{"id":3072,"date":"2020-08-26T11:00:00","date_gmt":"2020-08-26T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=3072"},"modified":"2020-08-24T11:06:59","modified_gmt":"2020-08-24T02:06:59","slug":"cfd-mv","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cfd-mv","title":{"rendered":"F\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\u306fF\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\u5c0e\u51fa\u65b9\u6cd5\u306f\u5927\u304d\u304f\uff12\u30d1\u30bf\u30fc\u30f3\u3042\u308a\u3001\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304b\u3089\u5c0e\u51fa\u3059\u308b\u65b9\u6cd5\u3068\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306e\u6027\u8cea\u304b\u3089\u6c42\u3081\u308b\u3082\u306e\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u8a18\u4e8b\u3067\u306f\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304b\u3089\u6c42\u3081\u308b\u65b9\u6cd5\u3092\u89e3\u8aac\u3057\u307e\u3059\u3002\u305d\u306e\u4ed6\u306eF\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u306f\uff1cF\u5206\u5e03\uff1e<\/a>\u306e\u8a18\u4e8b\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n\n

\n \n
\"F\u5206\u5e03\"<\/div>\n
\n
F\u5206\u5e03<\/div>\n
\u3000F\u5206\u5e03\uff08F distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002F\u5206\u5e03\u306f\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u6a19\u672c\u306e\u5206\u6563\u306b\u6bd4\u3092\u53d6\u3063\u305f\u308a\u3057\u305f\u3068\u304d\u306b\u898b\u3089\u308c\u308b\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n

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

F\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_{1},\\ m_{2}\\)\u306eF\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim F(m_{1},\\ m_{2})\\)\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{m_{2}}{m_{2}-2},\\ \\ \\ \\mathrm{Var}[X]=\\frac{2m_{2}^{2}(m_{1}+m_{2}-2)}{m_{1}(m_{2}-2)^{2}(m_{2}-4)}\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\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u7528\u3044\u308b\u5834\u5408\uff09<\/h3>\n\n\n\n

\u3000\u81ea\u7531\u5ea6\\(m_{1},\\ m_{2}\\)\u306eF\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f

\\begin{align}
f(x) = \\frac{1}{B\\left( \\frac{m_{1}}{2},\\ \\frac{m_{2}}{2} \\right)}\\cdot\\frac{ \\left( \\frac{m_{1}}{m_{2}} \\right)^{\\frac{m_{1}}{2}}x^{\\frac{m_{1}}{2}-1} }{ \\left( 1+\\frac{m_{1}}{m_{2}}x \\right)^{\\frac{m_{1}+m_{2}}{2}} }
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304c\u3053\u306e\u3088\u3046\u306b\u306a\u308b\u3053\u3068\u306f\uff1cF\u5206\u5e03\uff1e<\/a>\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002
\u3000\u671f\u5f85\u5024\u306e\u5b9a\u7fa9<\/a>\u304b\u3089\u3001\u76f4\u63a5\u8a08\u7b97\u3057\u307e\u3059\u3002
\\begin{align}
\\mathrm{E}[X] &= \\int_{0}^{\\infty}xf(x) dx\\\\
&= \\frac{ \\left( \\frac{m_{1}}{m_{2}} \\right)^{\\frac{m_{1}}{2}} }{B\\left( \\frac{m_{1}}{2},\\ \\frac{m_{2}}{2} \\right)}\\int_{0}^{\\infty} \\frac{ x^{\\frac{m_{1}}{2}} }{ \\left( 1+\\frac{m_{1}}{m_{2}}x \\right)^{\\frac{m_{1}+m_{2}}{2}} }dx
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u3053\u3053\u3067\\(u=\\frac{1}{1+\\frac{m_{1}}{m_{2}}x}\\)\u3068\u304a\u304f\u3068
\\begin{align}
x=\\frac{m_{2}}{m_{1}}(u^{-1}-1),\\ \\ \\ \\frac{dx}{du} = -\\frac{m_{2}}{m_{1}}u^{2}
\\end{align}<\/div>\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3001
\\begin{align}
\\mathrm{E}[X]
&=\\frac{ \\left( \\frac{m_{1}}{m_{2}} \\right)^{\\frac{m_{1}}{2}} }{B\\left( \\frac{m_{1}}{2},\\ \\frac{m_{2}}{2} \\right)}\\int_{1}^{0} \\left( \\frac{m_{2}}{m_{1}}(u^{-1}-1) \\right)^{\\frac{m_{1}}{2}}u^{\\frac{m_{1}+m_{2}}{2}}\\left( -\\frac{m_{2}}{m_{1}}u^{-2} \\right)du \\\\
&=\\frac{ \\frac{m_{2}}{m_{1}} }{B\\left( \\frac{m_{1}}{2},\\ \\frac{m_{2}}{2} \\right)}\\int_{0}^{1}(u^{-1}-1)^{\\frac{m_{1}}{2}}u^{\\frac{m_{1}+m_{2}}{2}-2} du \\\\
&=\\frac{ \\frac{m_{2}}{m_{1}} }{B\\left( \\frac{m_{1}}{2},\\ \\frac{m_{2}}{2} \\right)}\\int_{0}^{1}(1-u)^{\\left(\\frac{m_{1}}{2}+1\\right)-1}u^{\\left( \\frac{m_{2}}{2}-1 \\right)-1} du \\\\
&=\\frac{ \\frac{m_{2}}{m_{1}} }{B\\left( \\frac{m_{1}}{2},\\ \\frac{m_{2}}{2} \\right)}B\\left( \\frac{m_{1}}{2}+1,\\ \\frac{m_{2}}{2}-1 \\right)
\\end{align}<\/div>\u304c\u6210\u7acb\u3057\u307e\u3059\u3002\u30d9\u30fc\u30bf\u95a2\u6570\u306e\u6027\u8cea\u3088\u308a
\\begin{align}B(\\alpha,\\ \\beta)=\\frac{ \\Gamma(\\alpha)\\Gamma(\\beta) }{ \\Gamma(\\alpha,\\ \\beta) }\\end{align}<\/div>\u304c\u6210\u308a\u7acb\u3061\u3001\u3055\u3089\u306b\u30ac\u30f3\u30de\u95a2\u6570\u306e\u6027\u8cea\u3088\u308a\u3001
\\begin{align}\\Gamma(\\alpha)=(\\alpha-1)\\Gamma(\\alpha)\\end{align}<\/div>\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u304b\u3089\u3001\u3064\u304e\u306e\u3088\u3046\u306a\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002
\\begin{align}B(\\alpha+1,\\ \\beta-1)=\\frac{ \\alpha }{ \\beta-1 }B\\left(\\alpha,\\ \\beta\\right)\\end{align}<\/div>\u3088\u3063\u3066\u3001
\\begin{align} B\\left( \\frac{m_{1}}{2}+1,\\ \\frac{m_{2}}{2}-1 \\right) = \\frac{ m_{1} }{ m_{2}-2 }B\\left(\\frac{m_{1}}{2},\\ \\frac{\\ m_{2}}{2}\\right) \\end{align}<\/div>\u3068\u306a\u308b\u3053\u3068\u3092\u7528\u3044\u308b\u3068\u3001\u671f\u5f85\u5024\u306f
\\begin{align}\\mathrm{E}[X] &= \\frac{ \\frac{m_{2}}{m_{1}} }{B\\left( \\frac{m_{1}}{2},\\ \\frac{m_{2}}{2} \\right)}\\frac{ m_{1} }{ m_{2}-2 }B\\left(\\frac{m_{1}}{2},\\ \\frac{\\ m_{2}}{2}\\right) \\\\
&= \\frac{m_{2}}{m_{2}-2}\\end{align}<\/div>\u3068\u306a\u308a\u307e\u3059\u3002
\u540c\u69d8\u306b\u3057\u3066\u5206\u6563\u3082\u6c42\u3081\u3066\u3044\u304d\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
\\begin{align}
\\mathrm{E}[X^{2}] &= \\int_{0}^{\\infty}x^{2}f(x) dx\\\\
&= \\frac{ \\left( \\frac{m_{1}}{m_{2}} \\right)^{\\frac{m_{1}}{2}} }{B\\left( \\frac{m_{1}}{2},\\ \\frac{m_{2}}{2} \\right)}\\int_{0}^{\\infty} \\frac{ x^{\\frac{m_{1}}{2}+1} }{ \\left( 1+\\frac{m_{1}}{m_{2}}x \\right)^{\\frac{m_{1}+m_{2}}{2}} }dx \\\\
&=\\frac{ \\left( \\frac{m_{1}}{m_{2}} \\right)^{\\frac{m_{1}}{2}} }{B\\left( \\frac{m_{1}}{2},\\ \\frac{m_{2}}{2} \\right)}\\int_{1}^{0} \\left( \\frac{m_{2}}{m_{1}}(u^{-1}-1) \\right)^{\\frac{m_{1}}{2}+1}u^{\\frac{m_{1}+m_{2}}{2}}\\left( -\\frac{m_{2}}{m_{1}}u^{-2} \\right)du \\\\
&=\\frac{ \\frac{m_{2}^{2}}{m_{1}^{2}} }{B\\left( \\frac{m_{1}}{2},\\ \\frac{m_{2}}{2} \\right)}\\int_{0}^{1}(u^{-1}-1)^{\\frac{m_{1}}{2}+1}u^{\\frac{m_{1}+m_{2}}{2}-2} du \\\\
&=\\frac{ \\frac{m_{2}^{2}}{m_{1}^{2}} }{B\\left( \\frac{m_{1}}{2},\\ \\frac{m_{2}}{2} \\right)}\\int_{0}^{1}(1-u)^{\\left(\\frac{m_{1}}{2}+2\\right)-1}u^{\\left( \\frac{m_{2}}{2}-2 \\right)-1} du \\\\
&=\\frac{ \\frac{m_{2}^{2}}{m_{1}^{2}} }{B\\left( \\frac{m_{1}}{2},\\ \\frac{m_{2}}{2} \\right)}B\\left( \\frac{m_{1}}{2}+2,\\ \\frac{m_{2}}{2}-2 \\right)\\\\
&=\\frac{ \\frac{m_{2}^{2}}{m_{1}^{2}} }{B\\left( \\frac{m_{1}}{2},\\ \\frac{m_{2}}{2} \\right)}\\frac{m_{1}+2}{m_{2}-4}\u00a0 \\frac{m_{1}}{m_{2}-2}B\\left( \\frac{m_{1}}{2},\\ \\frac{m_{2}}{2} \\right)\\\\
&= \\frac{ m_{2}^{2}(m_{1}+2) }{ m_{1}(m_{2}-2)(m_{2}-4) }
\\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{ m_{2}^{2}(m_{1}+2) }{ m_{1}(m_{2}-2)(m_{2}-4) }+\\left( \\frac{m_{2}}{m_{2}-2} \\right)^{2} \\\\
&= \\frac{2m_{2}^{2}(m_{1}+m_{2}-2)}{m_{1}(m_{2}-2)^{2}(m_{2}-4)}
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

<\/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\u306fF\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":3060,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[74,72],"tags":[31,73],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/F\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570m1\u56fa\u5b9a\uff09.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/3072"}],"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=3072"}],"version-history":[{"count":31,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/3072\/revisions"}],"predecessor-version":[{"id":3107,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/3072\/revisions\/3107"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/3060"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=3072"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=3072"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=3072"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}