{"id":1952,"date":"2020-07-16T11:00:00","date_gmt":"2020-07-16T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1952"},"modified":"2020-07-19T20:40:24","modified_gmt":"2020-07-19T11:40:24","slug":"negative-binomial-distribution","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/negative-binomial-distribution","title":{"rendered":"\u8ca0\u306e\uff12\u9805\u5206\u5e03\uff08\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff09"},"content":{"rendered":"\n\n\n\n\n<p>\u8ca0\u306e\uff12\u9805\u5206\u5e03\uff08negative binomial distribution\uff09\u306f\u96e2\u6563\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3001\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\\(k\\)\u56de\u6210\u529f\u3059\u308b\u307e\u3067\u8a66\u884c\u3092\u884c\u3046\u30c7\u30fc\u30bf\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u6210\u529f\u78ba\u7387\\(p\\)\u306e\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\\(k\\)\u56de\u6210\u529f\u3059\u308b\u307e\u3067\u306b\u5931\u6557\u3059\u308b\u56de\u6570\u3092\\(X\\)\u3068\u3059\u308b\u3068\u3001\\(X\\)\u306f\u8ca0\u306e\uff12\u9805\u5206\u5e03\u306b\u5f93\u3046\u3068\u3044\u3044\u307e\u3059\u3002\u3053\u306e\u8a18\u4e8b\u3067\u306f\u3001\u8ca0\u306e\uff12\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u3068\u4e3b\u306a\u6027\u8cea\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<div id=\"toc_container\" class=\"toc_light_blue no_bullets\"><p class=\"toc_title\">\u76ee\u6b21<\/p><ul class=\"toc_list\"><li><a href=\"#i\"><span class=\"toc_number toc_depth_1\">1<\/span> \u8ca0\u306e\uff12\u9805\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\u95a2\u6570<\/a><\/li><\/ul><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_1\">2<\/span> \u5e7e\u4f55\u5206\u5e03\u3068\u8ca0\u306e\uff12\u9805\u5206\u5e03<\/a><\/li><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_1\">3<\/span> \u8ca0\u306e\uff12\u9805\u5206\u5e03\u306e\u518d\u751f\u6027<\/a><\/li><li><a href=\"#i-5\"><span class=\"toc_number toc_depth_1\">4<\/span> \u8ca0\u306e\uff12\u9805\u5206\u5e03\u3068\u30dd\u30a2\u30bd\u30f3\u5206\u5e03<\/a><\/li><li><a href=\"#i-6\"><span class=\"toc_number toc_depth_1\">5<\/span> \u30d1\u30e9\u30e1\u30fc\u30bf\u306e\u63a8\u5b9a<\/a><\/li><li><a href=\"#i-7\"><span class=\"toc_number toc_depth_1\">6<\/span> \u8ca0\u306e\uff12\u9805\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u5206\u5e03<\/a><\/li><\/ul><\/div>\n<h2><span id=\"i\">\u8ca0\u306e\uff12\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831<\/span><\/h2>\n\n\n\n<p>\u203b\u3000\u8868\u306f\u6a2a\u306b\u30b9\u30af\u30ed\u30fc\u30eb\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-table\">\n<table style=\"width: 100%; height: 1041px;\">\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; p &lt; 1\\)\uff08\\(p\\)\u306f\u6210\u529f\u78ba\u7387\u3092\u8868\u3057\u307e\u3059\u3002\uff09<br \/>\\(0 &lt; k\\)\uff08\\(k\\)\u306f\u6210\u529f\u56de\u6570\u3092\u8868\u3057\u307e\u3059\u3002\uff09<\/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;\">\\(\\sum_{i=1}^{x}\\ _{k+i-1}C_{k-1}p^{k}(1-p)^{i}\\)<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 36px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/probability-function\">\u78ba\u7387\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(_{k+x-1}C_{k-1}p^{k}(1-p)^{x}\\)<br \/>\u305f\u3060\u3057\\(k\\)\u306f\u6574\u6570<\/td>\n<\/tr>\n<tr style=\"height: 153px;\">\n<td style=\"width: 17.5862%; height: 153px;\">\n<p><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/probability-function\">\u78ba\u7387\u95a2\u6570<\/a><br \/>(\\(x\\)\u304c\u6574\u6570\u3067\u306f\u306a\u3044\u3068\u304d)<\/p>\n<\/td>\n<td style=\"width: 82.2414%; height: 153px;\">\\(\\displaystyle\\frac{\\Gamma(k+x)}{\\Gamma{(k)}x!}p^{k}(1-p)^{x}\\)<br \/>\u305f\u3060\u3057\\(k\\)\u306f\u6574\u6570<\/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\\frac{p^{k}}{\\{1-(1-p)e^{t}\\}^{k}}\\)<\/td>\n<\/tr>\n<tr style=\"height: 174px;\">\n<td style=\"width: 17.5862%; height: 174px;\"><a href=\"http:\/\/develop-chronos.com\/statistics-top\/statistics\/moment\">\u968e\u4e57\u7a4d\u7387\u6bcd\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 174px;\">\\(\\displaystyle\\left( 1-\\frac{(1-p)^{t}}{p} \\right)^{-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\/pgf\">\u78ba\u7387\u6bcd\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\displaystyle\\frac{p^{k}}{\\{1-(1-p)t\\}^{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\/cf\">\u7279\u6027\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 36px;\">\\(\\displaystyle\\frac{p^{k}}{\\{1-(1-p)e^{it}\\}^{k}}\\)<\/td>\n<\/tr>\n<tr style=\"height: 56px;\">\n<td style=\"width: 17.5862%; height: 56px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cgf\">\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570<\/a><\/td>\n<td style=\"width: 82.2414%; height: 56px;\">\\( k\\log p-k\\log\\{1-(1-p)e^{t}\\} \\)<\/td>\n<\/tr>\n<tr style=\"height: 156px;\">\n<td style=\"width: 17.5862%; height: 156px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/moment\">\u30e2\u30fc\u30e1\u30f3\u30c8\uff08\u5e73\u5747\u307e\u308f\u308a\uff09<\/a><\/td>\n<td style=\"width: 82.2414%; height: 156px;\"><div style=\"overflow-x: auto;\">\\begin{align}<br \/>\u30002\u6b21\uff08\u5206\u6563\uff09 &amp; \\displaystyle \\frac{ k(1-p) }{ p^{2} } \\\\<br \/>\u30003\u6b21\u3000\u3000\u3000 \u3000&amp; \\displaystyle \\frac{ k(1-p)\\left\\{1+(1-p)\\right\\} }{ p^{2} } \\\\<br \/>\u30004\u6b21\u3000\u3000\u3000 \u3000&amp; \\displaystyle \\frac{ k(1-p)\\left\\{ 1+4(1-p)+(1-p)^{2}+3k(1-p) \\right\\} }{ p^{4} }<br \/>\\end{align}<\/div><\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 17.5862%; height: 38px;\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mean-definition\">\u671f\u5f85\u5024<\/a><\/td>\n<td style=\"width: 82.2414%; height: 38px;\">\\( \\displaystyle \\frac{k(1-p)}{p} \\)<\/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 \\frac{ k(1-p) }{ p^{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 \\frac{1+(1-p)}{\\sqrt{k(1-p)}}\\)<\/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{6}{k}+\\frac{p^{3}}{k(1-p)} \\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/figure>\n\n\n\n<h3>\u8a3c\u660e\u4e00\u89a7<\/h3>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/nbiod-mv\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff08p0.4\uff09-100x100.png\" alt=\"\u8ca0\u306e\uff12\u9805\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u8ca0\u306e\uff12\u9805\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011<\/div>\n    <div class=\"blogcard_excerpt\">\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u3053\u306e\u8a18\u4e8b\u3067\u306f\u8ca0\u306e\uff12\u9805\u5206\u5e03\uff08\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff09\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\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\/nbiod-mc\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff08p0.4\uff09-100x100.png\" alt=\"\u8ca0\u306e\uff12\u9805\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u8ca0\u306e\uff12\u9805\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9<\/div>\n    <div class=\"blogcard_excerpt\">\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u3053\u306e\u8a18\u4e8b\u3067\u306f\u8ca0\u306e\uff12\u9805\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u3092\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h3><span id=\"i-2\">\u78ba\u7387\u95a2\u6570<\/span><\/h3>\n\n\n\n<p>\\(k=2\\)\u3067\u56fa\u5b9a\u3057\u305f\u5834\u5408\u306e\u78ba\u7387\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=\"979\" height=\"618\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff08k2\uff09.png\" alt=\"\" class=\"wp-image-1959\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff08k2\uff09.png 979w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff08k2\uff09-300x189.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff08k2\uff09-768x485.png 768w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff08k2\uff09-320x202.png 320w\" sizes=\"(max-width: 979px) 100vw, 979px\" \/><\/figure>\n\n\n\n<p>\\(p=0.4\\)\u3067\u56fa\u5b9a\u3057\u305f\u5834\u5408\u306e\u78ba\u7387\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\/\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff08p0.4\uff09.png\" alt=\"\" class=\"wp-image-1960\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff08p0.4\uff09.png 980w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff08p0.4\uff09-300x189.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff08p0.4\uff09-768x484.png 768w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff08p0.4\uff09-320x202.png 320w\" sizes=\"(max-width: 980px) 100vw, 980px\" \/><\/figure>\n\n\n\n<h2><span id=\"i-3\">\u5e7e\u4f55\u5206\u5e03\u3068\u8ca0\u306e\uff12\u9805\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u3000\\(k\\)\u500b\u306e\u72ec\u7acb\u306a\u5e7e\u4f55\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\u548c\u306f\u3001\\(k\\)\u6b21\u306e\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f\u3001\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u4f7f\u3063\u3066\u78ba\u8a8d\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br>\u3000\\(X_{1},\\cdots,X_{k}\\sim Geo(p)\\)\u3068\u3057\u307e\u3059\u3002\u5e7e\u4f55\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>M_{X_{i}}(t) = \\frac{ pe^{t} }{ 1-(1-p)e^{t} }<br>\\end{align}<\/div><br>\u3067\u3057\u305f\u3002\u3053\u306e\u3053\u3068\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/geometric-distribution\">\uff1c\u5e7e\u4f55\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\uff1e<\/a>\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002\u3053\u306e\u3068\u304d\u3001\\(Y=X_{1}+\\cdots+X_{k}-k\\)\u306b\u3064\u3044\u3066\u8003\u3048\u3066\u307f\u307e\u3059\u3002\u3053\u306e\\(Y\\)\u306f\\(k\\)\u56de\u6210\u529f\u3059\u308b\u307e\u3067\u306b\u3001\u5931\u6557\u3059\u308b\u56de\u6570\u3092\u8868\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\\(Y\\)\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>M_{Y}(t) &amp;= \\mathrm{E}\\left[ e^{tY} \\right] \\\\<br>&amp;= \\mathrm{E}\\left[ e^{t(X_{1}+\\cdots+X_{k}-k)} \\right] \\\\<br>&amp;= \\prod_{i=1}^{k}\\mathrm{E}\\left[ e^{tX_{i}} \\right]\\cdot\\mathrm{E}\\left[ e^{-tk} \\right] \\\\<br>&amp;= \\prod_{i=1}^{k}\\frac{ pe^{t} }{ 1-(1-p)e^{t} }\\cdot e^{-tk} \\\\<br>&amp;= \\frac{ p^{k}e^{tk} }{ \\left\\{1-(1-p)e^{t}\\right\\}^{k} }\\cdot e^{-tk}  \\\\<br>&amp;= \\frac{ p^{k} }{ \\left\\{1-(1-p)e^{t}\\right\\}^{k} } \\\\<br>\\end{align}<\/div><\/p>\n\n\n\n<p>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u306f\u8ca0\u306e\uff12\u9805\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u3068\u4e00\u81f4\u3057\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u306e\u6027\u8cea\u304b\u3089\u3001\u3053\u306e\\(Y\\)\u306f\u8ca0\u306e\uff12\u9805\u5206\u5e03\u306b\u5f93\u3046\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u306b\u3064\u3044\u3066\u306f\u4e0b\u8a18\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-4\">\u8ca0\u306e\uff12\u9805\u5206\u5e03\u306e\u518d\u751f\u6027<\/span><\/h2>\n\n\n\n<p>\u3000\u78ba\u7387\u5909\u6570\\(X_{1},\\cdots,X_{n}\\)\u304c\u4e92\u3044\u306b\u72ec\u7acb\u306b\u30d1\u30e9\u30e1\u30fc\u30bf\\(k_{i},p\\)\u306e\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u306b\u5f93\u3063\u3066\u3044\u308b\u3068\u304d\u3001\u78ba\u7387\u5909\u6570\u306e\u548c\\(X_{1}+\\cdots+X_{n}\\)\u3082\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\sum_{i=1}^{n}k_{i},p\\)\u306e\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f\u3001\u4e0a\u306e\u8a3c\u660e\u3068\u540c\u69d8\u3001\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u7528\u3044\u3066\u8a3c\u660e\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<h2><span id=\"i-5\">\u8ca0\u306e\uff12\u9805\u5206\u5e03\u3068\u30dd\u30a2\u30bd\u30f3\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\\(\\displaystyle\\frac{k(1-p)}{p}=\\lambda\\)\u3092\u4e00\u5b9a\u306b\u4fdd\u3061\u306a\u304c\u3089\u3001\\(k\\rightarrow\\infty,\\ p\\rightarrow 1\\)\u306e\u3068\u304d\u3001\u8ca0\u306e\uff12\u9805\u5206\u5e03\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\lambda\\)\u306e\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306b\u53ce\u675f\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<h2><span id=\"i-6\">\u30d1\u30e9\u30e1\u30fc\u30bf\u306e\u63a8\u5b9a<\/span><\/h2>\n\n\n\n<p>\u30fb\\(k\\)\u304c\u65e2\u77e5\u306e\u3068\u304d\uff08\u6700\u5c24\u6cd5\uff09<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\widehat{p} = \\frac{k}{k+\\bar{X}}<br>\\end{align}<\/div><br>\u4e0d\u504f\u63a8\u5b9a\u91cf\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\frac{k-\\displaystyle\\frac{1}{n}}{k+\\bar{X}-\\displaystyle\\frac{1}{n}}<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u30fb\\(k\\)\u304c\u672a\u77e5\u306e\u3068\u304d\uff08\u30e2\u30fc\u30e1\u30f3\u30c8\u6cd5\uff09<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\widehat{p} &amp;= \\frac{\\bar{X}}{\\widehat{\\sigma}^{2}} \\\\<br>\\widehat{k} &amp;= \\frac{\\bar{X}^{2}}{\\widehat{\\sigma}^{2}-\\bar{X}}<br>\\end{align}<\/div><\/p>\n\n\n\n<p>\u305f\u3060\u3057\u3001<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\bar{X} = \\frac{1}{n}\\sum_{i=1}^{n}X_{i},\\ \\ \\ \\widehat{\\sigma}^{2} = \\frac{1}{n}\\sum_{i=1}^{n}(X_{i}-\\bar{X})^{2}<br>\\end{align}<\/div><br>\u3067\u3059\u3002<\/p>\n\n\n\n<h2><span id=\"i-7\">\u8ca0\u306e\uff12\u9805\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u8ca0\u306e\uff12\u9805\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u5206\u5e03\u3092\u56f3\u30fb\u8868\u3067\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002\u5404\u5206\u5e03\u306e\u8a73\u3057\u3044\u60c5\u5831\u306f\u8868\u306e\u4e2d\u306e\u30ea\u30f3\u30af\u304b\u3089\u304a\u9858\u3044\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"738\" height=\"514\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7.png\" alt=\"\" class=\"wp-image-2055\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7.png 738w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7-300x209.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7-320x223.png 320w\" sizes=\"(max-width: 738px) 100vw, 738px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/bernoulli-distribution\">\u30d9\u30eb\u30cc\u30fc\u30a4\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\uff11\u56de\u884c\u3046\u3068\u304d\u306e\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/categorical-distribution\">\u30ab\u30c6\u30b4\u30ea\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">1\u56de\u306e\u8a66\u884c\u3067\\(k\\)\u901a\u308a\u306e\u30d1\u30bf\u30fc\u30f3\u306e\u4e2d\u304b\u3089\u3069\u308c\u304b\u304c\u5f97\u3089\u308c\u308b\u53ef\u80fd\u6027\u304c\u3042\u308b\u8a66\u884c\u3092\u8868\u3059\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/bind\">\uff12\u9805\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\\(n\\)\u56de\u306e\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3067\\(x\\)\u56de\u6210\u529f\u3059\u308b\u3068\u304d\u306e\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/dgeod\">\u5e7e\u4f55\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u8907\u6570\u56de\u884c\u3046\u4e0a\u3067\u3001\u521d\u3081\u3066\u6210\u529f\u3059\u308b\u307e\u3067\u306e<br>\u8a66\u884c\u56de\u6570\u3092\u8868\u3059\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/dhgeod\">\u8d85\u5e7e\u4f55\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u8907\u6570\u56de\u884c\u3063\u3066\u3044\u304f\u4e0a\u3067\u3001\u305d\u306e\u90fd\u5ea6\u6210\u529f\u78ba\u7387\u304c<br>\u5909\u5316\u3059\u308b\u5206\u5e03\uff08\u6709\u9650\u500b\u306e\u30a2\u30bf\u30ea\u304f\u3058\u306a\u3069\uff09<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/nbiob\">\u8ca0\u306e\uff12\u9805\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u8907\u6570\u56de\u884c\u3046\u4e0a\u3067\u3001\\(k\\)\u56de\u6210\u529f\u3059\u308b\u307e\u3067\u306e<br>\u5931\u6557\u3059\u308b\u56de\u6570\u3092\u8868\u3059\u5206\u5e03\uff08\\(k=1\\)\u306e\u5834\u5408\u3001\u5e7e\u4f55\u5206\u5e03\u306b\u306a\u308a\u307e\u3059\uff09<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/dmuld\">\u591a\u9805\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u6210\u529f\u30fb\u5931\u6557\u306e\uff12\u7a2e\u985e\u3060\u3051\u3067\u306a\u304f\u3001\u8a66\u884c\u306e\u7d50\u679c\u304c<br>\u8907\u6570\u500b\u3042\u308b\u5834\u5408\u306e\u5206\u5e03\u3092\u8868\u3057\u307e\u3059<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/dpoid\">\u30dd\u30a2\u30bd\u30f3\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u6210\u529f\u78ba\u7387\u304c\u6975\u7aef\u306b\u5c0f\u3055\u304f\uff08\u3064\u307e\u308a\u6ec5\u591a\u306b\u8d77\u3053\u3089\u306a\u3044\uff09\u3001<br>\u8a66\u884c\u56de\u6570\u304c\u6975\u7aef\u306b\u5927\u304d\u3044\u5834\u5408\u306e\u5206\u5e03<\/td><\/tr><\/tbody><\/table><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>\u8ca0\u306e\uff12\u9805\u5206\u5e03\uff08negative binomial distribution\uff09\u306f\u96e2\u6563\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3001\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\\(k\\)\u56de\u6210\u529f\u3059\u308b\u307e\u3067\u8a66 &#8230; <\/p>\n","protected":false},"author":1,"featured_media":1959,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[84],"tags":[71,105],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff08k2\uff09.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1952"}],"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=1952"}],"version-history":[{"count":20,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1952\/revisions"}],"predecessor-version":[{"id":2148,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1952\/revisions\/2148"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/1959"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=1952"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=1952"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=1952"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}