{"id":2774,"date":"2020-08-19T11:00:00","date_gmt":"2020-08-19T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2774"},"modified":"2020-08-04T15:42:44","modified_gmt":"2020-08-04T06:42:44","slug":"cchisqd-mv","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cchisqd-mv","title":{"rendered":"\u30ab\u30a4\uff12\u4e57\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

\u3000<\/p>\n\n\n\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\u306f\u30ab\u30a4\uff12\u4e57\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\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u306f\uff1c\u30ab\u30a4\uff12\u4e57\u5206\u5e03\uff1e<\/a>\u306e\u8a18\u4e8b\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n\n

\n \n
\"\u30ab\u30a4\uff12\u4e57\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
\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011<\/div>\n
\u3000 \u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u3053\u306e\u8a18\u4e8b\u3067\u306f\u30ab\u30a4\uff12\u4e57\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\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n

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

\u30ab\u30a4\uff12\u4e57\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\\(n\\)\u306e\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim \\chi^{2}(n)\\)\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]=n,\\ \\ \\ \\mathrm{Var}[X]=2n\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

\u3000\u81ea\u7531\u5ea6\\(n\\)\u306e\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f

\\begin{align}
f(x) = \\frac{ x^{\\frac{n}{2}-1}e^{-\\frac{x}{2}} }{ \\Gamma\\left( \\frac{n}{2} \\right)2^{\\frac{n}{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\uff1c\u30ab\u30a4\uff12\u4e57\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\\\\
&= \\int_{0}^{\\infty}x\\frac{ x^{\\frac{n}{2}-1}e^{-\\frac{x}{2}} }{ \\Gamma\\left( \\frac{n}{2} \\right)2^{\\frac{n}{2}} } dx\\\\
&= \\int_{0}^{\\infty}\\frac{ x^{\\left(\\frac{n}{2}+1\\right)-1}e^{-\\frac{x}{2}} }{ \\Gamma\\left( \\frac{n}{2} \\right)2^{\\frac{n}{2}} } dx\\\\
&= \\frac{n}{2}\\cdot 2\\int_{0}^{\\infty}\\frac{ x^{\\left(\\frac{n}{2}+1\\right)-1}e^{-\\frac{x}{2}} }{ \\Gamma\\left( \\frac{n}{2}+1 \\right)2^{\\frac{n}{2}+1} } dx\\\\
&=n
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u6700\u5f8c\u306e\u884c\u306e\u5f0f\u5909\u5f62\u306f\u3001\u7a4d\u5206\u306e\u4e2d\u8eab\u306e\u5f0f
\\begin{align}
\\frac{ x^{\\left(\\frac{n}{2}+1\\right)-1}e^{-\\frac{x}{2}} }{ \\Gamma\\left( \\frac{n}{2}+1 \\right)2^{\\frac{n}{2}+1} }
\\end{align}<\/div>\u3092\u898b\u3066\u307f\u308b\u3068\u3001\u3053\u306e\u5f0f\u306f\u81ea\u7531\u5ea6\\(n+2\\)\u306e\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304b\u3089\u3001\u7a4d\u5206\u304c\\(1\\)\u3068\u306a\u308b\u3053\u3068\u3092\u7528\u3044\u307e\u3057\u305f\u3002\u3055\u3089\u306b\u3001\u9014\u4e2d\u306e\u5f0f\u5909\u5f62\u3067\u306f\u3001\u30ac\u30f3\u30de\u95a2\u6570\u306e\u6027\u8cea
\\begin{align}\\Gamma(\\alpha)=(\\alpha-1)\\Gamma(\\alpha-1) \\Rightarrow \\Gamma(\\alpha-1)=\\frac{\\Gamma(\\alpha)}{\\alpha-1}\\end{align}<\/div>\u3092\u7528\u3044\u307e\u3057\u305f\u3002
\u5206\u6563\u3082\u540c\u69d8\u306b\u3057\u3066\u6c42\u3081\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\u3088\u3063\u3066
\\begin{align}
\\mathrm{E}[X^{2}] &= \\int_{0}^{\\infty}x^{2}f(x) dx\\\\
&= \\int_{0}^{\\infty}x^{2}\\frac{ x^{\\frac{n}{2}-1}e^{-\\frac{x}{2}} }{ \\Gamma\\left( \\frac{n}{2} \\right)2^{\\frac{n}{2}} } dx\\\\
&= \\int_{0}^{\\infty}\\frac{ x^{\\left(\\frac{n}{2}+2\\right)-1}e^{-\\frac{x}{2}} }{ \\Gamma\\left( \\frac{n}{2} \\right)2^{\\frac{n}{2}} } dx\\\\
&= \\frac{n}{2}\\left(\\frac{n}{2}+1\\right)2^{2}\\int_{0}^{\\infty}\\frac{ x^{\\left(\\frac{n}{2}+2\\right)-1}e^{-\\frac{x}{2}} }{ \\Gamma\\left( \\frac{n}{2}+2 \\right)2^{\\frac{n}{2}+2} } dx\\\\
&= n^{2}+2n
\\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] &= (n^{2}-2n)-n^{2} \\\\
&= 2n
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

\u3000 \u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u3053\u306e\u8a18\u4e8b\u3067\u306f\u30ab\u30a4\uff12\u4e57\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 … <\/p>\n","protected":false},"author":1,"featured_media":2759,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[74,72,84],"tags":[118,31,73],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/\u30ab\u30a42\u4e57\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\/2774"}],"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=2774"}],"version-history":[{"count":13,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2774\/revisions"}],"predecessor-version":[{"id":3073,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2774\/revisions\/3073"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2759"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2774"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2774"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2774"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}