{"id":2088,"date":"2020-07-23T11:00:00","date_gmt":"2020-07-23T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2088"},"modified":"2020-07-20T22:24:01","modified_gmt":"2020-07-20T13:24:01","slug":"poisson-distribution","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/poisson-distribution","title":{"rendered":"\u30dd\u30a2\u30bd\u30f3\u5206\u5e03"},"content":{"rendered":"\n\n\n\n\n<p>\u3000\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08Poisson distribution\uff09\u306f\u96e2\u6563\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306f\u4e00\u5b9a\u9593\u9694\u306e\u6642\u9593\u3067\u767a\u751f\u3059\u308b\u4e8b\u8c61\u306e\u6570\u3092\u8868\u3057\u307e\u3059\u3002\u3082\u3046\u5c11\u3057\u5177\u4f53\u7684\u306b\u306f\u300c\u3042\u308b\u6642\u9593\u306e\u9593\u306b\u5e73\u5747\u3067\\(\\lambda\\)\u56de\u767a\u751f\u3059\u308b\u4e8b\u8c61\u304c\u3042\u3063\u305f\u3068\u304d\u3001\u3053\u306e\u4e8b\u8c61\u304c\\(X=k\\)\u56de\u767a\u751f\u3059\u308b\u300d\u3068\u3044\u3046\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u78ba\u7387\u5909\u6570\\(X\\)\u306f\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306b\u5f93\u3046\u3068\u3044\u3044\u3001\u8a18\u53f7\\(X\\sim Po(\\lambda)\\)\u3068\u8868\u3057\u307e\u3059\u3002\u4f3c\u305f\u3088\u3046\u306a\u5206\u5e03\u306b\u6307\u6570\u5206\u5e03\u3068\u3044\u3046\u3082\u306e\u304c\u3042\u308a\u307e\u3059\u304c\u3001\u3053\u3061\u3089\u306f\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3001\u6b21\u306b\u4e8b\u8c61\u304c\u306f\u3063\u305b\u3044\u3059\u308b\u307e\u3067\u306e\u671f\u9593\u3092\u3042\u3089\u308f\u3059\u78ba\u7387\u5206\u5e03\u306b\u306a\u308a\u307e\u3059\u3002<br>\u3000\u3082\u3046\u5c11\u3057\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306e\u5177\u4f53\u4f8b\u3092\u53d6\u308a\u4e0a\u3052\u308b\u3068\u3001\u300c10\u500b\u88fd\u54c1\u3092\u5236\u4f5c\u3057\u305f\u6642\u3001\u5e73\u5747\u3057\u30661\u500b\u4e0d\u826f\u54c1\u304c\u51fa\u3066\u304f\u308b\u300d\u3068\u3044\u3046\u6a5f\u68b0\u304c\u3042\u3063\u305f\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u6a5f\u68b0\u3092\u4f7f\u3063\u30661000\u500b\u88fd\u54c1\u3092\u5236\u4f5c\u3057\u305f\u6642\u3001\u51fa\u3066\u304f\u308b\u4e0d\u826f\u54c1\u306e\u6570\u3092\u8003\u3048\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u4e0d\u826f\u54c1\u306e\u6570\u3092\\(X\\)\u3068\u3057\u305f\u3068\u304d\u3001\\(\\lambda=100\\)\u306e\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<div id=\"toc_container\" class=\"toc_light_blue no_bullets\"><p class=\"toc_title\">\u76ee\u6b21<\/p><ul class=\"toc_list\"><li><a href=\"#i\"><span class=\"toc_number toc_depth_1\">1<\/span> \u30dd\u30a2\u30bd\u30f3\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> \u6f38\u5316\u5f0f\u306b\u3088\u308b\u30dd\u30a2\u30bd\u30f3\u78ba\u7387\u306e\u8a08\u7b97<\/a><\/li><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_1\">3<\/span> \uff12\u9805\u5206\u5e03\u3068\u30dd\u30a2\u30bd\u30f3\u5206\u5e03<\/a><\/li><li><a href=\"#i-5\"><span class=\"toc_number toc_depth_1\">4<\/span> \u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u3068\u6b63\u898f\u5206\u5e03<\/a><\/li><li><a href=\"#i-6\"><span class=\"toc_number toc_depth_1\">5<\/span> \u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306e\u518d\u751f\u6027<\/a><\/li><li><a href=\"#i-7\"><span class=\"toc_number toc_depth_1\">6<\/span> \u30d1\u30e9\u30e1\u30fc\u30bf\u63a8\u5b9a<\/a><\/li><li><a href=\"#i-8\"><span class=\"toc_number toc_depth_1\">7<\/span> \u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u5206\u5e03<\/a><\/li><\/ul><\/div>\n<h2><span id=\"i\">\u30dd\u30a2\u30bd\u30f3\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: 614px;\">\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;\">\\(0 &lt; \\lambda\\)<\/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;\">\\(0\\leq x \\)\u3000\u3000\\(x\\)\u306f\u6574\u6570<\/td>\n<\/tr>\n<tr style=\"height: 56px;\">\n<td style=\"width: 17.5862%; height: 56px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cumulative-distribution-function\">\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 56px;\">\\(\\displaystyle\\sum_{k=0}^{x}\\frac{\\lambda^{k}e^{-\\lambda}}{k!}\\)<\/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\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\displaystyle\\frac{\\lambda^{x}e^{-\\lambda}}{x!}\\)<\/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\" data-wplink-edit=\"true\">\u7a4d\u7387\u6bcd\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\displaystyle\\exp\\left[ \\lambda\\left\\{ \\exp[t]-1 \\right\\} \\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\/pgf\">\u78ba\u7387\u6bcd\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\displaystyle\\exp\\left[ \\lambda(t-1) \\right]\\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 31px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cf\">\u7279\u6027\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 31px;\">\\(\\displaystyle\\exp\\left[ \\lambda\\left\\{ \\exp[it]-1 \\right\\} \\right]\\)<\/td>\n<\/tr>\n<tr style=\"height: 56px;\">\n<td style=\"width: 17.5862%; height: 63px;\"><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: 63px;\">\\( \\displaystyle\\lambda\\left( \\exp[t]-1 \\right) \\)<\/td>\n<\/tr>\n<tr style=\"height: 56px;\">\n<td style=\"width: 17.5862%; height: 56px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cgf\">\\(r\\)\u6b21\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8<\/a><\/td>\n<td style=\"width: 82.2414%; height: 56px;\">\\( \\lambda \\)<\/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;\">\\( \\lambda \\)<\/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;\">\\( \\lambda \\)<\/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 \\frac{1}{\\sqrt{\\lambda}}\\)<\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/kurtosis\">\u5c16\u5ea6<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\( 3+\\displaystyle\\frac{1}{\\lambda} \\)<\/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\/dpoid-mv\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570\uff09-100x100.png\" alt=\"\u30dd\u30a2\u30bd\u30f3\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\">\u30dd\u30a2\u30bd\u30f3\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\u30dd\u30a2\u30bd\u30f3\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\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\/dpoid-mc\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09-100x100.png\" alt=\"\u30dd\u30a2\u30bd\u30f3\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\">\u30dd\u30a2\u30bd\u30f3\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 \u3053\u306e\u8a18\u4e8b\u3067\u306f\u30dd\u30a2\u30bd\u30f3\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\u3068\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570<\/span><\/h3>\n\n\n\n<p>\\(\\lambda=0.5,\\ 1,\\ 5,\\ 10\\)\u306b\u5bfe\u5fdc\u3059\u308b\u30dd\u30a2\u30bd\u30f3\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<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"980\" height=\"618\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570\uff09.png\" alt=\"\" class=\"wp-image-2094\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570\uff09.png 980w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570\uff09-300x189.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570\uff09-768x484.png 768w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570\uff09-320x202.png 320w\" sizes=\"(max-width: 980px) 100vw, 980px\" \/><\/figure>\n\n\n\n<p>\\(\\lambda=0.5,\\ 1,\\ 5,\\ 10\\)\u306b\u5bfe\u5fdc\u3059\u308b\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306e\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"980\" height=\"618\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09.png\" alt=\"\" class=\"wp-image-2095\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09.png 980w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09-300x189.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09-768x484.png 768w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09-320x202.png 320w\" sizes=\"(max-width: 980px) 100vw, 980px\" \/><\/figure>\n\n\n\n<h2><span id=\"i-3\">\u6f38\u5316\u5f0f\u306b\u3088\u308b\u30dd\u30a2\u30bd\u30f3\u78ba\u7387\u306e\u8a08\u7b97<\/span><\/h2>\n\n\n\n<p>\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim Po(\\lambda)\\)\u306b\u3064\u3044\u3066\u3001\\(\\mathrm{P}(X=k+1)\\)\u306f\\(\\mathrm{P}(X=k)\\)\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>\\mathrm{P}(X=k+1) = \\frac{\\lambda}{x+1}\\mathrm{P}(X=k),\\ \\ \\ \\ \\mathrm{P}(X=0) = e^{-\\lambda}<br>\\end{align}<\/div><br><\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2><span id=\"i-4\">\uff12\u9805\u5206\u5e03\u3068\u30dd\u30a2\u30bd\u30f3\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\uff12\u9805\u5206\u5e03\u306b\u5f93\u3063\u3066\u3044\u308b\u78ba\u7387\u5909\u6570\\(X\\sim Bin(n,p)\\)\u306e\u78ba\u7387\u95a2\u6570\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>_{n}C_{x}p^{x}(1-p)^{n-x}<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\uff12\u9805\u5206\u5e03\u306b\u304a\u3044\u3066\\(\\lambda=np\\)\u3068\u304a\u304d\u3001\\(\\lambda\\)\u3092\u4e00\u5b9a\u306b\u4fdd\u3061\u306a\u304c\u3089\u3001\\(n\\rightarrow \\infty\\)\u3068\u3057\u305f\u3082\u306e\u306f\u3001\uff12\u9805\u5206\u5e03\u306e\u78ba\u7387\u95a2\u6570\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\frac{e^{-\\lambda}\\lambda^{x}}{x!}<br>\\end{align}<\/div><br>\u306b\u53ce\u675f\u3057\u3001\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306e\u78ba\u7387\u95a2\u6570\u3068\u4e00\u81f4\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p> <\/p>\n\n\n\n<h2><span id=\"i-5\">\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u3068\u6b63\u898f\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\lambda\\)\u306e\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306f\u3001\\(\\lambda\\rightarrow\\infty\\)\u3068\u306a\u3063\u305f\u3068\u304d\u3001\u671f\u5f85\u5024\\(\\lambda\\)\u3001\u5206\u6563\\(\\lambda\\)\u306e\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h2><span id=\"i-6\">\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306e\u518d\u751f\u6027<\/span><\/h2>\n\n\n\n<p>\u78ba\u7387\u5909\u6570\\(X_{1},\\cdots,X_{n}\\)\u304c\u4e92\u3044\u306b\u72ec\u7acb\u306b\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\lambda\\)\u306e\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306b\u5f93\u3063\u3066\u3044\u308b\u3068\u304d\u3001<br><div style=\"overflow-x: auto;\">\\begin{align}<br>X_{1}+X_{2}+\\cdots+X_{n}<br>\\end{align}<\/div><br>\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\displaystyle\\sum_{i=1}^{n}\\lambda_{i}\\)\u306e\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u203b\u7a4d\u7387\u6bcd\u95a2\u6570\u306b\u3064\u3044\u3066\u306f\u4ee5\u4e0b\u306e\u8a18\u4e8b\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/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<h3>\u8a3c\u660e<\/h3>\n\n\n\n<p>\u3000\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u4f7f\u3063\u3066\u8a3c\u660e\u3057\u3066\u3044\u304d\u307e\u3057\u3087\u3046\u3002\\(X_{i}\\sim Po(\\lambda_{i}),\\ i=1,\\cdots,n\\)\u306b\u3064\u3044\u3066\u305d\u308c\u305e\u308c\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>M_{X_{i}}(t) &amp;= \\exp\\left[ \\lambda_{i}\\left( \\exp[t]-1 \\right) \\right],\\ \\ \\ i=1,\\cdots,n<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u3001<br><div style=\"overflow-x: auto;\">\\begin{align}<br>S &amp;= X_{1}+X_{2}+\\cdots+X_{n}<br>\\end{align}<\/div><br>\u3068\u3059\u308b\u3068\u3001\\(S\\)\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>M_{S}(t) = \\mathrm{E}[\\exp[St]] \\\\<br>&amp;= \\mathrm{E}[\\exp[(X_{1}+X_{2}+\\cdots+X_{n})t]] \\\\<br>&amp;= \\mathrm{E}\\left[\\prod_{i=1}^{n}\\exp[X_{i}t]\\right] \\\\<br>&amp;= \\prod_{i=1}^{n} \\mathrm{E}\\left[\\exp[X_{i}t]\\right] \\\\<br>&amp;= \\prod_{i=1}^{n}M_{X_{i}}(t) \\\\<br>&amp;= \\prod_{i=1}^{n} \\exp\\left[ \\lambda_{i}\\left( \\exp[t]-1 \\right) \\right] \\\\<br>&amp;= \\exp\\left[ \\sum_{i=1}^{n}\\lambda_{i}\\left( \\exp[t]-1 \\right) \\right]<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\displaystyle\\sum_{i=1}^{n}\\lambda_{i}\\)\u306e\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u306e\u6027\u8cea\u304b\u3089\u3001\\(S\\)\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\displaystyle\\sum_{i=1}^{n}\\lambda_{i}\\)\u306e\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"has-text-align-right\">\u25a1<\/p>\n\n\n\n<p>\u203b\u7a4d\u7387\u6bcd\u95a2\u6570\u306e\u6027\u8cea\u306f\u4ee5\u4e0b\u306e\u8a18\u4e8b\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/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<h2><span id=\"i-7\">\u30d1\u30e9\u30e1\u30fc\u30bf\u63a8\u5b9a<\/span><\/h2>\n\n\n\n<p>\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\lambda\\)\u306e\u6700\u5c24\u63a8\u5b9a\u91cf\\(\\widehat{\\lambda}\\)\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\widehat{p} &amp;= \\bar{X}<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2><span id=\"i-8\">\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u5206\u5e03\u3092\u56f3\u30fb\u8868\u3067\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002\u5404\u5206\u5e03\u306e\u8a73\u3057\u3044\u60c5\u5831\u306f\u8868\u306e\u4e2d\u306e\u30ea\u30f3\u30af\u304b\u3089\u304a\u9858\u3044\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"738\" height=\"514\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7.png\" alt=\"\" class=\"wp-image-2055\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7.png 738w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7-300x209.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7-320x223.png 320w\" sizes=\"(max-width: 738px) 100vw, 738px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/bernoulli-distribution\">\u30d9\u30eb\u30cc\u30fc\u30a4\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\uff11\u56de\u884c\u3046\u3068\u304d\u306e\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/categorical-distribution\">\u30ab\u30c6\u30b4\u30ea\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">1\u56de\u306e\u8a66\u884c\u3067\\(k\\)\u901a\u308a\u306e\u30d1\u30bf\u30fc\u30f3\u306e\u4e2d\u304b\u3089\u3069\u308c\u304b\u304c\u5f97\u3089\u308c\u308b\u53ef\u80fd\u6027\u304c\u3042\u308b\u8a66\u884c\u3092\u8868\u3059\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/bind\">\uff12\u9805\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\\(n\\)\u56de\u306e\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3067\\(x\\)\u56de\u6210\u529f\u3059\u308b\u3068\u304d\u306e\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/dgeod\">\u5e7e\u4f55\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u8907\u6570\u56de\u884c\u3046\u4e0a\u3067\u3001\u521d\u3081\u3066\u6210\u529f\u3059\u308b\u307e\u3067\u306e<br>\u8a66\u884c\u56de\u6570\u3092\u8868\u3059\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/dhgeod\">\u8d85\u5e7e\u4f55\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u8907\u6570\u56de\u884c\u3063\u3066\u3044\u304f\u4e0a\u3067\u3001\u305d\u306e\u90fd\u5ea6\u6210\u529f\u78ba\u7387\u304c<br>\u5909\u5316\u3059\u308b\u5206\u5e03\uff08\u6709\u9650\u500b\u306e\u30a2\u30bf\u30ea\u304f\u3058\u306a\u3069\uff09<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/nbiob\">\u8ca0\u306e\uff12\u9805\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u8907\u6570\u56de\u884c\u3046\u4e0a\u3067\u3001\\(k\\)\u56de\u6210\u529f\u3059\u308b\u307e\u3067\u306e<br>\u5931\u6557\u3059\u308b\u56de\u6570\u3092\u8868\u3059\u5206\u5e03\uff08\\(k=1\\)\u306e\u5834\u5408\u3001\u5e7e\u4f55\u5206\u5e03\u306b\u306a\u308a\u307e\u3059\uff09<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/dmuld\">\u591a\u9805\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u6210\u529f\u30fb\u5931\u6557\u306e\uff12\u7a2e\u985e\u3060\u3051\u3067\u306a\u304f\u3001\u8a66\u884c\u306e\u7d50\u679c\u304c<br>\u8907\u6570\u500b\u3042\u308b\u5834\u5408\u306e\u5206\u5e03\u3092\u8868\u3057\u307e\u3059<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/dpoid\">\u30dd\u30a2\u30bd\u30f3\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u6210\u529f\u78ba\u7387\u304c\u6975\u7aef\u306b\u5c0f\u3055\u304f\uff08\u3064\u307e\u308a\u6ec5\u591a\u306b\u8d77\u3053\u3089\u306a\u3044\uff09\u3001<br>\u8a66\u884c\u56de\u6570\u304c\u6975\u7aef\u306b\u5927\u304d\u3044\u5834\u5408\u306e\u5206\u5e03<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u3000\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08Poisson distribution\uff09\u306f\u96e2\u6563\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306f\u4e00\u5b9a\u9593\u9694\u306e\u6642\u9593\u3067\u767a\u751f\u3059\u308b\u4e8b\u8c61\u306e\u6570\u3092\u8868\u3057\u307e\u3059\u3002 &#8230; <\/p>\n","protected":false},"author":1,"featured_media":2094,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[84],"tags":[108,71],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570\uff09.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2088"}],"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=2088"}],"version-history":[{"count":17,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2088\/revisions"}],"predecessor-version":[{"id":2173,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2088\/revisions\/2173"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2094"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2088"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2088"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2088"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}