{"id":2020,"date":"2020-07-20T11:00:00","date_gmt":"2020-07-20T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2020"},"modified":"2020-08-11T11:14:32","modified_gmt":"2020-08-11T02:14:32","slug":"dmuld-mv","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/dmuld-mv","title":{"rendered":"\u591a\u9805\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\u306f\u591a\u9805\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\u591a\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u306f\uff1c\u591a\u9805\u5206\u5e03\uff1e<\/a>\u306e\u8a18\u4e8b\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n\n

\n \n
\"\u591a\u9805\u5206\u5e03\"<\/div>\n
\n
\u591a\u9805\u5206\u5e03<\/div>\n
\u591a\u9805\u5206\u5e03\uff08multinomial distribution\uff09\u306f\u96e2\u6563\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\\(1\\)\u56de\u306e\u8a66\u884c\u3067\\(k\\)\u901a\u308a\u306e\u30d1\u30bf\u30fc\u30f3\\(A_{1},\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n

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

\u591a\u9805\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563<\/h2>\n\n\n\n
\u671f\u5f85\u5024\u3068\u5206\u6563<\/strong><\/div>\n
\\(1\\)\u56de\u306e\u8a66\u884c\u3067\\(k\\)\u901a\u308a\u306e\u30d1\u30bf\u30fc\u30f3\\(A_{1},\\cdots,A_{k}\\)\u304c\u5f97\u3089\u308c\u308b\u53ef\u80fd\u6027\u304c\u3042\u308a\u3001\u305d\u308c\u305e\u308c\u306e\u30d1\u30bf\u30fc\u30f3\u304c\u5f97\u3089\u308c\u308b\u78ba\u7387\u3092\\(p_{1},\\cdots,p_{k}\\)\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\\(n\\)\u56de\u306e\u8a66\u884c\u306b\u3088\u308b\u30d1\u30bf\u30fc\u30f3\\(A_{i}\\)\u304c\u5f97\u305f\u308c\u308b\u56de\u6570\\(X_{i}\\)\uff08\u591a\u9805\u5206\u5e03\uff09\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_{i}]=np_{i},\\ \\ \\ \\mathrm{Var}[X_{i}]=np_{i}(1-p_{i})\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<\/h4>\n\n\n\n

\\(X_{1},\\cdots,X_{k}\\)\u306e\u540c\u6642\u78ba\u7387\u95a2\u6570\u306f

\\begin{align}
f(x_{1},\\cdots,x_{k})= n!\\displaystyle\\prod_{i=1}^{k}\\left( \\frac{p_{i}^{x_{i}}}{x_{i}!} \\right)
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f<\/a><\/a><\/a>\uff1c\u591a\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\uff1e<\/a>\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002
\u3000\u307e\u305a\u3001\u671f\u5f85\u5024\u3092\u6c42\u3081\u3066\u3044\u304d\u307e\u3059\u3002\u671f\u5f85\u5024\u306e\u5b9a\u7fa9\u304b\u3089
\\begin{align}
\\mathrm{E}[X_{i}] &= \\sum_{j=1}^{k}x_{i}f(x_{1},\\cdots,x_{k}) \\\\
&= \\sum_{j=1}^{k}x_{i}n!\\displaystyle\\prod_{i=1}^{k}\\left( \\frac{p_{i}^{x_{i}}}{x_{i}!} \\right) \\\\
&= \\sum_{j=1}^{k}x_{i}n! \\frac{p_{1}^{x_{1}}p_{2}^{x_{2}}\\cdots p_{k}^{x_{k}}}{x_{1}!x_{2}!\\cdots x_{k}!} \\\\
&= np_{i}\\sum_{j=1}^{k}x_{i}(n-1)! \\frac{p_{1}^{x_{1}}p_{2}^{x_{2}}\\cdots p_{i}^{(x_{i}-1)}\\cdots p_{k}^{x_{k}}}{x_{1}!x_{2}!\\cdots(x_{i}-1)!\\cdots x_{k}!} \\\\
&= np_{i}
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067\u6700\u5f8c\u306e\u884c\u306e\u5909\u5f62\u306f\u3001\u7dcf\u548c\u306e\u4e2d\u306b\u3064\u3044\u3066\u898b\u3066\u307f\u308b\u3068\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\u304c\u7570\u306a\u308b\u591a\u9805\u5206\u5e03\u306e\u78ba\u7387\u95a2\u6570\u306b\u306a\u3063\u3066\u3044\u308b\u306e\u3067\u3001\u78ba\u7387\u95a2\u6570\u306e\u7dcf\u548c\u304c\\(1\\)\u3068\u306a\u308b\u3053\u3068\u3092\u7528\u3044\u307e\u3057\u305f\u3002\u3053\u306e\u3053\u3068\u306f\u5206\u6563\u3092\u6c42\u3081\u308b\u3068\u304d\u306b\u3082\u4f7f\u7528\u3057\u307e\u3059\u3002
\u3000\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_{i}^{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_{i}^{2}] &= \\sum_{j=1}^{k}x_{i}^{2}f(x_{1},\\cdots,x_{k}) \\\\
&= \\sum_{j=1}^{k}\\{x_{i}(x_{i}-1)+x_{i}\\}n!\\displaystyle\\prod_{i=1}^{k}\\left( \\frac{p_{i}^{x_{i}}}{x_{i}!} \\right) \\\\
&= \\sum_{j=1}^{k}x_{i}(x_{i}-1)n! \\frac{p_{1}^{x_{1}}p_{2}^{x_{2}}\\cdots p_{k}^{x_{k}}}{x_{1}!x_{2}!\\cdots x_{k}!}+ \\sum_{j=1}^{k}x_{i}n!\\displaystyle\\prod_{i=1}^{k}\\left( \\frac{p_{i}^{x_{i}}}{x_{i}!} \\right) \\\\
&= \\sum_{j=1}^{k}x_{i}(x_{i}-1)n! \\frac{p_{1}^{x_{1}}p_{2}^{x_{2}}\\cdots p_{k}^{x_{k}}}{x_{1}!x_{2}!\\cdots x_{k}!}+ np_{i} \\\\
&= n(n-1)p_{i}^{2} \\sum_{j=1}^{k}(n-2)! \\frac{p_{1}^{x_{1}}p_{2}^{x_{2}}\\cdots p_{i}^{(x_{i}-2)}\\cdots p_{k}^{x_{k}}}{x_{1}!x_{2}!\\cdots (x_{i}-2)!\\cdots x_{k}!} +np_{i}\\\\
&= n(n-1)p_{i}^{2} + np_{i}
\\end{align}<\/div>
\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u6c42\u3081\u305f\u3044\u5206\u6563\u3092\u5c0e\u304f\u3053\u3068\u304c\u3067\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\u2606\u2606\u2606 \u3053\u306e\u8a18\u4e8b\u3067\u306f\u591a\u9805\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 … <\/p>\n","protected":false},"author":1,"featured_media":801,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[74,72],"tags":[31,106,73],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/05\/PAK39_toumeinadice_TP_V.jpg","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2020"}],"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=2020"}],"version-history":[{"count":14,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2020\/revisions"}],"predecessor-version":[{"id":2035,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2020\/revisions\/2035"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/801"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2020"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2020"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2020"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}