{"id":2041,"date":"2020-07-21T11:00:00","date_gmt":"2020-07-21T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2041"},"modified":"2020-08-11T11:14:53","modified_gmt":"2020-08-11T02:14:53","slug":"dmuld-mc","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/dmuld-mc","title":{"rendered":"\u591a\u9805\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9"},"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\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\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\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

\u591a\u9805\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570<\/h2>\n\n\n\n

\u591a\u9805\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u3092\u5c0e\u304f\u524d\u306b\u6b21\u306e\u88dc\u984c\u3092\u793a\u3057\u307e\u3059\u3002\u88dc\u984c\u306b\u306f\u30ab\u30c6\u30b4\u30ea\u5206\u5e03\u3092\u4f7f\u7528\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u30ab\u30c6\u30b4\u30ea\u5206\u5e03\u3068\u591a\u9805\u5206\u5e03<\/strong><\/div>\n
\u3000\\(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\\(\\mathrm{P}(A_{i})=p_{i}\\)\u3068\u3057\u307e\u3059\u3002<\/br>\n\u3000\u78ba\u7387\u5909\u6570\\(Y_{1},\\cdots,Y_{n}\\)\u3092\u4e92\u3044\u306b\u72ec\u7acb\u306b\u30ab\u30c6\u30b4\u30ea\u5206\u5e03\u306b\u5f93\u3046\u3068\u3057\u307e\u3059\uff08\u305d\u308c\u305e\u308c\u306e\u78ba\u7387\u5909\u6570\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u306f\\(p_{1},\\cdots,p_{k}\\)\u3067\u3059\uff09\u3002<\/br>\n\u3000\u78ba\u7387\u5909\u6570\\(X\\)\u3092\u30d1\u30e9\u30e1\u30fc\u30bf\\(p_{1},\\cdots,p_{k}\\)\u306e\u8a66\u884c\u56de\u6570\\(n\\)\u56de\u306e\u591a\u9805\u5206\u5e03\u306b\u5f93\u3063\u3066\u3044\u308b\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\n
\\begin{align}\nX = Y_{1}+Y_{2}+\\cdots +Y_{n}\n\\end{align}<\/div>\n\u304c\u6210\u7acb\u3057\u307e\u3059\u3002\n<\/div>\n\n\n\n

\u3000\u3053\u306e\u88dc\u984c\u306e\u8a3c\u660e\u306f\u7701\u7565\u3057\u307e\u3059\u304c\u3001\u78ba\u7387\u95a2\u6570\u3092\u4f7f\u3063\u3066\\(\\mathrm{P}(Y_{1}+Y_{2}+\\cdots +Y_{n})\\)\u3092\u8a08\u7b97\u3059\u308c\u3070\u78ba\u8a8d\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570<\/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\u591a\u9805\u5206\u5e03\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)=\\left( \\sum_{i=1}^{k}p_{i}\\exp[t_{i}]\\right)^{n},\\ \\ \\ \\phi_{X}(t)=\\left( \\sum_{i=1}^{k}p_{i}\\exp[it_{i}]\\right)^{n}\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

\u307e\u305a\u4e0a\u306e\u88dc\u984c\u304b\u3089\u30d1\u30e9\u30e1\u30fc\u30bf\\(p_{1},\\cdots,p_{k}\\)\u306b\u5f93\u3046\u4e92\u3044\u306b\u72ec\u7acb\u306b\u30ab\u30c6\u30b4\u30ea\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(Y_{1},\\cdots,Y_{n}\\)\u7528\u3044\u3066

\\begin{align}
X=Y_{1}+\\cdots+Y_{n}
\\end{align}<\/div>
\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u30ab\u30c6\u30b4\u30ea\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(Y_{i}\\)\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\\(M_{Y_{i}}(t)\\)\u306f
\\begin{align}
M_{Y_{i}}(t) &= \\sum_{j=1}^{k}p_{j}\\exp[t_{j}]
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u305f\u3060\u3057\\(t_{j}\\)\u306f\u30d9\u30af\u30c8\u30eb\\(t\\)\u306e\\(j\\)\u756a\u76ee\u306e\u5024\u3067\u3059\u3002\u3053\u306e\u3053\u3068\u306f\uff1c\u30ab\u30c6\u30b4\u30ea\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\uff1e<\/a>\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002
\u3000\u3053\u306e\u3053\u3068\u3092\u4f7f\u3063\u3066\u3001\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u6c42\u3081\u3066\u3044\u304d\u307e\u3059\u3002
\\begin{align}
M_{X}(t) &= \\mathrm{E}[\\exp[tX]] \\\\
&= \\mathrm{E}[\\exp[t(Y_{1}+\\cdots+Y_{n})]] \\\\
&= \\prod_{\\ell=1}^{n}\\mathrm{E}[\\exp[tY_{\\ell}]] \\\\
&= \\prod_{\\ell=1}^{n} \\sum_{j=1}^{k}p_{j}\\exp[t_{j}] \\\\
&= \\left( \\sum_{j=1}^{k}p_{j}\\exp[t_{j}] \\right)^{n}
\\end{align}<\/div>
\u304c\u6210\u7acb\u3057\u307e\u3059\u3002
\u00a0\u540c\u69d8\u306b\u3057\u3066\u7279\u6027\u95a2\u6570\u3082\u6c42\u3081\u307e\u3059\u3002\u30ab\u30c6\u30b4\u30ea\u5206\u5e03\u306e\u7279\u6027\u95a2\u6570\u3092\u7528\u3044\u3066
\\begin{align}
\\phi_{X}(t) &= \\mathrm{E}[\\exp[itX]] \\\\
&= \\mathrm{E}[\\exp[it(Y_{1}+\\cdots+Y_{n})]] \\\\
&= \\prod_{\\ell=1}^{n}\\mathrm{E}[\\exp[itY_{\\ell}]] \\\\
&= \\prod_{\\ell=1}^{n} \\sum_{j=1}^{k}p_{j}\\exp[it_{j}] \\\\
&= \\left( \\sum_{j=1}^{k}p_{j}\\exp[it_{j}] \\right)^{n}
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n\n\n\n

\u25a1<\/p>\n","protected":false},"excerpt":{"rendered":"

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