{"id":2788,"date":"2020-08-20T11:00:00","date_gmt":"2020-08-20T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2788"},"modified":"2020-08-09T11:08:35","modified_gmt":"2020-08-09T02:08:35","slug":"cchisqd-mc","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cchisqd-mc","title":{"rendered":"\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\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\u306f\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u3092\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u3067\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u3092\u7528\u3044\u308b\u6a5f\u4f1a\u306f\u5c11\u306a\u3044\u3067\u3059\u304c\u8a08\u7b97\u306f\u5bb9\u6613\u306b\u3067\u304d\u308b\u306e\u3067\u3001\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\u30a42\u4e57\u5206\u5e03\"<\/div>\n
\n
\u30ab\u30a42\u4e57\u5206\u5e03<\/div>\n
\u3000\u30ab\u30a42\u4e57\u5206\u5e03\uff08chi square distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u591a\u304f\u306e\u66f8\u7c4d\u3067\u306f\u3001\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306e\u5e73\u65b9\u548c\u3067\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\u30ab\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n

\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570<\/h2>\n\n\n\n
\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570<\/strong><\/div>\n
\u81ea\u7531\u5ea6\\(n\\)\u306e\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\\(M_{X}(t)\\)\u3068\u7279\u6027\u95a2\u6570\\(\\phi_{X}(t)\\)\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\n
\\begin{align}\nM_{X}(t)=(1-2t)^{-\\frac{n}{2}}\\ \uff08t<\\frac{1}{2}\uff09,\\ \\ \\ \\phi_{X}(t)=(1-2it)^{-\\frac{n}{2}}\n\\end{align}<\/div>\n<\/div>\n\n\n\n

\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u3092\u6c42\u3081\u308b\u969b\u306b\u306f\uff1c\u7a4d\u7387\u6bcd\u95a2\u6570\u306e\u5b9a\u7fa9\uff1e<\/a>\u304a\u3088\u3073\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

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

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

\u81ea\u7531\u5ea6\\(n\\)\u306e\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f

\\begin{align}
f(x)= \\frac{ 1 }{ 2^{\\frac{1}{2}n}\\Gamma\\left( \\frac{n}{2} \\right) }e^{-\\frac{x}{2}}x^{\\frac{n}{2}-1}
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f\uff1c\u30ab\u30a4\uff12\u4e57\u5206\u5e03\uff1e<\/a>\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002
\u3000\u307e\u305a\u3001\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u6c42\u3081\u3066\u3044\u304d\u307e\u3059\u3002
\\begin{align}
M_{X}(t) &= \\mathrm{E}[\\exp[tX]] \\\\
&= \\int_{0}^{\\infty} e^{tx}\\frac{ 1 }{ 2^{\\frac{1}{2}n}\\Gamma\\left( \\frac{n}{2} \\right) }e^{-\\frac{x}{2}}x^{\\frac{n}{2}-1}dx\\\\
&= \\int_{0}^{\\infty} \\frac{ 1 }{ 2^{\\frac{1}{2}n}\\Gamma\\left( \\frac{n}{2} \\right) }e^{\\left( \\frac{1-2t}{2} \\right)x}x^{\\frac{n}{2}-1}dx\\\\
&= (1-2t)^{-\\frac{n}{2}} \\int_{0}^{\\infty} \\frac{ 1 }{ \\left( \\frac{2}{1-2t} \\right)^{\\frac{1}{2}n}\\Gamma\\left( \\frac{n}{2} \\right) }e^{\\left( \\frac{1-2t}{2} \\right)x}x^{\\frac{n}{2}-1}dx\\\\
&=(1-2t)^{-\\frac{n}{2}}
\\end{align}<\/div>
\u304c\u6210\u7acb\u3057\u307e\u3059\u3002\u6700\u5f8c\u306e\u5f0f\u5909\u5f62\u306b\u3064\u3044\u3066
\\begin{align}
\\frac{ 1 }{ \\left( \\frac{2}{1-2t} \\right)^{\\frac{1}{2}n}\\Gamma\\left( \\frac{n}{2} \\right) }e^{\\left( \\frac{1-2t}{2} \\right)x}x^{\\frac{n}{2}-1}
\\end{align}<\/div>
\u306b\u3064\u3044\u3066\u898b\u3066\u307f\u308b\u3068\u3001\u3053\u308c\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\alpha=\\frac{n}{2}\\)\u3001\\(\\beta=\\frac{2}{1-2t}\\)\u306e\u30ac\u30f3\u30de\u5206\u5e03<\/a>\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306b\u306a\u3063\u3066\u3044\u308b\u306e\u3067\u3053\u306e\u7dcf\u548c\u306f\\(1\\)\u306b\u306a\u308b\u3053\u3068\u3092\u7528\u3044\u307e\u3057\u305f\u3002
\u203b\u30ac\u30f3\u30de\u5206\u5e03\u306b\u3064\u3044\u3066\u306f\u4ee5\u4e0b\u306e\u30ea\u30f3\u30af\u304b\u3089\u304a\u9858\u3044\u3057\u307e\u3059\u3002<\/p>\n\n\n\n\n
\n \n
\"\u30ac\u30f3\u30de\u5206\u5e03\"<\/div>\n
\n
\u30ac\u30f3\u30de\u5206\u5e03<\/div>\n
\u3000\u30ac\u30f3\u30de\u5206\u5e03\uff08gamma distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u30ac\u30f3\u30de\u5206\u5e03\u306f\u300c\u3042\u308b\u671f\u9593\\(f\\)\u306e\u9593\u306b\\(1\\)\u56de\u8d77\u3053\u308b\u4e8b\u8c61\u304c\u3001\\\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n


\u3000\u540c\u69d8\u306b\u3057\u3066\u7279\u6027\u95a2\u6570\u3082\u6c42\u3081\u307e\u3059\u3002\u7279\u6027\u95a2\u6570\u306e\u5b9a\u7fa9\u304b\u3089

\\begin{align}
\\phi_{X}(t) &= \\mathrm{E}[\\exp[itX]] \\\\
&= \\int_{0}^{\\infty} e^{itx}\\frac{ 1 }{ 2^{\\frac{1}{2}n}\\Gamma\\left( \\frac{n}{2} \\right) }e^{-\\frac{x}{2}}x^{\\frac{n}{2}-1}dx\\\\
&= \\int_{0}^{\\infty} \\frac{ 1 }{ 2^{\\frac{1}{2}n}\\Gamma\\left( \\frac{n}{2} \\right) }e^{\\left( \\frac{1-2it}{2} \\right)x}x^{\\frac{n}{2}-1}dx\\\\
&= (1-2it)^{-\\frac{n}{2}} \\int_{0}^{\\infty} \\frac{ 1 }{ \\left( \\frac{2}{1-2it} \\right)^{\\frac{1}{2}n}\\Gamma\\left( \\frac{n}{2} \\right) }e^{\\left( \\frac{1-2it}{2} \\right)x}x^{\\frac{n}{2}-1}dx\\\\
&=(1-2it)^{-\\frac{n}{2}}
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3061\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\u306f\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u3092\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u30ab\u30a4\uff12\u4e57\u5206\u5e03\u3067\u7a4d\u7387\u6bcd\u95a2\u6570 … <\/p>\n","protected":false},"author":1,"featured_media":2760,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[76,84],"tags":[118,80,79],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/08\/\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff08\u30ab\u30a42\u4e57\u5206\u5e03\uff09.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2788"}],"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=2788"}],"version-history":[{"count":14,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2788\/revisions"}],"predecessor-version":[{"id":2805,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2788\/revisions\/2805"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2760"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2788"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2788"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2788"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}