{"id":2196,"date":"2020-07-27T11:00:00","date_gmt":"2020-07-27T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2196"},"modified":"2020-08-11T11:16:14","modified_gmt":"2020-08-11T02:16:14","slug":"cbetad-mv","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cbetad-mv","title":{"rendered":"\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

\u3053\u306e\u8a18\u4e8b\u3067\u306f\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\u30d9\u30fc\u30bf\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u306f\uff1c\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
\"\u30d9\u30fc\u30bf\u5206\u5e03\uff08\u7b2c\uff11\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\uff09\"<\/div>\n
\n
\u30d9\u30fc\u30bf\u5206\u5e03\uff08\u7b2c\uff11\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\uff09<\/div>\n
\u3000\u30d9\u30fc\u30bf\u5206\u5e03\uff08beta distribution\uff09\u306f\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u30d9\u30fc\u30bf\u5206\u5e03\u306b\u306f\u7b2c\uff11\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\u3068\u7b2c\uff12\u7a2e\u30d9\u30fc\u30bf\u5206\u5e03\u304c\u3042\u308a\u3001\u5358\u306b\u30d9\u30fc\u30bf\u5206\u5e03\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n

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

\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
\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\alpha,\\beta\\)\u306e\u30d9\u30fc\u30bf\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim Beta(\\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}{\\alpha+\\beta},\\ \\ \\ \\mathrm{Var}[X]=\\frac{\\alpha\\beta}{(\\alpha+\\beta)^{2}(\\alpha+\\beta+1)}\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

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

\u8a3c\u660e<\/h4>\n\n\n\n

\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\alpha,\\beta\\)\u306e\u30d9\u30fc\u30bf\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f

\\begin{align}
f(x)= \\frac{ x^{\\alpha-1}(1-x)^{\\beta-1} }{ B(\\alpha,\\beta) }
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u305f\u3060\u3057\u3001\\(B(\\alpha,\\beta)\\)\u306f\u30d9\u30fc\u30bf\u95a2\u6570\u3067\u3059\u3002
\\begin{align}
B(\\alpha,\\beta) = \\int_{0}^{1}u^{\\alpha-1}(1-u)^{\\beta-1}du
\\end{align}<\/div>
\u3053\u306e\u3053\u3068\u306f<\/a><\/a><\/a><\/a>\uff1c\u30d9\u30fc\u30bf\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\uff1e<\/a>\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002
\u3000\u307e\u305a\u3001\u671f\u5f85\u5024\u3092\u6c42\u3081\u3066\u3044\u304d\u307e\u3059\u3002\u671f\u5f85\u5024\u306e\u5b9a\u7fa9\u304b\u3089
\\begin{align}
\\mathrm{E}[X] &= \\int_{0}^{1}xf(x)dx \\\\
&=\\int_{0}^{1}x\\frac{ x^{\\alpha-1}(1-x)^{\\beta-1} }{ B(\\alpha,\\beta) }dx \\\\
&=\\frac{1}{B(\\alpha,\\beta)}\\int_{0}^{1}x^{\\alpha}(1-x)^{\\beta-1}dx \\\\
&= \\frac{1}{B(\\alpha,\\beta)}\\int_{0}^{1}x^{(\\alpha+1)-1}(1-x)^{\\beta-1}dx \\\\
&= \\frac{1}{B(\\alpha,\\beta)}B(\\alpha+1,\\beta) \\\\
&= \\frac{\\alpha}{\\alpha+\\beta}
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u6700\u5f8c\u306e\u884c\u306e\u5909\u5f62\u306b\u306f\u30d9\u30fc\u30bf\u95a2\u6570\u306e\u6027\u8cea\u3092\u7528\u3044\u307e\u3057\u305f\u3002\u30d9\u30fc\u30bf\u95a2\u6570\u306f\u6b21\u5f0f\u306e\u3088\u3046\u306a\u6027\u8cea\u304c\u3042\u308a\u307e\u3059\uff08\u3053\u306e\u5f0f\u306f\u90e8\u5206\u7a4d\u5206\u3092\u3059\u308c\u3070\u7c21\u5358\u306b\u78ba\u8a8d\u3067\u304d\u307e\u3059\uff09\u3002
\\begin{align}
B(\\alpha+1,\\beta)=\\frac{\\alpha}{\\alpha+\\beta}B(\\alpha,\\beta)
\\end{align}<\/div>
\u3053\u306e\u3088\u3046\u306a\u8003\u3048\u65b9\u306f\u5206\u6563\u3092\u6c42\u3081\u308b\u3068\u304d\u306b\u3082\u7528\u3044\u307e\u3059\u3002
\u3000\uff1c\u5206\u6563\u306e\u5b9a\u7fa9\uff1e<\/a>\u306e\u8a18\u4e8b\u304b\u3089\u5206\u6563\u306f
\\begin{align}
\\mathrm{Var}[X] &= \\mathrm{E}[X^{2}]-\\mathrm{E}[X]^{2}
\\end{align}<\/div>
\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\u3001\\(\\mathrm{E}[X^{2}]\\)\u3092\u6c42\u3081\u308c\u3070\u3088\u3044\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u3088\u3063\u3066
\\begin{align}
\\mathrm{E}[X^{2}] &= \\int_{0}^{1}x^{2}f(x)dx \\\\
&=\\int_{0}^{1}x^{2}\\frac{ x^{\\alpha-1}(1-x)^{\\beta-1} }{ B(\\alpha,\\beta) }dx \\\\
&=\\frac{1}{B(\\alpha,\\beta)}\\int_{0}^{1}x^{\\alpha+1}(1-x)^{\\beta-1}dx \\\\
&= \\frac{1}{B(\\alpha,\\beta)}\\int_{0}^{1}x^{(\\alpha+2)-1}(1-x)^{\\beta-1}dx \\\\
&= \\frac{1}{B(\\alpha,\\beta)}B(\\alpha+2,\\beta) \\\\
&= \\frac{1}{B(\\alpha,\\beta)}\\cdot\\frac{\\alpha+1}{\\alpha+\\beta+1}B(\\alpha+1,\\beta) \\\\
&= \\frac{\\alpha+1}{\\alpha+\\beta+1}\\cdot\\frac{\\alpha}{\\alpha+\\beta}
\\end{align}<\/div>
\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u6c42\u3081\u305f\u3044\u5206\u6563\u306f
\\begin{align}
\\mathrm{Var}[X] &= \\frac{\\alpha+1}{\\alpha+\\beta+1}\\cdot\\frac{\\alpha}{\\alpha+\\beta} – \\left( \\frac{\\alpha}{\\alpha+\\beta} \\right)^{2} \\\\
&=\\frac{\\alpha\\beta}{(\\alpha+\\beta)^{2}(\\alpha+\\beta+1)}
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u25a1<\/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 \u3053\u306e\u8a18\u4e8b\u3067\u306f\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 … <\/p>\n","protected":false},"author":1,"featured_media":2175,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[74,72],"tags":[109,31,73],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30fc\u30bf\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09\uff12.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2196"}],"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=2196"}],"version-history":[{"count":11,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2196\/revisions"}],"predecessor-version":[{"id":2247,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2196\/revisions\/2247"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2175"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2196"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2196"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2196"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}