{"id":2333,"date":"2020-08-02T11:00:00","date_gmt":"2020-08-02T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2333"},"modified":"2020-08-11T11:17:48","modified_gmt":"2020-08-11T02:17:48","slug":"cbetapri-mv","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cbetapri-mv","title":{"rendered":"\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011"},"content":{"rendered":"\n

\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

\u3000\u3053\u306e\u8a18\u4e8b\u3067\u306f\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\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\u662f\u975e\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002\u305d\u306e\u4ed6\u306e\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u306f\uff1c\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\uff1e<\/a>\u306e\u8a18\u4e8b\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n\n

\n \n
\"\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\uff08\u30d9\u30fc\u30bf\u30d7\u30e9\u30a4\u30e0\u5206\u5e03\uff09\"<\/div>\n
\n
\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\uff08\u30d9\u30fc\u30bf\u30d7\u30e9\u30a4\u30e0\u5206\u5e03\uff09<\/div>\n
\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\uff08\u53cd\u8ee2\u30d9\u30fc\u30bf\u5206\u5e03\u3001beta prime distribution\u3001beta distribution of the second\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n

\u3000<\/p>\n\n\n\n

\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563<\/h2>\n\n\n\n
\u671f\u5f85\u5024\u3068\u5206\u6563<\/strong><\/div>\n
\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim Beta^{\\prime}(\\alpha,\\beta)\\)\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\n
\\begin{align}\n\\mathrm{E}[X]=\\frac{\\alpha}{\\beta-1},\\ \\ \\ \\mathrm{Var}[X]=\\frac{ \\alpha(\\alpha+\\beta-1) }{ (\\beta-2)(\\beta-1)^{2} }\n\\end{align}<\/div>\n<\/div>\n\n\n\n

\u671f\u5f85\u5024\u30fb\u5206\u6563\u3092\u6c42\u3081\u308b\u969b\u306b\u306f\uff1c\u671f\u5f85\u5024\u306e\u5b9a\u7fa9\uff1e<\/a>\u304a\u3088\u3073\uff1c\u5206\u6563\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

\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u3092\u6c42\u3081\u308b\u306b\u3042\u305f\u3063\u3066\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\u306e\\(r\\)\u6b21\u306e\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u6c42\u3081\u307e\u3059\u3002<\/p>\n\n\n\n

\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\u306e\\(r\\)\u6b21\u306e\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8<\/strong><\/div>\n
\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim Beta^{\\prime}(\\alpha,\\beta)\\)\u306e\\(r\\)\u6b21\u306e\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\\(\\mu_{r}^{\\prime}\\)\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\n
\\begin{align}\n\\mu_{r}^{\\prime} = \\mathrm{E}[X^{r}] = \\prod_{k=1}^{r}\\frac{\\alpha+k-1}{\\beta-k},\\ \\ \\ \u305f\u3060\u3057\u3001\\alpha < r < \\beta\n\\end{align}<\/div>\n<\/div>\n\n\n\n

\u3000<\/p>\n\n\n\n

\u8a3c\u660e\uff08\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\u306e\\(r\\)\u6b21\u306e\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\uff09<\/h4>\n\n\n\n

\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\\(Beta^{\\prime}(\\alpha,\\beta)\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f

\\begin{align}
f(x) &= \\frac{ x^{\\alpha-1}(1+x)^{-\\alpha-\\beta} }{ B(\\alpha,\\beta) }
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f\uff1c\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\uff1e<\/a>\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002
\u3000\u307e\u305a\u306f\u3001\\(r\\)\u6b21\u306e\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u304c
\\begin{align}
\\mathrm{E}[X^{r}] = \\frac{B(\\alpha+r,\\beta-r)}{B(\\alpha,\\beta)}
\\end{align}<\/div>
\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3057\u307e\u3059\u3002\u671f\u5f85\u5024\u306e\u5b9a\u7fa9\u304b\u3089
\\begin{align}
\\mathrm{E}[X^{r}] &= \\int_{0}^{\\infty}x^{r}f(x)dx \\\\
&= \\int_{0}^{\\infty}x^{r}\\frac{ x^{\\alpha-1}(1+x)^{-\\alpha-\\beta} }{ B(\\alpha,\\beta) }dx \\\\
&= \\frac{1}{B(\\alpha,\\beta)} \\int_{0}^{\\infty} x^{\\alpha+r-1}(1+x)^{-\\alpha-\\beta} dx
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067\u3001\\(x=\\frac{u}{1-u}\\)\u306b\u5909\u6570\u5909\u63db\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u30e4\u30b3\u30d3\u30a2\u30f3\u3092\u8a08\u7b97\u3059\u308b\u3068\\(dx=(1-u)^{-2}du\\)\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3001
\\begin{align}
\\mathrm{E}[X^{r}] &= \\frac{1}{B(\\alpha,\\beta)} \\int_{0}^{1} \\left( \\frac{u}{1-u} \\right)^{\\alpha+r-1}\\left(1+\\frac{u}{1-u}\\right)^{-\\alpha-\\beta} (1-u)^{-2}du \\\\
&= \\frac{1}{B(\\alpha,\\beta)} \\int_{0}^{1} u^{(\\alpha+r)-1}\\left( 1-u \\right)^{-\\alpha-r+1}\\left(1-u\\right)^{\\alpha+\\beta} (1-u)^{-2}du \\\\
&= \\frac{1}{B(\\alpha,\\beta)} \\int_{0}^{1} u^{(\\alpha+r)-1}\\left( 1-u \\right)^{(\\beta-r)-1}du \\\\
&= \\frac{B(\\alpha+r,\\beta-r)}{B(\\alpha,\\beta)}
\\end{align}<\/div>
\u304c\u6210\u7acb\u3057\u307e\u3059\u3002\u3053\u306e\u5f0f\u3092\u3001\u30d9\u30fc\u30bf\u95a2\u6570\u306e\u57fa\u672c\u6027\u8cea\u3068\u30ac\u30f3\u30de\u95a2\u6570\u306e\u57fa\u672c\u6027\u8cea\u3092\u4f7f\u3063\u3066\u6574\u7406\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u5f0f\u5909\u5f62\u3067\u7528\u3044\u308b\u6027\u8cea\u306f
\\begin{align}
B(\\alpha,\\beta)&=\\frac{\\Gamma(\\alpha)\\Gamma(\\beta)}{\\Gamma(\\alpha+\\beta)} \\\\
\\Gamma(a)&=(a-1)\\Gamma(a-1)
\\end{align}<\/div>
\u3060\u3051\u3067\u3059\u3002\u65e9\u901f\u3001\u5f0f\u3092\u5909\u5f62\u3057\u3066\u3044\u304d\u307e\u3059\u3002
\\begin{align}
\\mathrm{E}[X^{r}] &= \\frac{\\Gamma(\\alpha+r)\\Gamma(\\beta-r)}{\\Gamma(\\alpha+\\beta)}\\cdot\\frac{\\Gamma(\\alpha+\\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \\\\
&= \\frac{\\Gamma(\\alpha+r)}{\\Gamma(\\alpha)}\\cdot\\frac{\\Gamma(\\beta-r)}{\\Gamma(\\beta)} \\\\
&= \\frac{\\alpha+r-1}{\\beta-1}\\times\\frac{\\Gamma(\\alpha+r-1)}{\\Gamma(\\alpha)}\\cdot\\frac{\\Gamma(\\beta-r)}{\\Gamma(\\beta-1)} \\\\
&= \\frac{\\alpha+r-1}{\\beta-1}\\frac{\\alpha+r-2}{\\beta-2}\\times\\frac{\\Gamma(\\alpha+r-2)}{\\Gamma(\\alpha)}\\cdot\\frac{\\Gamma(\\beta-r)}{\\Gamma(\\beta-2)} \\\\
&\u3000\u3000\\vdots\\\\
&= \\frac{\\alpha+r-1}{\\beta-1}\\frac{\\alpha+r-2}{\\beta-2}\\cdots\\frac{\\alpha}{\\beta} \\times\\frac{\\Gamma(\\alpha)}{\\Gamma(\\alpha)}\\cdot\\frac{\\Gamma(\\beta-r)}{\\Gamma(\\beta-r)} \\\\
&= \\prod_{k=1}^{r}\\frac{\\alpha+k-1}{\\beta-k}
\\end{align}<\/div>
\u3053\u306e\u3053\u3068\u304b\u3089\u3001\u6c42\u3081\u305f\u3044\\(r\\)\u6b21\u306e\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

\u25a1<\/p>\n\n\n\n

\u3000\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u7528\u3044\u308b\u3068\u671f\u5f85\u5024\u30fb\u5206\u6563\u306f\u7c21\u5358\u306b\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u307e\u305a\u671f\u5f85\u5024\u306f\\(r=1\\)\u3067\u3042\u308b\u304b\u3089

\\begin{align}
\\mathrm{E}[X^{1}] &= \\prod_{k=1}^{1}\\frac{\\alpha+k-1}{\\beta-k} \\\\
&= \\frac{\\alpha}{\\beta-1}
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3055\u3089\u306b\u3001\uff1c\u5206\u6563\u306e\u5b9a\u7fa9\uff1e<\/a>\u304b\u3089\u5206\u6563\u306f
\\begin{align}
\\mathrm{Var}[X] = \\mathrm{E}[X^{2}]-\\mathrm{E}[X]^{2}
\\end{align}<\/div>
\u3068\u8868\u305b\u307e\u3059\u3002\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u7528\u3044\u3066
\\begin{align}
\\mathrm{E}[X^{2}] &= \\prod_{k=1}^{2}\\frac{\\alpha+k-1}{\\beta-k} \\\\
&= \\frac{\\alpha}{\\beta-1}\\cdot\\frac{\\alpha+1}{\\beta-2}
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304b\u3089\u3001\u6c42\u3081\u305f\u3044\u5206\u6563\u306f
\\begin{align}
\\mathrm{Var}[X] &= \\frac{\\alpha}{\\beta-1}\\cdot\\frac{\\alpha+1}{\\beta-2}-\\left( \\frac{\\alpha}{\\beta-1} \\right)^{2} \\\\
&= \\frac{ \\alpha(\\alpha+\\beta-1) }{ (\\beta-2)(\\beta-1)^{2} }
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\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\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\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 … <\/p>\n","protected":false},"author":1,"featured_media":2312,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[74,72],"tags":[31,73,111],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2333"}],"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=2333"}],"version-history":[{"count":19,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2333\/revisions"}],"predecessor-version":[{"id":2755,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2333\/revisions\/2755"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2312"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2333"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2333"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2333"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}