{"id":1207,"date":"2020-06-19T11:00:00","date_gmt":"2020-06-19T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1207"},"modified":"2020-06-30T21:16:30","modified_gmt":"2020-06-30T12:16:30","slug":"pgf","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/pgf","title":{"rendered":"\u78ba\u7387\u6bcd\u95a2\u6570\u3068\u306f\u3010\u610f\u5473\u3068\u5b9a\u7fa9\u3092\u89e3\u8aac\u3011"},"content":{"rendered":"\n<p class=\"has-text-align-right\"><span style=\"text-decoration: underline;\">\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606<\/span><\/p>\n\n\n\n\n\n\n\n<p><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/discrete-continuous\">\u96e2\u6563\u578b<\/a>\u306e<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/random-variable\">\u78ba\u7387\u5909\u6570<\/a>\u306b\u3064\u3044\u3066<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/moment\">\u968e\u4e57\u30e2\u30fc\u30e1\u30f3\u30c8<\/a>\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u95a2\u6570\u306b\u78ba\u7387\u6bcd\u95a2\u6570\u3068\u3044\u3046\u3082\u306e\u304c\u3042\u308a\u307e\u3059\u3002\u3042\u307e\u308a\u898b\u304b\u3051\u308b\u3053\u3068\u306f\u306a\u3044\u3067\u3059\u304c\u3001\u78ba\u7387\u6bcd\u95a2\u6570\u3092\u6c42\u3081\u3089\u308c\u305f\u3089<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mean-definition\">\u671f\u5f85\u5024<\/a>\u3001<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/variance\">\u5206\u6563<\/a>\u306e\u8a08\u7b97\u304c\u975e\u5e38\u306b\u697d\u306b\u306a\u308a\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> \u78ba\u7387\u6bcd\u95a2\u6570\u306e\u5b9a\u7fa9<\/a><ul><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_2\">1.1<\/span> \u78ba\u7387\u6bcd\u95a2\u6570\u3068\u78ba\u7387<\/a><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_2\">1.2<\/span> \u78ba\u7387\u6bcd\u95a2\u6570\u3068\u968e\u4e57\u30e2\u30fc\u30e1\u30f3\u30c8<\/a><\/li><\/ul><\/li><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_1\">2<\/span> \u30e2\u30fc\u30e1\u30f3\u30c8\u306b\u95a2\u3059\u308b\u95a2\u9023\u8a18\u4e8b<\/a><\/li><\/ul><\/div>\n<h2><span id=\"i\">\u78ba\u7387\u6bcd\u95a2\u6570\u306e\u5b9a\u7fa9<\/span><\/h2>\n\n\n\n<p>\u96e2\u6563\u578b\u306e\u78ba\u7387\u5909\u6570\\(X\\)\u306f\\(0,1,2,\\cdots\\)\u3068\u3044\u3046\u975e\u8ca0\u306e\u6574\u6570\u5024\u3092\u3068\u308b\u3082\u306e\u3060\u3068\u3057\u307e\u3059\u3002\u5f0f\u3067\u8868\u3059\u3068<br><div style=\"overflow-x: auto;\">\\begin{align}<br>&amp; \\mathrm{P}(X=j) =p_{j},\\ \\ \\ \\ \\ j=0,1,2,\\cdots \\\\<br>&amp; \\sum_{j=0}^{\\infty}p_{j} = 1<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u3001\u78ba\u7387\u5909\u6570\u306b\u5bfe\u3057\u3066\u78ba\u7387\u6bcd\u95a2\u6570\u304c\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u78ba\u7387\u6bcd\u95a2\u6570(probability generating function(pgf))<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\n\u96e2\u6563\u578b\u306e\u78ba\u7387\u5909\u6570\\(X\\)\u306f\\(0,1,2,\\cdots\\)\u3068\u3044\u3046\u975e\u8ca0\u306e\u6574\u6570\u5024\u3092\u3068\u308a\n<div style=\"overflow-x: auto;\">\\begin{align}\n&amp; \\mathrm{P}(X=j) =p_{j},\\ \\ \\ \\ \\ j=0,1,2,\\cdots \\\\\n&amp; \\sum_{j=0}^{\\infty}p_{j} = 1\n\\end{align}<\/div>\n\u3092\u6e80\u305f\u3059\u3082\u306e\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u78ba\u7387\u5909\u6570\u3078\\(|t|\\leq 1\\)\u3092\u6e80\u305f\u3059\u4efb\u610f\u306e\\(t\\)\u306b\u5bfe\u3057\u3066\u78ba\u7387\u6bcd\u95a2\u6570\u304c\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\u78ba\u7387\u6bcd\u95a2\u6570\u306f\\(t\\)\u306b\u95a2\u3059\u308b\u95a2\u6570\u3067\u3042\u308a\u3001\u8a18\u53f7\\(G_{X}(t)\\)\u3067\u8868\u3057\u3001\u4e0b\u5f0f\u3067\u5b9a\u7fa9\u3057\u307e\u3059\u3002$$G_{X}(t) = \\mathrm{E}\\left[ t^{X} \\right] = \\sum_{j=0}^{\\infty}p_{j}t^{j}$$\n<\/div>\n<p>&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-liquid-speech-balloon liquid-speech-balloon-wrap liquid-speech-balloon-00  liquid-speech-balloon-short\"><div class=\"liquid-speech-balloon-avatar\"><\/div><div class=\"liquid-speech-balloon-text\" style=\"border-color:#d6ebf8;background-color:#d6ebf8;color:#000000\"><p>\u78ba\u7387\u6bcd\u95a2\u6570\u306b\u95a2\u3057\u3066\u306f\u6b21\u306e\u3088\u3046\u306a\u6027\u8cea\u304c\u3042\u308a\u307e\u3059\u3002<\/p><div class=\"liquid-speech-balloon-arrow\" style=\"border-color:transparent #d6ebf8 transparent transparent\"><\/div><\/div><\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<!-- \u78ba\u7387\u6bcd\u95a2\u6570 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1927349886186426\" data-ad-slot=\"8307158523\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<h3><span id=\"i-2\">\u78ba\u7387\u6bcd\u95a2\u6570\u3068\u78ba\u7387<\/span><\/h3>\n\n\n\n<p>\u78ba\u7387\u6bcd\u95a2\u6570\\(G_{X}(t)\\)\u3092\u7528\u3044\u3066\u3001\\(X=r\\)\u3068\u306a\u308b\u78ba\u7387\u3092\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u78ba\u7387\u6bcd\u95a2\u6570\u3068\u78ba\u7387<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\n\u78ba\u7387\u6bcd\u95a2\u6570\\(G_{X}(t)\\)\u3092\u7528\u3044\u3066\u3001\\(X=r\\)\u3068\u306a\u308b\u78ba\u7387\\(p_{r}\\)\u6b21\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002$$p_{r}= \\left.\\frac{1}{r!}\\cdot\\frac{d^{r}}{dt^{r}}G_{X}(t)\\right|_{t=0}$$\n<\/div>\n<p>&nbsp;<\/p>\n\n\n\n<p>\u3053\u306e\u5f0f\u306e\u8a08\u7b97\u65b9\u6cd5\u3067\u3059\u304c\u3001\u307e\u305a\\(G_{X}(t)\\)\u3092\\(t\\)\u3067\\(r\\)\u56de\u5fae\u5206\u3057\u305f\u5f8c\u306b\\(t\\)\u306b\\(0\\)\u3092\u4ee3\u5165\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"wp-block-liquid-speech-balloon liquid-speech-balloon-wrap liquid-speech-balloon-00  liquid-speech-balloon-short\"><div class=\"liquid-speech-balloon-avatar\"><\/div><div class=\"liquid-speech-balloon-text\" style=\"border-color:#d6ebf8;background-color:#d6ebf8;color:#000000\"><p>\u8a3c\u660e\u81ea\u4f53\u306f\u3068\u3063\u3066\u3082\u7c21\u5358\u3067\u3059\u3002\u5b9f\u969b\u306b\u53f3\u8fba\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p><div class=\"liquid-speech-balloon-arrow\" style=\"border-color:transparent #d6ebf8 transparent transparent\"><\/div><\/div><\/div>\n\n\n\n<p><div style=\"overflow-x: auto;\">\\begin{align}<br>\\left.\\frac{1}{r!}\\cdot\\frac{d^{r}}{dt^{j}}G_{X}(t)\\right|_{t=0} &amp;= \\frac{1}{r!}\\cdot r!p_{r}+\\left.\\frac{1}{r!}\\sum_{j=r+1}^{\\infty}j!\\cdot p_{j}t^{j-r}\\right|_{t=0} \\\\<br>&amp;= p_{r}<br>\\end{align}<\/div><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<!-- \u78ba\u7387\u6bcd\u95a2\u6570\uff12 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1927349886186426\" data-ad-slot=\"3054831842\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<h3><span id=\"i-3\">\u78ba\u7387\u6bcd\u95a2\u6570\u3068\u968e\u4e57\u30e2\u30fc\u30e1\u30f3\u30c8<\/span><\/h3>\n\n\n\n<p>\u78ba\u7387\u6bcd\u95a2\u6570\u3092\u7528\u3044\u3066\u3001\u968e\u4e57\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u78ba\u7387\u6bcd\u95a2\u6570\u3068\u968e\u4e57\u30e2\u30fc\u30e1\u30f3\u30c8<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\n\u78ba\u7387\u6bcd\u95a2\u6570\\(G_{X}(t)\\)\u3092\u7528\u3044\u3066\u3001\\(k\\)\u6b21\u306e\u968e\u4e57\u30e2\u30fc\u30e1\u30f3\u30c8\\(\\mu_{[k]}^{\\prime}\\)\u3092\u6b21\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002$$\\mu_{[k]}^{\\prime} = \\left.\\frac{d^{k}}{dt^{k}}G_{X}(t)\\right|_{t=1}$$\n<\/div>\n<p>&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-liquid-speech-balloon liquid-speech-balloon-wrap liquid-speech-balloon-00  liquid-speech-balloon-short\"><div class=\"liquid-speech-balloon-avatar\"><\/div><div class=\"liquid-speech-balloon-text\" style=\"border-color:#d6ebf8;background-color:#d6ebf8;color:#000000\"><p>\u3053\u306e\u5f0f\u3092<strong>\u968e\u4e57\u7a4d\u7387\u6bcd\u95a2\u6570<\/strong>\u3068\u547c\u3073\u307e\u3059\u3002<br>\u3053\u306e\u5f0f\u3092\u8a3c\u660e\u3057\u3066\u307f\u307e\u3057\u3087\u3046\uff01<\/p><div class=\"liquid-speech-balloon-arrow\" style=\"border-color:transparent #d6ebf8 transparent transparent\"><\/div><\/div><\/div>\n\n\n\n<p>\u307e\u305a\u3001\\(t^{X}\\)\u3092\\(t=1\\)\u306e\u307e\u308f\u308a\u3067\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3057\u3066\u307f\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>t^{X} &amp;= 1 + X(t-1) + \\frac{1}{2!}\\cdot X(X-1)(t-1)^{2}\\\\<br>&amp;\\ \\ \\ + \\frac{1}{3!}\\cdot X(X-1)(X-2)(t-1)^{3}+\\cdots<br>\\end{align}<\/div><br>\u3053\u306e\u5f0f\u3092\u7528\u3044\u308b\u3068\u3001\\(k\\)\u6b21\u306e\u968e\u4e57\u30e2\u30fc\u30e1\u30f3\u30c8(\\mu_{[k]}^{\\prime})\u3092\u4f7f\u3063\u3066<br><div style=\"overflow-x: auto;\">\\begin{align}<br>G_{X}(t) &amp;= \\mathrm{E}\\left[ t^{X} \\right] \\\\<br>&amp;=  1 + (t-1)\\mathrm{E}[X] + \\frac{1}{2!}\\cdot (t-1)^{2}\\mathrm{E}[X(X-1)]\\\\<br>&amp;\\ \\ \\ + \\frac{1}{3!}\\cdot (t-1)^{3}\\mathrm{E}[X(X-1)(X-2)]+\\cdots \\\\<br>&amp;= 1 + (t-1)\\mu_{[1]}^{\\prime} + \\frac{1}{2!}\\cdot (t-1)^{2}\\mu_{[2]}^{\\prime} + \\frac{1}{3!}\\cdot (t-1)^{3}\\mu_{[3]}^{\\prime}+\\cdots<br>\\end{align}<\/div><br>\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u306e\u5f0f\u304b\u3089\u53f3\u8fba\u3092\\(t\\)\u306b\u3064\u3044\u3066\\(k\\)\u56de\u5fae\u5206\u3059\u308b\u3068<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\frac{d^{k}}{dt^{k}}G_{X}(t) &amp;= \\mu_{[k]}^{\\prime}+(t-1)^{k-1}\\mu_{[k+1]}^{\\prime}+\\cdots<br>\\end{align}<\/div><br>\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3001\\(t=1\\)\u3092\u4ee3\u5165\u3059\u308b\u3068<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\left.\\frac{d^{k}}{dt^{k}}G_{X}(t)\\right|_{t=1} &amp;= \\mu_{[k]}^{\\prime}<br>\\end{align}<\/div><br>\u304c\u6210\u7acb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<h2><span id=\"i-4\">\u30e2\u30fc\u30e1\u30f3\u30c8\u306b\u95a2\u3059\u308b\u95a2\u9023\u8a18\u4e8b<\/span><\/h2>\n\n\n\n<p>\u30fb<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/moment\">\u30e2\u30fc\u30e1\u30f3\u30c8<\/a>\uff1a\u30e2\u30fc\u30e1\u30f3\u30c8\u306b\u3064\u3044\u3066\u5b9a\u7fa9\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u30fb<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mgf\">\u7a4d\u7387\u6bcd\u95a2\u6570<\/a>\uff1a\\(k\\)\u6b21\u306e\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u95a2\u6570\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u30fb<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cf\">\u7279\u6027\u95a2\u6570<\/a>\uff1a\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u4e00\u822c\u5316\u3057\u305f\u3082\u306e\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u30fb<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cgf\">\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570<\/a>\uff1a\u30ad\u30e5\u30df\u30e5\u30e9\u30f3\u30c8\u3092\u7528\u3044\u3066\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u30fb<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/gf-relationship\">\u6bcd\u95a2\u6570\u9593\u306e\u95a2\u4fc2\u306e\u307e\u3068\u3081<\/a>\uff1a\u6bcd\u95a2\u6570\u9593\u306e\u95a2\u4fc2\u3092\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u96e2\u6563\u578b\u306e\u78ba\u7387\u5909\u6570\u306b\u3064\u3044\u3066\u968e\u4e57\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u95a2\u6570\u306b\u78ba\u7387\u6bcd\u95a2\u6570\u3068\u3044\u3046\u3082\u306e\u304c\u3042\u308a\u307e\u3059\u3002\u3042\u307e\u308a\u898b\u304b &#8230; <\/p>\n","protected":false},"author":1,"featured_media":1202,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[76],"tags":[77,78,83],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/9fc17caf-9cad-4670-b857-eb43ec911ab9.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1207"}],"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=1207"}],"version-history":[{"count":29,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1207\/revisions"}],"predecessor-version":[{"id":1847,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1207\/revisions\/1847"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/1202"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=1207"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=1207"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=1207"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}