{"id":2229,"date":"2020-07-29T11:00:00","date_gmt":"2020-07-29T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2229"},"modified":"2020-08-11T11:16:52","modified_gmt":"2020-08-11T02:16:52","slug":"cbetad-cf","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cbetad-cf","title":{"rendered":"\u30d9\u30fc\u30bf\u5206\u5e03\u306e\u7279\u6027\u95a2\u6570\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\u7279\u6027\u95a2\u6570\u3092\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002\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\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\u7279\u6027\u95a2\u6570<\/h2>\n\n\n\n
\u7279\u6027\u95a2\u6570<\/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\u7279\u6027\u95a2\u6570\\(\\phi_{X}(t)\\)\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\n
\\begin{align}\n\\phi_{X}(t) &= _{1}\\!F_{1}(\\alpha,\\alpha+\\beta;it) =1+\\sum_{k=1}^{\\infty}\\left( \\prod_{r=0}^{k-1}\\frac{\\alpha+r}{\\alpha+\\beta+r} \\right)\\frac{(it)^{k}}{k!}\n\\end{align}<\/div>\n\u305f\u3060\u3057\u3001\\(_{1}F_{1}(\\alpha,\\alpha+\\beta;it)\\)\u306f\u5408\u6d41\u578b\u8d85\u5e7e\u4f55\u95a2\u6570\u3067\u3059\u3002\n<\/div>\n\n\n\n

\u203b\u5408\u6d41\u578b\u8d85\u5e7e\u4f55\u95a2\u6570\u306f\u4ee5\u4e0b\u3067\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u307e\u3059\u3002

\\begin{align}
_{1}F_{1}(a,b;z)&=\\frac{1}{B(a,b)}\\int_{0}^{1}e^{zt}t^{a-1}(1-t)^{b-a-1}dt
\\end{align}<\/div>
\u3053\u306e\u5f0f\u3092\u7528\u3044\u305a\u3068\u3082\u3001\u7279\u6027\u95a2\u6570\u3092\u8868\u3059\u3053\u3068\u306f\u3067\u304d\u308b\u306e\u3067\u3001\u5408\u6d41\u578b\u8d85\u5e7e\u4f55\u95a2\u6570\u81ea\u4f53\u306f\u3042\u307e\u308a\u91cd\u8981\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/p>\n\n\n\n

\u7279\u6027\u95a2\u6570\u3092\u6c42\u3081\u308b\u969b\u306b\u306f\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

\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\u7279\u6027\u95a2\u6570\u306e\u5b9a\u7fa9\u304b\u3089
\\begin{align}
\\phi_{X}(t) &= \\mathrm{E}[e^{itX}] \\\\
&= \\int_{0}^{1}e^{itx}f(x)dx \\\\
&=\\int_{0}^{1}e^{itx}\\frac{ x^{\\alpha-1}(1-x)^{\\beta-1} }{ B(\\alpha,\\beta) }dx \\\\
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067\\(e^{itx}\\)\u306b\u3064\u3044\u3066\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3092\u884c\u3044\u307e\u3059\u3002\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u306f\u95a2\u6570\\(f(x)\\)\u306b\u5bfe\u3057\u3066
\\begin{align}
f(x) = \\sum_{k=0}^{\\infty} \\frac{f^{(k)}(0)}{k!}x^{k}
\\end{align}<\/div>
\u3068\u3044\u3046\u5f62\u306b\u5909\u5f62\u3059\u308b\u3053\u3068\u3067\u3059\u3002\u3053\u3053\u3067\\(f^{(i)}(x)\\)\u306f\\(f(x)\\)\u3092\\(i\\)\u56de\u5fae\u5206\u3057\u305f\u3082\u306e\u3092\u8868\u3057\u307e\u3059\u3002\\(f(x)=e^{tx}\\)\u306b\u3064\u3044\u3066\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3092\u8003\u3048\u308b\u3068\u304d\u3001
\\begin{align}
f^{(1)}(0) = it,\\ \\ f^{(2)}(0) = (it)^{2},\\ \\ f^{(3)}(0) = (it)^{3},\\ \\ \\cdots\\ \\ ,f^{(n)}(0) = (it)^{n}
\\end{align}<\/div>
\u3068\u306a\u308b\u3053\u3068\u3092\u7528\u3044\u308b\u3068\u3001
\\begin{align}
e^{itx} &= \\sum_{k=0}^{\\infty} \\frac{(it)^{k}}{k!}x^{k}
\\end{align}<\/div>
\u304c\u6210\u7acb\u3057\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u3092\u7528\u3044\u3066\u3001\u7279\u6027\u95a2\u6570\u3092\u66f8\u304d\u63db\u3048\u308b\u3068
\\begin{align}
\\phi_{X}(t) &= \\int_{0}^{1}\\sum_{k=0}^{\\infty} \\frac{(it)^{k}}{k!}x^{k}\\frac{ x^{\\alpha-1}(1-x)^{\\beta-1} }{ B(\\alpha,\\beta) }dx \\\\
&= \\sum_{k=0}^{\\infty}\\frac{(it)^{k}}{k!}\\frac{1}{B(\\alpha,\\beta)}\\int_{0}^{1} x^{\\alpha+k-1}(1-x)^{\\beta-1} dx \\\\
&= \\sum_{k=0}^{\\infty}\\frac{(it)^{k}}{k!}\\frac{1} {B(\\alpha,\\beta)}B(\\alpha+k,\\beta) \\\\
&= \\frac{(it)^{0}}{0!}\\frac{1} {B(\\alpha,\\beta)}B(\\alpha,\\beta)+ \\sum_{k=1}^{\\infty}\\frac{(it)^{k}}{k!}\\frac{1} {B(\\alpha,\\beta)}B(\\alpha+k,\\beta) \\\\
&=1+ \\sum_{k=1}^{\\infty}\\frac{(it)^{k}}{k!}\\frac{1} {B(\\alpha,\\beta)}B(\\alpha+k,\\beta)
\\end{align}<\/div>
\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u3053\u3067\u3001\u30d9\u30fc\u30bf\u95a2\u6570\u306e\u6027\u8cea\u304b\u3089
\\begin{align}
B(\\alpha+1,\\beta)=\\frac{\\alpha}{\\alpha+\\beta}B(\\alpha,\\beta)
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\uff08\u3053\u306e\u3053\u3068\u306f\u90e8\u5206\u7a4d\u5206\u3092\u7528\u3044\u308c\u3070\u7c21\u5358\u306b\u78ba\u8a8d\u3067\u304d\u307e\u3059\uff09\u3002
\u3000\u3053\u306e\u6027\u8cea\u3092\\(B(\\alpha+k,\\beta)\\)\u306b\u7528\u3044\u308b\u3068
\\begin{align}
B(\\alpha+k,\\beta) &= \\frac{\\alpha+k-1}{(\\alpha+k-1)+\\beta}B(\\alpha+k-1,\\beta) \\\\
&= \\frac{\\alpha+k-1}{(\\alpha+k-1)+\\beta}\\cdot\\frac{\\alpha+k-2}{(\\alpha+k-2)+\\beta}B(\\alpha+k-2,\\beta) \\\\
&\u3000\u3000\u3000\\vdots \\\\
&= \\prod_{r=0}^{k-1}\\frac{\\alpha+r}{\\alpha+\\beta+r}B(\\alpha,\\beta)
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u7279\u6027\u95a2\u6570\u3092\u6574\u7406\u3059\u308b\u3068\u3001
\\begin{align}
\\phi_{X}(t) &= 1+ \\sum_{k=1}^{\\infty}\\frac{(it)^{k}}{k!}\\frac{1} {B(\\alpha,\\beta)}\\prod_{r=0}^{k-1}\\frac{\\alpha+r}{\\alpha+\\beta+r}B(\\alpha,\\beta) \\\\
&= 1+\\sum_{k=1}^{\\infty}\\left( \\prod_{r=0}^{k-1}\\frac{\\alpha+r}{\\alpha+\\beta+r} \\right)\\frac{(it)^{k}}{k!}
\\end{align}<\/div>
\u3068\u306a\u308a\u3001\u6c42\u3081\u305f\u3044\u5f0f\u304c\u51fa\u3066\u304d\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\u7279\u6027\u95a2\u6570\u3092\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9\u304c\u5206\u304b\u3089\u306a\u3044\u65b9\u306f\u662f\u975e\u304a … <\/p>\n","protected":false},"author":1,"featured_media":2176,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[76],"tags":[109,80],"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\uff13.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2229"}],"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=2229"}],"version-history":[{"count":12,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2229\/revisions"}],"predecessor-version":[{"id":2244,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2229\/revisions\/2244"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2176"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2229"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2229"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2229"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}