{"id":2472,"date":"2020-08-10T11:00:00","date_gmt":"2020-08-10T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2472"},"modified":"2020-08-11T11:20:03","modified_gmt":"2020-08-11T02:20:03","slug":"cgamd-mc","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cgamd-mc","title":{"rendered":"\u30ac\u30f3\u30de\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\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\u2605\u2606\u2606<\/span><\/p>\n\n\n\n\n\n\n\n<p>\u3000\u3053\u306e\u8a18\u4e8b\u3067\u306f\u30ac\u30f3\u30de\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\u5206\u304b\u3089\u306a\u3044\u65b9\u306f\u662f\u975e\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002\u305d\u306e\u4ed6\u306e\u30ac\u30f3\u30de\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/gamma-distribution\">\uff1c\u30ac\u30f3\u30de\u5206\u5e03\uff1e<\/a>\u306e\u8a18\u4e8b\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/gamma-distribution\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30ac\u30f3\u30de\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u03b14\uff09-100x100.png\" alt=\"\u30ac\u30f3\u30de\u5206\u5e03\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u30ac\u30f3\u30de\u5206\u5e03<\/div>\n    <div class=\"blogcard_excerpt\">\u3000\u30ac\u30f3\u30de\u5206\u5e03\uff08gamma distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u30ac\u30f3\u30de\u5206\u5e03\u306f\u300c\u3042\u308b\u671f\u9593\\(f\\)\u306e\u9593\u306b\\(1\\)\u56de\u8d77\u3053\u308b\u4e8b\u8c61\u304c\u3001\\\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h2>\u30ac\u30f3\u30de\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570<\/h2>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\alpha,\\beta\\)\u306e\u30ac\u30f3\u30de\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim Gam(\\alpha,\\beta)\\)\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\n<div style=\"overflow-x: auto;\">\\begin{align}\nM_{X}(t)=(1-\\beta t)^{-\\alpha}\\ \\ \uff08 t<\\frac{1}{\\beta} \uff09,\\ \\ \\ \\phi_{X}(t)=(1-i\\beta t)^{-\\alpha}\n\\end{align}<\/div>\n<\/div>\n\n\n\n<p>\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u3092\u6c42\u3081\u308b\u969b\u306b\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mgf\">\uff1c\u7a4d\u7387\u6bcd\u95a2\u6570\u306e\u5b9a\u7fa9\uff1e<\/a>\u304a\u3088\u3073<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cf\">\uff1c\u7279\u6027\u95a2\u6570\u306e\u5b9a\u7fa9\uff1e<\/a>\u3092\u4f7f\u7528\u3059\u308b\u306e\u3067\u3001\u899a\u3048\u3066\u3044\u306a\u3044\u65b9\u306f\u8a3c\u660e\u3092\u8aad\u3080\u524d\u306b\u4e00\u5ea6\u3001\u76ee\u3092\u901a\u3057\u3066\u304a\u3044\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h4>\u8a3c\u660e<\/h4>\n\n\n\n<p>\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\alpha,\\beta\\)\u306e\u30ac\u30f3\u30de\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f<br data-rich-text-line-break=\"true\" \/><div style=\"overflow-x: auto;\">\\begin{align}<br data-rich-text-line-break=\"true\" \/>f(x)= \\frac{x^{\\alpha-1}\\exp\\left[ -\\frac{x}{\\beta} \\right]}{\\beta^{\\alpha}\\Gamma(\\alpha)}<br data-rich-text-line-break=\"true\" \/>\\end{align}<\/div><br data-rich-text-line-break=\"true\" \/>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/gamma-distribution\">\uff1c\u30ac\u30f3\u30de\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\uff1e<\/a>\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<br data-rich-text-line-break=\"true\" \/>\u3000\u307e\u305a\u3001\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u6c42\u3081\u3066\u3044\u304d\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u306e\u5b9a\u7fa9\u304b\u3089\u76f4\u63a5\u8a08\u7b97\u3059\u308b\u3068\u3001<br \/><div style=\"overflow-x: auto;\">\\begin{align}<br \/>M_{X}(t) &amp;= \\mathrm{E}\\left[ \\exp[tX] \\right] \\\\<br \/>&amp;= \\int_{0}^{\\infty} \\exp[tx] \\frac{x^{\\alpha-1}\\exp\\left[ -\\frac{x}{\\beta} \\right]}{\\beta^{\\alpha}\\Gamma(\\alpha)} dx \\\\<br \/>&amp;= \\int_{0}^{\\infty} \\frac{x^{\\alpha-1}\\exp\\left[ \\left( t-\\frac{1}{\\beta}\\right)x \\right]}{\\beta^{\\alpha}\\Gamma(\\alpha)} dx \\\\<br \/>&amp;= \\int_{0}^{\\infty} \\frac{x^{\\alpha-1}\\exp\\left[ -\\frac{1-\\beta t}{\\beta} x \\right]}{(1-\\beta t)^{\\alpha}\\left( \\frac{\\beta}{1-\\beta t} \\right)^{\\alpha}\\Gamma(\\alpha)} dx \\\\<br \/>&amp;= (1-\\beta t)^{-\\alpha}\\int_{0}^{\\infty} \\frac{x^{\\alpha-1}\\exp\\left[ -\\frac{1-\\beta t}{\\beta} x \\right]}{\\left( \\frac{\\beta}{1-\\beta t} \\right)^{\\alpha}\\Gamma(\\alpha)} dx \\\\<br \/>&amp;= (1-\\beta t)^{-\\alpha}<br \/>\\end{align}<\/div><br \/>\u3068\u306a\u308a\u307e\u3059\u3002\u6700\u5f8c\u306e\u5f0f\u5909\u5f62\u306f<br \/><div style=\"overflow-x: auto;\">\\begin{align}<br data-rich-text-line-break=\"true\" \/>\\int_{0}^{\\infty} \\frac{x^{\\alpha-1}\\exp\\left[ -\\frac{1-\\beta t}{\\beta} x \\right]}{\\left( \\frac{\\beta}{1-\\beta t} \\right)^{\\alpha}\\Gamma(\\alpha)} dx<br data-rich-text-line-break=\"true\" \/>\\end{align}<\/div><br data-rich-text-line-break=\"true\" \/>\u306b\u3064\u3044\u3066\u898b\u3066\u307f\u308b\u3068\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\alpha,\\ \\frac{\\beta}{1-\\beta t}\\)\u306e\u30ac\u30f3\u30de\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306e\u7a4d\u5206\u306b\u306a\u3063\u3066\u3044\u308b\u306e\u3067\u3001\\(1\\)\u3068\u306a\u308a\u307e\u3059\u3002<br \/>\u3000\u540c\u69d8\u306b\u3057\u3066\u7279\u6027\u95a2\u6570\u3092\u6c42\u3081\u307e\u3059\u3002\u7279\u6027\u95a2\u6570\u306e\u5b9a\u7fa9\u304b\u3089<br \/><div style=\"overflow-x: auto;\">\\begin{align}<br \/>\\phi_{X}(t) &amp;= \\mathrm{E}\\left[ \\exp[itX] \\right] \\\\<br \/>&amp;= \\int_{0}^{\\infty} \\exp[itx] \\frac{x^{\\alpha-1}\\exp\\left[ -\\frac{x}{\\beta} \\right]}{\\beta^{\\alpha}\\Gamma(\\alpha)} dx \\\\<br \/>&amp;= \\int_{0}^{\\infty} \\frac{x^{\\alpha-1}\\exp\\left[ \\left( it-\\frac{1}{\\beta}\\right)x \\right]}{\\beta^{\\alpha}\\Gamma(\\alpha)} dx \\\\<br \/>&amp;= \\int_{0}^{\\infty} \\frac{x^{\\alpha-1}\\exp\\left[ -\\frac{1-i\\beta t}{\\beta} x \\right]}{(1-i\\beta t)^{\\alpha}\\left( \\frac{\\beta}{1-\\beta t} \\right)^{\\alpha}\\Gamma(\\alpha)} dx \\\\<br \/>&amp;= (1-i\\beta t)^{-\\alpha}\\int_{0}^{\\infty} \\frac{x^{\\alpha-1}\\exp\\left[ -\\frac{1-i\\beta t}{\\beta} x \\right]}{\\left( \\frac{\\beta}{1-i\\beta t} \\right)^{\\alpha}\\Gamma(\\alpha)} dx \\\\<br \/>&amp;= (1-i\\beta t)^{-\\alpha}<br \/>\\end{align}<\/div><br \/>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"has-text-align-right\">\u25a1<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2605\u2606\u2606 \u3000\u3053\u306e\u8a18\u4e8b\u3067\u306f\u30ac\u30f3\u30de\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 &#8230; <\/p>\n","protected":false},"author":1,"featured_media":2453,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[76],"tags":[114,80,79],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30ac\u30f3\u30de\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u03b21\uff09.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2472"}],"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=2472"}],"version-history":[{"count":9,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2472\/revisions"}],"predecessor-version":[{"id":2482,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2472\/revisions\/2482"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2453"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2472"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2472"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2472"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}