{"id":1271,"date":"2020-06-22T11:00:00","date_gmt":"2020-06-22T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1271"},"modified":"2020-06-14T21:20:55","modified_gmt":"2020-06-14T12:20:55","slug":"cgf","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cgf","title":{"rendered":"\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570\u3068\u306f\u3010\u5b9a\u7fa9\u304b\u3089\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>\u5206\u5e03\u306e\u7279\u5fb4\u3092\u8868\u3059\u3082\u306e\u306b\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u3068\u3044\u3046\u3082\u306e\u304c\u3042\u308a\u307e\u3059\u3002\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u3068\u30e2\u30fc\u30e1\u30f3\u30c8\u306e\u9593\u306b\u306f\u4e00\u5b9a\u306e\u95a2\u4fc2\u304c\u3042\u308a\u307e\u3059\u3002\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u306f\u7d71\u8a08\u5b66\u3060\u3051\u3067\u306a\u304f\u3001\u7269\u7406\u5b66\u306b\u3082\u5229\u7528\u3055\u308c\u307e\u3059\u3002\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u3092\u7528\u3044\u308b\u3068\u4e2d\u5fc3\u30e2\u30fc\u30e1\u30f3\u30c8\u304a\u3088\u3073\u5e73\u5747\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u5e45\u5e83\u3044\u77e5\u898b\u306e\u305f\u3081\u306b\u3082\u662f\u975e\u3068\u3082\u6700\u5f8c\u307e\u3067\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\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> \u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570\u306e\u5b9a\u7fa9<\/a><ul><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_2\">1.1<\/span> \u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570\u3068\u7a4d\u7387\u6bcd\u95a2\u6570<\/a><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_2\">1.2<\/span> \u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u3068\u30e2\u30fc\u30e1\u30f3\u30c8<\/a><\/li><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_2\">1.3<\/span> \u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u3068\u4e2d\u5fc3\u30e2\u30fc\u30e1\u30f3\u30c8<\/a><\/li><\/ul><\/li><li><a href=\"#i-5\"><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<h3><span id=\"i\">\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570\u306e\u5b9a\u7fa9<\/span><\/h3>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570(cumulant generating function(cgf))<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\n\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\\(M_{X}(t)\\)\u3068\u3059\u308b\u3068\u304d\u3001\u7a4d\u7387\u6bcd\u95a2\u6570\u306b\u5bfe\u6570\u3092\u3068\u3063\u305f\u3082\u306e\u3092\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570\\(K_{X}(t)\\)\u3068\u3044\u3044\u307e\u3059\u3002$$K_{X}(t) = \\log M_{X}(t)$$\n<\/div>\n<p>&nbsp;<\/p>\n\n\n\n<p> \u203b\u3000\u7a4d\u7387\u6bcd\u95a2\u6570\u306b\u3064\u3044\u3066\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mgf\">\uff1c\u7a4d\u7387\u6bcd\u95a2\u6570\u306e\u8a18\u4e8b\uff1e<\/a>\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\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<p>\u7a4d\u7387\u6bcd\u95a2\u6570\u306b\u3064\u3044\u3066\u3001\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3092\u7528\u3044\u308b\u3068<br><div style=\"overflow-x: auto;\">\\begin{align}<br>M_{X}(t) &amp;= 1+t\\mu_{1}^{\\prime}+\\frac{t^{2}}{2!}\\mu_{2}^{\\prime} + \\frac{t^{3}}{3!}\\mu_{3}^{\\prime}+\\cdots\\\\<br>&amp;= \\sum_{j=0}^{\\infty}\\frac{t^{j}}{j!}\\mu_{j}^{\\prime}<br>\\end{align}<\/div><br>\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\uff08\u8a73\u3057\u304f\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mgf\">\uff1c\u7a4d\u7387\u6bcd\u95a2\u6570\u306e\u8a18\u4e8b\uff1e<\/a>\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\uff09\u3002\u3053\u306e\u5f0f\u306b\u3064\u3044\u3066\\(t^{j}\/j!\\)\u306e\u4fc2\u6570\u306f\\(\\mu_{j}^{\\prime}\\)\u3068\u306a\u308a\u307e\u3059\u3002<br>\u3000\u540c\u69d8\u306b\u3057\u3066\u3001\u5bfe\u6570\u3092\u3068\u3063\u305f\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u5c55\u958b\u3057\u305f\u6642\u3092<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\log M_{X}(t) &amp;=  \\sum_{j=1}^{\\infty}\\frac{t^{j}}{j!}\\kappa_{j}<br>\\end{align}<\/div><br>\u3068\u8868\u3057\u305f\u3068\u304d\u3001\\(t^{j}\/j!\\)\u306e\u4fc2\u6570\u306b\u3042\u305f\u308b\\(\\kappa_{j}\\)\u3092<strong>\\(j\\)\u6b21\u306e\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8<\/strong>\u3068\u3044\u3044\u307e\u3059\u3002\u3053\u3053\u3067\\(j=0\\)\u306e\u3068\u304d\u306b\u306f\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u304c\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u306a\u3044\u3053\u3068\u306b\u306f\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3042\u308b\u3002<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h4><span id=\"i-2\">\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570\u3068\u7a4d\u7387\u6bcd\u95a2\u6570<\/span><\/h4>\n\n\n\n<p>\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570\u306e\u5b9a\u7fa9\u304b\u3089\u3001\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\\(M_{X}(t)\\)\u3068\\(K_{X}(t)\\)\u306e\u9593\u306b\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\sum_{j=0}^{\\infty}\\frac{t^{j}}{j!}\\mu_{j}^{\\prime} &amp;= M_{X}(t) = \\exp\\left[ K_{X}(t) \\right] \\\\<br>&amp;= \\exp\\left[ \\sum_{j=1}^{\\infty}\\frac{t^{j}}{j!}\\kappa_{j} \\right]<br>\\end{align}<\/div><br>\u3068\u3044\u3046\u95a2\u4fc2\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<a href=\"https:\/\/px.a8.net\/svt\/ejp?a8mat=3BDSVQ+36B8XE+50+2HM5Z5\" rel=\"nofollow\">\n<img loading=\"lazy\" border=\"0\" width=\"728\" height=\"90\" alt=\"\" src=\"https:\/\/www28.a8.net\/svt\/bgt?aid=200518262192&amp;wid=001&amp;eno=01&amp;mid=s00000000018015052000&amp;mc=1\"><\/a>\n<img loading=\"lazy\" border=\"0\" width=\"1\" height=\"1\" src=\"https:\/\/www18.a8.net\/0.gif?a8mat=3BDSVQ+36B8XE+50+2HM5Z5\" alt=\"\">\n\n\n\n<h4><span id=\"i-3\">\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u3068\u30e2\u30fc\u30e1\u30f3\u30c8<\/span><\/h4>\n\n\n\n<p>\u7a4d\u7387\u6bcd\u95a2\u6570\u3068\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570\u3068\u306e\u95a2\u4fc2\u304b\u3089\u3001\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\\(\\kappa_{j}\\)\u3068\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\\(\\mu_{j}^{\\prime}\\)\u306b\u3064\u3044\u3066\u6b21\u306e\u3088\u3046\u306a\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\kappa_{1} &amp;= \\mu_{1}^{\\prime} \\\\<br>\\kappa_{2} &amp;= \\mu_{2}^{\\prime}-\\mu_{1}^{\\prime\\ 2} \\\\<br>\\kappa_{3} &amp;= \\mu_{3}^{\\prime}-3\\mu_{2}^{\\prime}\\mu_{1}^{\\prime}+2\\mu_{1}^{\\prime\\ 3} \\\\<br>\\kappa_{4} &amp;= \\mu_{4}^{\\prime}-4\\mu_{3}^{\\prime}\\mu_{1}^{\\prime}-3\\mu_{2}^{\\prime\\ 2}+12\\mu_{2}^{\\prime}\\mu_{1}^{\\prime\\ 2}-6\\mu_{1}^{\\prime\\ 4} \\\\<br>\\kappa_{5} &amp;= \\mu_{5}^{\\prime}-5\\mu_{4}^{\\prime}\\mu_{1}^{\\prime}-10\\mu_{3}^{\\prime}\\mu_{2}^{\\prime}+20\\mu_{3}^{\\prime}\\mu_{1}^{\\prime\\ 2}+30\\mu_{2}^{\\prime\\ 2}\\mu_{1}^{\\prime}-60\\mu_{2}^{\\prime}\\mu_{1}^{\\prime\\ 2}+24\\mu_{1}^{\\prime\\ 5} \\\\<br>&amp;\\vdots<br>\\end{align}<\/div><\/p>\n\n\n\n<p>\u3053\u306e\u3053\u3068\u3092\u3001\u9006\u306b\u7528\u3044\u308c\u3070\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mu_{1}^{\\prime} &amp;= \\kappa_{1} \\\\<br>\\mu_{2}^{\\prime} &amp;= \\kappa_{2}+\\kappa_{1}^{2} \\\\<br>\\mu_{3}^{\\prime} &amp;= \\kappa_{3}+3\\kappa_{2}\\kappa_{1}+\\kappa_{1}^{3} \\\\ <br>\\mu_{4}^{\\prime} &amp;= \\kappa_{4}+4\\kappa_{3}\\kappa_{1}+3\\kappa_{2}^{2}+6\\kappa_{2}\\kappa_{1}^{2}+\\kappa_{1}^{4} \\\\ <br>\\mu_{5}^{\\prime} &amp;= \\kappa_{5}+5\\kappa_{4}\\kappa_{1}+10\\kappa_{3}\\kappa_{2}+10\\kappa_{3}\\kappa_{1}^{2}+15\\kappa_{2}^{2}\\kappa_{1}+10\\kappa_{2}\\kappa_{1}^{3}+\\kappa_{1}^{5} \\\\<br>&amp;\\vdots<br>\\end{align}<\/div><\/p>\n\n\n\n<p>\u3088\u308a\u4e00\u822c\u7684\u306b\u306f\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u3068\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u3068\u306f<br><strong><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mu_{r}^{\\prime} = \\sum_{j=1}^{r}\\ _{r-1}C_{j-1}\\mu_{r-j}^{\\prime}\\kappa_{j}<br>\\end{align}<\/div><\/strong><br>\u3068\u3044\u3046\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n\n\n\n<a href=\"https:\/\/px.a8.net\/svt\/ejp?a8mat=3BDWTF+G23X6A+3JTE+62U35\" rel=\"nofollow\">\n<img loading=\"lazy\" border=\"0\" width=\"728\" height=\"90\" alt=\"\" src=\"https:\/\/www29.a8.net\/svt\/bgt?aid=200523363971&amp;wid=001&amp;eno=01&amp;mid=s00000016565001021000&amp;mc=1\"><\/a>\n<img loading=\"lazy\" border=\"0\" width=\"1\" height=\"1\" src=\"https:\/\/www18.a8.net\/0.gif?a8mat=3BDWTF+G23X6A+3JTE+62U35\" alt=\"\">\n\n\n\n<h4><span id=\"i-4\">\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u3068\u4e2d\u5fc3\u30e2\u30fc\u30e1\u30f3\u30c8<\/span><\/h4>\n\n\n\n<p>\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u3068\u540c\u69d8\u306b\u3001\u4e2d\u5fc3\u30e2\u30fc\u30e1\u30f3\u30c8\\(\\mu_{j}\\)\u3082\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\\(\\kappa_{j}\\)\u3092\u7528\u3044\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u3053\u3067\u306f\u30015\u6b21\u307e\u3067\u306e\u4e2d\u5fc3\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u7d39\u4ecb\u3057\u3066\u3044\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mu_{2} &amp;= \\kappa_{2}\\\\<br>\\mu_{3} &amp;= \\kappa_{3}\\\\<br>\\mu_{4} &amp;= \\kappa_{4}+3\\kappa_{2}^{2}\\\\<br>\\mu_{5} &amp;= \\kappa_{5}+10\\kappa_{3}\\kappa_{2}\\\\<br>&amp;\\vdots<br>\\end{align}<\/div><\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h3><span id=\"i-5\">\u30e2\u30fc\u30e1\u30f3\u30c8\u306b\u95a2\u3059\u308b\u95a2\u9023\u8a18\u4e8b<\/span><\/h3>\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\/pgf\">\u78ba\u7387\u6bcd\u95a2\u6570<\/a>\uff1a\u96e2\u6563\u578b\u78ba\u7387\u5909\u6570\u306b\u3064\u3044\u3066\\(k\\)\u6b21\u306e\u968e\u4e57\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\/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\/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\n\n\n<p><\/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 \u5206\u5e03\u306e\u7279\u5fb4\u3092\u8868\u3059\u3082\u306e\u306b\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u3068\u3044\u3046\u3082\u306e\u304c\u3042\u308a\u307e\u3059\u3002\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u3068\u30e2\u30fc\u30e1\u30f3\u30c8\u306e\u9593\u306b\u306f\u4e00\u5b9a\u306e\u95a2\u4fc2\u304c &#8230; 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