387\u5909\u6570\u306e\u7bc4\u56f2<\/td>\n<td style=\"width: 82.2414%; height: 56px;\">\\(-\\infty&lt; x &lt;\\infty \\)<\/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\/probability-function\">\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\displaystyle \\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2\\sigma^{2}}(x-\\mu)^{2} \\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 \\exp\\left[ \\mu t+\\frac{1}{2}\\sigma^{2}t^{2} \\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\/cf\">\u7279\u6027\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\displaystyle \\exp\\left[ i\\mu t-\\frac{1}{2}\\sigma^{2}t^{2} \\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\/cgf\">\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\displaystyle \\mu t+\\frac{1}{2}\\sigma^{2}t^{2}\\)<\/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\/cgf\">\\(r\\)\u6b21\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\kappa_{1}=\\mu,\\ \\kappa_{2}=\\sigma^{2},\\ \\kappa_{r}=0\uff08r&gt;2\uff09\\)<\/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\/moment\">\\(r\\)\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8<\/a><br \/>\uff08\u5e73\u5747\u307e\u308f\u308a\uff09<\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\displaystyle \\left\\{\\begin{array}{cc}<br \/>\\mu_{r}=0 &amp; r\u304c\u5947\u6570 \\\\<br \/>\\mu_{r}=\\frac{\\sigma^{r}r!}{2^{\\frac{r}{2}}\\left( \\frac{r}{2} \\right)!} &amp; r\u304c\u5076\u6570\\end{array}\\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\/mean-definition\">\u671f\u5f85\u5024<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\( \\displaystyle \\mu \\)<\/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 \\sigma^{2} \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 36px;\">\u4e2d\u4f4d\u6570<\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\( \\displaystyle \\mu \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 35px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/05231022\">\u30e2\u30fc\u30c9<\/a><\/td>\n<td style=\"width: 82.2414%; height: 35px;\">\\( \\displaystyle \\mu \\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 35px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/05242107-2\">\u5e73\u5747\u504f\u5dee<\/a><\/td>\n<td style=\"width: 82.2414%; height: 35px;\">\\( \\displaystyle \\sigma\\left( \\frac{2}{\\pi} \\right)^{\\frac{1}{2}} \\)<\/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\/skewness\">\u6b6a\u5ea6<\/a><\/td>\n<td style=\"width: 82.2414%; height: 100px;\">\n<p>\\(\\displaystyle 0\\)<\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 29px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/kurtosis\">\u5c16\u5ea6<\/a><\/td>\n<td style=\"width: 82.2414%; height: 29px;\">\\( \\displaystyle 3\\)<\/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\/cnormd-mv1\">\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\u03c31-100x100.png\" alt=\"\u6b63\u898f\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\">\u6b63\u898f\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\u6b63\u898f\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\/cnormd-mc\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09\u03bc0-100x100.png\" alt=\"\u6b63\u898f\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\">\u6b63\u898f\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\u2606\u2606\u2606 \u3000\u3053\u306e\u8a18\u4e8b\u3067\u306f\u6b63\u898f\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\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\/cnormd-sk\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09\u03c3\uff1d1-100x100.png\" alt=\"\u6b63\u898f\u5206\u5e03\u306e\u6b6a\u5ea6\u30fb\u5c16\u5ea6\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\">\u6b63\u898f\u5206\u5e03\u306e\u6b6a\u5ea6\u30fb\u5c16\u5ea6\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 \u3000\u3053\u306e\u8a18\u4e8b\u3067\u306f\u6b63\u898f\u5206\u5e03\u306e\u6b6a\u5ea6\u30fb\u5c16\u5ea6\u3092\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u6b6a\u5ea6\u30fb\u5c16\u5ea6\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<h3><span id=\"i-2\">\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3068\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570<\/span><\/h3>\n\n\n\n<p>\u69d8\u3005\u306a\\(\\mu,\\sigma\\)\u306b\u5bfe\u5fdc\u3059\u308b\u6b63\u898f\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>\u30fb\\(\\mu=0\\)\u3067\u56fa\u5b9a\u3057\u305f\u5834\u5408\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3068\u7d2f\u7a4d\u5206\u5e03\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\/\u6b63\u898f\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09\u03bc\uff1d\uff10.png\" alt=\"\" class=\"wp-image-2511\" srcset=\"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.png 977w, 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\uff10-300x216.png 300w, 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\uff10-768x554.png 768w, 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\uff10-320x231.png 320w\" sizes=\"(max-width: 977px) 100vw, 977px\" \/><\/figure>\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\/\u6b63\u898f\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09\u03bc0.png\" alt=\"\" class=\"wp-image-2513\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09\u03bc0.png 977w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09\u03bc0-300x216.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09\u03bc0-768x554.png 768w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09\u03bc0-320x231.png 320w\" sizes=\"(max-width: 977px) 100vw, 977px\" \/><\/figure>\n\n\n\n<p>\u30fb\\(\\sigma^{2}=1\\)\u3067\u56fa\u5b9a\u3057\u305f\u5834\u5408\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3068\u7d2f\u7a4d\u5206\u5e03\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\/\u6b63\u898f\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09\u03c31.png\" alt=\"\" class=\"wp-image-2512\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09\u03c31.png 977w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09\u03c31-300x216.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09\u03c31-768x554.png 768w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09\u03c31-320x231.png 320w\" sizes=\"(max-width: 977px) 100vw, 977px\" \/><\/figure>\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\/\u6b63\u898f\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09\u03c3\uff1d1.png\" alt=\"\" class=\"wp-image-2514\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09\u03c3\uff1d1.png 977w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09\u03c3\uff1d1-300x216.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09\u03c3\uff1d1-768x554.png 768w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09\u03c3\uff1d1-320x231.png 320w\" sizes=\"(max-width: 977px) 100vw, 977px\" \/><\/figure>\n\n\n\n<h2><span id=\"i-3\">\u6b63\u898f\u5206\u5e03\u306e\u6027\u8cea<\/span><\/h2>\n\n\n\n<p>\u3000\u6b63\u898f\u5206\u5e03\u304c\u6301\u3064\u3001\u91cd\u8981\u306a\u6027\u8cea\u3092\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h3><span id=\"i-4\">\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306e\u898b\u5206\u3051\u65b9<\/span><\/h3>\n\n\n\n<p>\u3000\u3042\u308b\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304c\\(f(x)\\)\u3067\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u78ba\u7387\u5909\u6570\u304c\u9023\u7d9a\u578b\u3067\u3042\u308a\u3001\u5fae\u5206\u53ef\u80fd\u3067\u3042\u308b\u3068\u304d\u3001\u3053\u306e\\(f(x)\\)\u306e\u5c0e\u95a2\u6570\u304c\u3001\u3042\u308b\\(\\gamma&gt;0,\\ \\delta\\)\u304c\u5b58\u5728\u3057\u3066<br><div style=\"overflow-x: auto;\">\\begin{align}<br>f^{\\prime}(x) = -f(x)(\\gamma x+\\delta)<br>\\end{align}<\/div><br>\u3068\u8868\u305b\u308b\u3068\u304d\u3001\u78ba\u7387\u5909\u6570\\(X\\)\u306f\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002\u307e\u305f\u3001\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u306a\u3089\u3070\u3001\u3053\u306e\u5f0f\u3092\u6e80\u305f\u3059\u3068\u304d\u306e\u307f\u306b\u9650\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u3057\u304b\u3057\u3001\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304c\u308f\u304b\u3063\u3066\u3044\u308b\u5834\u5408\u306f\u304b\u306a\u308a\u9650\u3089\u308c\u307e\u3059\u3002\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304c\u672a\u77e5\u306e\u3068\u304d\u3001\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3063\u3066\u3044\u308b\u304b\u3069\u3046\u304b\u306f\u3001QQ\u30d7\u30ed\u30c3\u30c8\u306a\u3069\u306e\u65b9\u6cd5\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h3><span id=\"i-5\">\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\u7dda\u5f62\u95a2\u6570<\/span><\/h3>\n\n\n\n<p>\u3000\u78ba\u7387\u5909\u6570\\(X\\)\u304c\u671f\u5f85\u5024\\(\\mu\\)\u3001\u5206\u6563\\(\\sigma^{2}\\)\u306e\u6b63\u898f\u5206\u5e03\\(X\\sim N(\\mu,\\sigma^{2})\\)\u306b\u5f93\u3046\u3068\u304d\u3001\u5b9a\u6570\\(a,b\\)\u3092\u7528\u3044\u3066\\(Y=a+bX\\)\u3068\u5909\u63db\u3057\u305f\u3068\u304d\u3001\u5909\u63db\u3055\u308c\u305f\u78ba\u7387\u5909\u6570\\(Y\\)\u3082\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3044<br><div style=\"overflow-x: auto;\">\\begin{align}<br>Y=a+bX \\sim N(a+b\\mu,b^{2}\\sigma^{2})<br>\\end{align}<\/div><br>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<br>\u3000\u3053\u306e\u6027\u8cea\u3092\u7528\u3044\u3066\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\u6a19\u6e96\u5316\u306f\u4ee5\u4e0b\u306e\u5f0f\u3067\u884c\u3046\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>Z = \\frac{X-\\mu}{\\sigma}\\sim N(0,1)<br>\\end{align}<\/div><\/p>\n\n\n\n<h3><span id=\"i-6\">\u6b63\u898f\u5206\u5e03\u306e\u518d\u751f\u6027<\/span><\/h3>\n\n\n\n<p>\\(X_{1},\\cdots,X_{n}\\)\u304c\u4e92\u3044\u306b\u72ec\u7acb\u306b\u3001\u671f\u5f85\u5024\\(\\mu_{i}\\)\u3001\u5206\u6563\\(\\sigma^{2}\\)\u306e\u6b63\u898f\u5206\u5e03\\(X_{i}\\sim N(mu_{i},\\sigma_{i}^{2})\\)\uff08\\(i=1,\\cdots,n\\)\uff09\u306b\u5f93\u3046\u3068\u304d\u3001\u3053\u306e\u78ba\u7387\u5909\u6570\u306e\u548c\u3082\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3044<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\sum_{i=1}^{n}X_{i} = X_{1}+\\cdots+X_{n}\\sim N\\left( \\sum_{i}^{n}\\mu_{i}, \\sum_{i=1}^{2}\\sigma_{i}^{2} \\right)<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002\u7279\u306b\u3001\\(X_{1},\\cdots,X_{n}\\)\u304c\u540c\u3058\u671f\u5f85\u5024\\(\\mu\\)\u3068\u5206\u6563\\(\\sigma^{2}\\)\u3092\u6301\u3064\u3068\u304d\u3001<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\sum_{i=1}^{n}X_{i} = X_{1}+\\cdots+X_{n}\\sim N(n\\mu,n\\sigma^{2})<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002<br>\u3000\u4e0a\u3067\u7d39\u4ecb\u3057\u305f\u300c\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\u7dda\u5f62\u95a2\u6570\u300d\u306e\u6027\u8cea\u3092\u7528\u3044\u308c\u3070\u3001\u78ba\u7387\u5909\u6570\\(X_{1},\\cdots,X_{n}\\)\u3092\u305d\u308c\u305e\u308c\u3001\u5b9a\u6570\\(c_{1},\\cdots,c_{n}\\)\u3067\u5b9a\u6570\u500d\u3057\u305f\u3082\u306e\u306e\u548c\u3082\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3044\u3001<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\sum_{i=1}^{n}c_{i}X_{i} =c_{1}X_{1}+\\cdots+c_{n}X_{n} \\sim N\\left(\\sum_{i=1}^{n}c_{i}\\mu_{i},\\sum_{i=1}^{n}c_{i}^{2}\\sigma_{i}^{2}\\right)<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h3><span id=\"i-7\">\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306e\u30a8\u30eb\u30df\u30fc\u30c8\u591a\u9805\u5f0f\u306b\u3088\u308b\u5c55\u958b<\/span><\/h3>\n\n\n\n<p>\u3000\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim N(0,1)\\)\u306e\u5206\u5e03\u95a2\u6570\\(\\Phi(x)\\)\u306b\u95a2\u3057\u3066\u3001\u5206\u5e03\u95a2\u6570\u306f\u30a8\u30eb\u30df\u30fc\u30c8\u591a\u9805\u5f0f\u3092\u7528\u3044\u3066\u6b21\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\Phi (x) &amp;= \\int_{-\\infty}^{x}\\frac{1}{\\sqrt{2\\pi}}\\exp\\left[-\\frac{u^{2}}{2}\\right]du \\\\<br>&amp;= \\frac{1}{2}+\\int_{0}^{x}\\frac{1}{2\\pi}\\exp\\left[ -\\frac{u^{2}}{2} \\right]du \\\\<br>&amp;= \\frac{1}{2} + \\frac{1}{\\sqrt{2\\pi}}x\\exp\\left[ -\\frac{x^{2}}{8} \\right]\\sum_{j=0}^{\\infty}\\frac{H_{2j}\\left( \\frac{1}{2} \\right)}{(2j+1)!}\\left( \\frac{1}{2}x \\right)^{2j+1}<br>\\end{align}<\/div><br>\u3053\u3053\u3067\u3001\\(H_{r}(x)\\)\u306f\u30c1\u30a7\u30d3\u30b7\u30a7\u30d5-\u30a8\u30eb\u30df\u30fc\u30c8\u591a\u9805\u5f0f\u3067\u3059\u3002\u30c1\u30a7\u30d3\u30b7\u30a7\u30d5-\u30a8\u30eb\u30df\u30fc\u30c8\u591a\u9805\u5f0f\u306f\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\\(r\\)\u56de\u5fae\u5206\u3057\u305f\u6642\u306b\u5f97\u3089\u308c\u308b\u4fc2\u6570\u306b\u3042\u305f\u308b\u3082\u306e\u3067\u3059\u3002\u6b63\u78ba\u306b\u5b9a\u7fa9\u3059\u308b\u3068\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\\(f(x)\\)\u3068\u3059\u308b\u3068\u3001\u3053\u306e\u95a2\u6570\u306e\u5fae\u5206\u306b\u3064\u3044\u3066\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\frac{d}{dx}f(x) &amp;= (-1)xf(x) \\\\<br>\\frac{d^{2}}{dx^{2}}f(x) &amp;= (-1)^{2}(x^{2}-1)f(x) \\\\<br>\\frac{d^{3}}{dx^{3}}f(x) &amp;= (-1)^{3}(x^{3}-3x)f(x) \\\\<br>\\frac{d^{4}}{dx^{4}}f(x) &amp;= (-1)^{4}(x^{4}-6x^{2}+3)f(x) \\\\<br>&amp;\\vdots <br>\\end{align}<\/div><br>\u3053\u308c\u3089\u306e\u7d50\u679c\u3092\u4e00\u822c\u7684\u306b<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\frac{d^{r}}{dx^{r}}f(x) &amp;= (-1)^{r}H_{r}(x)f(x)<br>\\end{align}<\/div><br>\u3067\u8868\u3057\u307e\u3059\u3002\u3053\u306e\\(H_{r}(x)\\)\u3092\u30c1\u30a7\u30d3\u30b7\u30a7\u30d5-\u30a8\u30eb\u30df\u30fc\u30c8\u591a\u9805\u5f0f\u3068\u547c\u3073\u5177\u4f53\u7684\u306b\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>H_{0}(x) &amp;= 1 \\\\<br>H_{1}(x) &amp;= x \\\\<br>H_{2}(x) &amp;= x^{2}-1 \\\\<br>H_{3}(x) &amp;= x^{3}-3x \\\\<br>H_{4}(x) &amp;= x^{4}-6x^{2}+3 \\\\<br>&amp;\\vdots<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h3><span id=\"i-8\">\u6b63\u898f\u5206\u5e03\u306e\u5e73\u5747\u504f\u5dee\u30e2\u30fc\u30e1\u30f3\u30c8<\/span><\/h3>\n\n\n\n<p>\u78ba\u7387\u5909\u6570\\(X\\)\u304c\u6b63\u898f\u5206\u5e03\\(X\\sim N(\\mu,\\sigma^{2})\\)\u306b\u5f93\u3046\u3068\u304d\u3001\u5e73\u5747\u504f\u5dee\u306e\\(r\\)\u6b21\u306e\u5e73\u5747\u504f\u5dee\u30e2\u30fc\u30e1\u30f3\u30c8\u3092<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\upsilon_{r} &amp;= \\mathrm{E}\\left[ |X-\\mu|^{r} \\right]<br>\\end{align}<\/div><br>\u3068\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\upsilon_{1} &amp;= \\sigma\\left( \\frac{2}{\\pi} \\right)^{\\frac{1}{2}} \\\\<br>\\upsilon_{2} &amp;= \\sigma^{2} \\\\<br>\\upsilon_{3} &amp;= 2\\sigma^{3}\\left( \\frac{2}{\\pi} \\right)^{\\frac{1}{2}} \\\\<br>\\upsilon_{4} &amp;= 3\\sigma^{4} \\\\<br>\\upsilon_{5} &amp;= 8\\sigma^{5}\\left( \\frac{2}{\\pi} \\right)^{\\frac{1}{2}}<br>\\end{align}<\/div><br>\u7279\u306b\u3001\u4e00\u822c\u7684\u306b\u306f\u3001<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\upsilon_{r} &amp;= 2^{\\frac{r-1}{2}}\\Gamma\\left( \\frac{r+1}{2} \\right)\\sigma^{r}\\left( \\frac{2}{\\pi} \\right)^{\\frac{1}{2}}<br>\\end{align}<\/div><br>\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u203b\u30e2\u30fc\u30e1\u30f3\u30c8\u306b\u3064\u3044\u3066\u306f\u4ee5\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\/moment\">\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=\"\u30e2\u30fc\u30e1\u30f3\u30c8\u3068\u306f\u3010\u5b9a\u7fa9\u304b\u3089\u89e3\u8aac\u3011\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u30e2\u30fc\u30e1\u30f3\u30c8\u3068\u306f\u3010\u5b9a\u7fa9\u304b\u3089\u89e3\u8aac\u3011<\/div>\n    <div class=\"blogcard_excerpt\">\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u9ad8\u6821\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u5e73\u5747\u3084\u5206\u6563\u306a\u3069\u3092\u4e00\u822c\u5316\u3057\u305f\u3082\u306e\u304c\u30e2\u30fc\u30e1\u30f3\u30c8\uff08\u7a4d\u7387\uff09\u3067\u3059\u3002\u30e2\u30fc\u30e1\u30f3\u30c8\u306b\u306f[\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8]\u3068[\u5e73\u5747\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h2><span id=\"i-9\">\u6b63\u898f\u5206\u5e03\u3068\u4ed6\u306e\u5206\u5e03\u3068\u306e\u95a2\u4fc2<\/span><\/h2>\n\n\n\n<p>\u3000\u6b63\u898f\u5206\u5e03\u306f\u4ed6\u306e\u5206\u5e03\u3068\u305f\u304f\u3055\u3093\u306e\u95a2\u4fc2\u304c\u3042\u308a\u307e\u3059\u3002\u3059\u3079\u3066\u6319\u3052\u308b\u3068\u304d\u308a\u304c\u306a\u3044\u306e\u3067\u3001\u4ee3\u8868\u7684\u306a\u3082\u306e\u3060\u3051\u3092\u307e\u3068\u3081\u3066\u3044\u304d\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h3><span id=\"i-10\">\u30ab\u30a4\uff12\u4e57\u5206\u5e03<\/span><\/h3>\n\n\n\n<p>\u3000\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim N(0,1)\\)\u306b\u3064\u3044\u3066\u3001\u3053\u306e\u78ba\u7387\u5909\u6570\u306e\\(2\\)\u4e57\u306f\u81ea\u7531\u5ea6\\(1\\)\u306e\u30ab\u30a4\uff12\u4e57\u5206\u5e03<br><div style=\"overflow-x: auto;\">\\begin{align}<br>X^{2}\\sim \\chi^{2}(1)<br>\\end{align}<\/div><br>\u306b\u5f93\u3044\u307e\u3059\u3002\u307e\u305f\\(X_{1},\\cdots,X_{n}\\sim N(0,1)\\)\u3068\u4e92\u3044\u306b\u72ec\u7acb\u306b\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3063\u3066\u3044\u308b\u3068\u304d\u3001\uff12\u4e57\u548c\u306f\u81ea\u7531\u5ea6\\(n\\)\u306e\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306b\u5f93\u3044<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\sum_{i=1}^{n}X_{i}^{2} \\sim \\chi^{2}(n)<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h3><span id=\"i-11\">\u975e\u5fc3\u30ab\u30a4\uff12\u4e57\u5206\u5e03<\/span><\/h3>\n\n\n\n<p>\u3000\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim N(\\delta,1)\\)\u306b\u3064\u3044\u3066\u3001\u3053\u306e\u78ba\u7387\u5909\u6570\u306e\\(2\\)\u4e57\u306f\u81ea\u7531\u5ea6\\(1\\)\u3001\u975e\u5fc3\u5ea6\\(\\delta^{2}\\)\u306e\u975e\u5fc3\u30ab\u30a4\uff12\u4e57\u5206\u5e03<br><div style=\"overflow-x: auto;\">\\begin{align}<br>X^{2}\\sim \\chi^{2}(1,\\delta^{2})<br>\\end{align}<\/div><br>\u306b\u5f93\u3044\u307e\u3059\u3002\u307e\u305f\\(X_{i}\\sim N(\\delta_{i},1)\\)\uff08\\(i=1,\\cdots,n\\)\uff09\u3068\u4e92\u3044\u306b\u72ec\u7acb\u306b\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3063\u3066\u3044\u308b\u3068\u304d\u3001\uff12\u4e57\u548c\u306f\u81ea\u7531\u5ea6\\(n\\)\u3001\u975e\u5fc3\u5ea6\\(\\sum_{i=1}^{n}\\delta_{i}^{2}\\)\u306e\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306b\u5f93\u3044<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\sum_{i=1}^{n}X_{i}^{2} \\sim \\chi^{2}\\left(n,\\sum_{i}^{n}\\delta_{i}^{2}\\right)<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h3><span id=\"i-12\">\u5bfe\u6570\u6b63\u898f\u5206\u5e03<\/span><\/h3>\n\n\n\n<p>\u3000\u3042\u308b\u78ba\u7387\u5909\u6570\\(X\\)\u306b\u5bfe\u6570\u3092\u3068\u3063\u305f\u3082\u306e\u3001\\(\\log X\\)\u304c\u671f\u5f85\u5024\\(\\mu\\)\u3001\u5206\u6563\\(\\sigma^{2}\\)\u306e\u6b63\u898f\u5206\u5e03\\(\\log X\\sim N(\\mu,\\sigma^{2})\\)\u306b\u5f93\u3046\u3068\u304d\u3001\u78ba\u7387\u5909\u6570\\(X\\)\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\mu,\\sigma^{2}\\)\u306e\u5bfe\u6570\u6b63\u898f\u5206\u5e03<br><div style=\"overflow-x: auto;\">\\begin{align}<br>X \\sim LN(\\mu,\\sigma^{2})<br>\\end{align}<\/div><br>\u306b\u5f93\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h3><span id=\"i-13\">\u6b63\u898f\u5206\u5e03\u3068\u30e9\u30d7\u30e9\u30b9\u5206\u5e03<\/span><\/h3>\n\n\n\n<p>\u3000\u78ba\u7387\u5909\u6570\\(X_{1},X_{2},X_{3},X_{4}\\)\u304c\u4e92\u3044\u306b\u72ec\u7acb\u306b\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\\(N(0,1)\\)\u306b\u5f93\u3046\u3068\u304d\u3001<br><div style=\"overflow-x: auto;\">\\begin{align}<br>Y &amp;= X_{1}X_{2}+X_{3}X_{4}<br>\\end{align}<\/div><br>\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\\(0,1\\)\u306e\u30e9\u30d7\u30e9\u30b9\u5206\u5e03<br><div style=\"overflow-x: auto;\">\\begin{align}<br>Y \\sim Laplace(0,1)<br>\\end{align}<\/div><br>\u306b\u5f93\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h3><span id=\"i-14\">\u975e\u5bfe\u79f0\u6b63\u898f\u5206\u5e03<\/span><\/h3>\n\n\n\n<p>\u3000\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\\(f(x)\\)\u3001\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\\(\\Phi(x)\\)\u3068\u3057\u305f\u3068\u304d\u3001\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304c<br><div style=\"overflow-x: auto;\">\\begin{align}<br>f(x) &amp;= 2f(x)\\Phi(\\lambda x),\\ \\ \\ \\ -\\infty &lt;\\lambda &lt;\\infty<br>\\end{align}<\/div><br>\u3068\u306a\u308b\u3068\u304d\u3001\u78ba\u7387\u5909\u6570\\(X\\)\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\lambda\\)\u306e\u975e\u5bfe\u79f0\u6b63\u898f\u5206\u5e03\uff08skew-normal distribution\uff09\u306b\u5f93\u3046\u3068\u3044\u3044\u3001\u8a18\u53f7\\(X\\sim SN(\\lambda)\\)\u3068\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<h3><span id=\"i-15\">\u30d0\u30fc\u30f3\u30d0\u30a6\u30e0-\u30b5\u30f3\u30c0\u30fc\u30b9\u5206\u5e03<\/span><\/h3>\n\n\n\n<p>\u78ba\u7387\u5909\u6570\\(Z\\)\u304c\u6a19\u6e96\u6b63\u898f\u5206\u5e03\\(Z\\sim N(0,1)\\)\u306b\u5f93\u3046\u3068\u304d\u3001\u78ba\u7387\u5909\u6570\\(X\\)\u304c<br><div style=\"overflow-x: auto;\">\\begin{align}<br>X &amp;= \\beta\\left[ \\frac{1}{2}Z\\alpha+\\left\\{ 1+\\left( \\frac{1}{2}Z\\alpha \\right)^{2} \\right\\}^{\\frac{1}{2}} \\right]^{2}<br>\\end{align}<\/div><br>\u306f\u30d0\u30fc\u30f3\u30d0\u30a6\u30e0-\u30b5\u30f3\u30c0\u30fc\u30b9\u5206\u5e03\uff08Birnbaum-Saunders distribution\uff09\u3068\u547c\u3070\u308c\u3001\\(X\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\frac{\\exp[-\\alpha^{2}]}{2\\alpha\\sqrt{2\\pi\\beta}}x^{-\\frac{3}{2}}(x+\\beta)\\exp\\left[ -\\frac{1}{2\\alpha^{2}}\\left(  \\frac{x}{\\beta}+\\frac{\\beta}{x}\\right) \\right]<br>\\end{align}<\/div><br>\u3067\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002\u307e\u305f\u78ba\u7387\u5909\u6570\\(Z\\)\u304c<br><div style=\"overflow-x: auto;\">\\begin{align}<br>Z &amp;= \\frac{1}{\\alpha}\\left( \\sqrt{\\frac{X}{\\beta}}-\\sqrt{\\frac{\\beta}{X}} \\right) \\sim N(0,1)<br>\\end{align}<\/div><br>\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u305f\u3068\u304d\u3001\u78ba\u7387\u5909\u6570\\(X\\)\u306f\u30d0\u30fc\u30f3\u30d0\u30a6\u30e0-\u30b5\u30f3\u30c0\u30fc\u30b9\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u203b\u3000\u3053\u306e\u30d0\u30fc\u30f3\u30d0\u30a6\u30e0-\u30b5\u30f3\u30c0\u30fc\u30b9\u5206\u5e03\u306f\u3042\u308b\u6750\u6599\u306e\u5bff\u547d\u3092\u8868\u3059\u5206\u5e03\u3068\u3057\u3066\u63d0\u6848\u3055\u308c\u305f\u5206\u5e03\u3067\u3059\u3002\u4e3b\u306b\u6750\u6599\u79d1\u5b66\u306b\u304a\u3051\u308b\u30de\u30a4\u30ca\u30fc\u5247\u3092\u9069\u5fdc\u3055\u305b\u305f\u3068\u304d\u3001\u3053\u306e\u30e2\u30c7\u30eb\u304c\u59a5\u5f53\u3067\u3042\u308b\u3068\u8003\u3048\u3089\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h3><span id=\"i-16\">\u9006\u6b63\u898f\u5206\u5e03<\/span><\/h3>\n\n\n\n<p>\u3000\u78ba\u7387\u5909\u6570\\(Y\\)\u304c\u6b63\u898f\u5206\u5e03\\(Y\\sim N(\\mu,\\sigma^{2})\\)\u306b\u5f93\u3046\u3068\u304d\u3001\u78ba\u7387\u5909\u6570\u306e\u9006\u6570\\(X=1\/Y\\)\u306f\u9006\u6b63\u898f\u5206\u5e03\uff08inverse normal distribution\uff09\u306b\u5f93\u3046\u3068\u3044\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h2><span id=\"i-17\">\u6b63\u898f\u5206\u5e03\u306b\u53ce\u675f\u3059\u308b\u78ba\u7387\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u3000\u78ba\u7387\u5206\u5e03\u306f\u305f\u304f\u3055\u3093\u306e\u7a2e\u985e\u304c\u3042\u308a\u307e\u3059\u304c\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\u306b\u6761\u4ef6\u4ed8\u3051\u305f\u308a\u3001\u6975\u9650\u3092\u3068\u3063\u305f\u308a\u3059\u308b\u3068\u3001\u6b63\u898f\u5206\u5e03\u306b\u53ce\u675f\u3059\u308b\u5206\u5e03\u304c\u3044\u304f\u3064\u304b\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u7ae0\u3067\u306f\u6b63\u898f\u5206\u5e03\u306b\u53ce\u675f\u3059\u308b\u78ba\u7387\u5206\u5e03\u3092\u3044\u304f\u3064\u304b\u7d39\u4ecb\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u306a\u304a\u3001\u4ee5\u4e0b\u3067\u7d39\u4ecb\u3059\u308b\u3082\u306e\u306f\u3059\u3079\u3066\u6a19\u6e96\u5316\u3057\u305f\u4e0a\u3067\u53ce\u675f\u3057\u3066\u3044\u308b\u306e\u3067\u3001\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044\u3002\uff08\u6a19\u6e96\u5316\u3068\u306f\u671f\u5f85\u5024\u3067\u5f15\u3044\u3066\u6a19\u6e96\u504f\u5dee\u3067\u5272\u308b\u64cd\u4f5c\u306e\u3053\u3068\u3067\u3059\u3002\uff09<\/p>\n\n\n\n<ul><li>\u30ab\u30a4\uff12\u4e57\u5206\u5e03\\(\\chi^{2}(n)\\)\u3001\u671f\u5f85\u5024\\(m\\)\u3001\u5206\u6563\\(2m\\)<br>\\(n\\rightarrow\\infty\\)\u306e\u3068\u304d\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\\(N(0,1)\\)\u306b\u53ce\u675f\u3057\u307e\u3059\u3002<br>\u3000<\/li><li>\u30ac\u30f3\u30de\u5206\u5e03\\(Gam(\\alpha,\\beta)\\)\u3001\u671f\u5f85\u5024\\(\\alpha\\beta\\)\u3001\u5206\u6563\\(\\alpha\\beta^{2}\\)<br>\\(\\alpha\\rightarrow\\infty\\)\u306e\u3068\u304d\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\\(N(0,1)\\)\u306b\u53ce\u675f\u3057\u307e\u3059\u3002<br>\u3000<\/li><li>\\(t\\)\u5206\u5e03\\(t(n)\\)\u3001\u671f\u5f85\u5024\\(\\displaystyle 0\\)\u3001\u5206\u6563\\(\\displaystyle\\frac{n}{n-2}\\)\uff08\\(n &gt;2\\)\uff09<br>\\(n\\rightarrow\\infty\\)\u306e\u3068\u304d\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\\(N(0,1)\\)\u306b\u53ce\u675f\u3057\u307e\u3059\u3002<br>\u3000<\/li><li>\u5bfe\u6570\u6b63\u898f\u5206\u5e03\\(LN(\\mu,\\sigma^{2})\\)\u3001\u671f\u5f85\u5024\\(\\displaystyle\\exp\\left[ \\mu+\\frac{1}{2}\\sigma^{2} \\right]\\)\u3001\u5206\u6563\\(\\displaystyle\\exp\\left[ 2\\mu+\\sigma^{2} \\right](\\exp[\\sigma^{2}]-1)\\)<br>\\(\\sigma\\rightarrow\\infty\\)\u306e\u3068\u304d\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\\(N(0,1)\\)\u306b\u5f93\u3044\u307e\u3059\u3002<br>\u3000<\/li><li>\uff12\u9805\u5206\u5e03\\(Bin(n,p)\\)\u3001\u671f\u5f85\u5024\\(np\\)\u3001\u5206\u6563\\(np(1-p)\\)<br>\\(n\\rightarrow\\infty\\)\u306e\u3068\u304d\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\\(N(0,1)\\)\u306b\u5f93\u3044\u307e\u3059\u3002<br>\u3000<\/li><li>\u30d9\u30fc\u30bf\u5206\u5e03\\(Beta(\\alpha,\\beta)\\)\u3001\u671f\u5f85\u5024\\(\\frac{\\alpha}{\\alpha+\\beta}\\)\u3001\u5206\u6563\\(\\frac{\\alpha\\beta}{(\\alpha+\\beta)^{2}(\\alpha+\\beta+1)}\\)<br>\\(\\frac{\\alpha}{\\beta}\\)\u3092\u56fa\u5b9a\u3057\u306a\u304c\u3089\u3001\\(\\alpha\\rightarrow\\infty\\)\u304b\u3064\\(\\beta\\rightarrow\\infty\\)\u3068\u306a\u308b\u3068\u304d\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\\(N(0,1)\\)\u306b\u53ce\u675f\u3057\u307e\u3059\u3002<br>\u3000<\/li><li>\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\\(Po(\\lambda)\\)\u3001\u671f\u5f85\u5024\\(\\lambda\\)\u3001\u5206\u6563\\(\\lambda\\)<br>\\(\\lambda\\rightarrow\\infty\\)\u306e\u3068\u304d\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\\(N(0,1)\\)\u306b\u53ce\u675f\u3057\u307e\u3059\u3002<\/li><\/ul>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2><span id=\"i-18\">\u6b63\u898f\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u78ba\u7387\u5206\u5e03<\/span><\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"738\" height=\"404\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\u306e\u95a2\u9023\u5206\u5e03\uff08\u30de\u30c3\u30d7\uff09.png\" alt=\"\" class=\"wp-image-2555\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\u306e\u95a2\u9023\u5206\u5e03\uff08\u30de\u30c3\u30d7\uff09.png 738w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\u306e\u95a2\u9023\u5206\u5e03\uff08\u30de\u30c3\u30d7\uff09-300x164.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\u306e\u95a2\u9023\u5206\u5e03\uff08\u30de\u30c3\u30d7\uff09-320x175.png 320w\" sizes=\"(max-width: 738px) 100vw, 738px\" \/><\/figure>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\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 &#8230; <\/p>\n","protected":false},"author":1,"featured_media":2511,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[84],"tags":[115,71],"jetpack_featured_media_url":"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.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2483"}],"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=2483"}],"version-history":[{"count":46,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2483\/revisions"}],"predecessor-version":[{"id":2612,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2483\/revisions\/2612"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2511"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2483"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2483"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2483"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}