{"id":3002,"date":"2020-08-24T11:00:08","date_gmt":"2020-08-24T02:00:08","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=3002"},"modified":"2020-08-20T21:02:52","modified_gmt":"2020-08-20T12:02:52","slug":"cstutd-mv","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cstutd-mv","title":{"rendered":"t\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\u2606\u2606\u2606<\/span><\/p>\n\n\n\n\n\n\n\n

\u3053\u306e\u8a18\u4e8b\u3067\u306ft\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\u306et\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u306f\uff1ct\u5206\u5e03\uff1e<\/a>\u306e\u8a18\u4e8b\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n\n

\n \n
\"t\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011\"<\/div>\n
\n
t\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011<\/div>\n
\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u3053\u306e\u8a18\u4e8b\u3067\u306ft\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\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n

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

t\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563<\/h2>\n\n\n\n
\u671f\u5f85\u5024\u3068\u5206\u6563<\/strong><\/div>\n
\u81ea\u7531\u5ea6\\(m\\)\u306et\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim t(m)\\)\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]=0,\\ \\ \\ \\mathrm{Var}[X]=\\frac{m}{m-2}\\ \\ \\ m>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

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

\u8a3c\u660e<\/h3>\n\n\n\n

\u671f\u5f85\u5024\u306e\u5c0e\u51fa\u306f\u3068\u3066\u3082\u7c21\u5358\u3067\u3059\u3002\u81ea\u7531\u5ea6\\(m\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\\(f(x)\\)\u306f

\\begin{align}f(x)=\\frac{ \\Gamma\\left( \\frac{m+1}{2} \\right) }{ (\\pi m)^{\\frac{1}{2}}\\Gamma\\left( \\frac{m}{2} \\right)\\left( 1+\\frac{x^{2}}{m} \\right)^{\\frac{m+1}{2}} } \\end{align}<\/div>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f\uff1ct\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\uff1e<\/a>\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002
\u3000\u3053\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f\u5076\u95a2\u6570\u306b\u306a\u308a\u307e\u3059\uff08\u5076\u95a2\u6570\u3068\u306f\\(f(x)=f(-x)\\)\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3067\u3059\uff09\u3002\u3088\u3063\u3066\u671f\u5f85\u5024\u306f
\\begin{align}
\\mathrm{E}[X]&=\\int_{-\\infty}^{\\infty}xf(x)dx \\\\
&= \\int_{-\\infty}^{0}xf(x)dx+\\int_{0}^{\\infty}xf(x)dx \\\\
&= -\\int_{0}^{\\infty}xf(-x)dx+\\int_{0}^{\\infty}xf(x)dx\\\\
&= 0
\\end{align}<\/div>\u3068\u306a\u308a\u307e\u3059\u3002\u5206\u6563\u306f\u3061\u3087\u3063\u3068\u6c42\u3081\u308b\u306e\u304c\u96e3\u3057\u304f\u306a\u308a\u307e\u3059\u3002\u307e\u305a\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\u5b9a\u7fa9\u901a\u308a\u306b\u8a08\u7b97\u3059\u308b\u3068
\\begin{align}
\\mathrm{E}[X^{2}] &= \\int_{-\\infty}^{\\infty}x^{2}f(x) dx\\\\
&= \\int_{-\\infty}^{\\infty}x^{2}\\frac{ \\Gamma\\left( \\frac{m+1}{2} \\right) }{ (\\pi m)^{\\frac{1}{2}}\\Gamma\\left( \\frac{m}{2} \\right)\\left( 1+\\frac{x^{2}}{m} \\right)^{\\frac{m+1}{2}} } dx\\\\
&=\\frac{ \\Gamma\\left( \\frac{m+1}{2} \\right) }{ (\\pi m)^{\\frac{1}{2}}\\Gamma\\left( \\frac{m}{2} \\right)} \\int_{-\\infty}^{\\infty}x^{2}\\left( 1+\\frac{x^{2}}{m} \\right)^{-\\frac{m+1}{2}}\u00a0 dx\\\\
&=\\frac{ 2\\Gamma\\left( \\frac{m+1}{2} \\right) }{ (\\pi m)^{\\frac{1}{2}}\\Gamma\\left( \\frac{m}{2} \\right)} \\int_{0}^{\\infty}x^{2}\\left( 1+\\frac{x^{2}}{m} \\right)^{-\\frac{m+1}{2}}\u00a0 dx\\\\
\\end{align}<\/div>\u304c\u6210\u7acb\u3057\u307e\u3059\u3002\u3053\u3053\u3067\u3001\u5f0f\u5909\u5f62\u3067\u306f\u7a4d\u5206\u5185\u304c\u5076\u95a2\u6570\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u3092\u5229\u7528\u3057\u307e\u3057\u305f\u3002\u3053\u3053\u3067\u7a4d\u5206\u306b\u6ce8\u76ee\u3057\u307e\u3059\u3002\u6b21\u306e\u3088\u3046\u306a\u5909\u6570\u5909\u63db\u3092\u884c\u3044\u307e\u3059\u3002
\\begin{align}u=\\left( 1+\\frac{x^{2}}{m} \\right)^{-1}\\end{align}<\/div>\u3053\u306e\u5909\u6570\u5909\u63db\u304b\u3089
\\begin{align}
x&=\\sqrt{m(u^{-1}-1)}\\\\
\\frac{dx}{du} &= -\\frac{\\sqrt{m}}{2}(u^{-1}-1)^{-\\frac{1}{2}}u^{-2}
\\end{align}<\/div>\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3001
\\begin{align}\u00a0
\\int_{0}^{\\infty}x^{2}\\left( 1+\\frac{x^{2}}{m} \\right)^{-\\frac{m+1}{2}}\u00a0 dx &= \\int_{1}^{0} m(u^{-1}-1)u^{\\frac{m+1}{2}}\\left( -\\frac{\\sqrt{m}}{2}(u^{-1}-1)^{-\\frac{1}{2}}u^{-2} \\right) du\\\\
&= \\frac{m^{\\frac{3}{2}}}{2}\\int_{0}^{1}u^{\\frac{m}{2}-2}(1-u)^{\\frac{1}{2}}du \\\\
&= \\frac{m^{\\frac{3}{2}}}{2}\\int_{0}^{1}u^{(\\frac{m}{2}-1)-1}(1-u)^{\\frac{3}{2}-1}du \\\\
&= \\frac{m^{\\frac{3}{2}}}{2}B\\left( \\frac{m}{2}-1,\\ \\frac{3}{2} \\right)\\\\
&= \\frac{m^{\\frac{3}{2}}}{2}\\frac{ \\Gamma\\left( \\frac{m}{2}-1 \\right)\\Gamma\\left( \\frac{3}{2} \\right) }{ \\Gamma\\left( \\frac{m+1}{2}\\right) }
\\end{align}<\/div>\u304c\u6210\u7acb\u3057\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u3092\u7528\u3044\u308b\u3068
\\begin{align}
\\mathrm{E}[X^{2}] &= \\frac{ 2\\Gamma\\left( \\frac{m+1}{2} \\right) }{ (\\pi m)^{\\frac{1}{2}}\\Gamma\\left( \\frac{m}{2} \\right)}\\times \\frac{m^{\\frac{3}{2}}}{2}\\frac{ \\Gamma\\left( \\frac{m}{2}-1 \\right)\\Gamma\\left( \\frac{3}{2} \\right) }{ \\Gamma\\left( \\frac{m+1}{2}\\right) } \\\\
&= \\frac{ 2\\Gamma\\left( \\frac{m+1}{2} \\right) }{ (\\pi m)^{\\frac{1}{2}}\\left( \\frac{m-2}{2} \\right)\\Gamma\\left( \\frac{m}{2}-1 \\right)}\\times \\frac{m^{\\frac{3}{2}}}{2}\\frac{ \\Gamma\\left( \\frac{m}{2}-1 \\right)\\frac{1}{2}\\sqrt{\\pi}\u00a0 }{ \\Gamma\\left( \\frac{m+1}{2}\\right) } \\\\
&= \\frac{m}{m-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]= \\mathrm{E}[X^{2}]-\\mathrm{E}[X]^{2}=\\frac{m}{m-2}\\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\u2606\u2606\u2606 \u3053\u306e\u8a18\u4e8b\u3067\u306ft\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 … <\/p>\n","protected":false},"author":1,"featured_media":2999,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[74,72],"tags":[124,31,73],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/t\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/3002"}],"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=3002"}],"version-history":[{"count":3,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/3002\/revisions"}],"predecessor-version":[{"id":3009,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/3002\/revisions\/3009"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2999"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=3002"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=3002"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=3002"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}