{"id":1852,"date":"2020-07-10T11:00:00","date_gmt":"2020-07-10T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1852"},"modified":"2020-08-11T11:12:12","modified_gmt":"2020-08-11T02:12:12","slug":"dbd-mc","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/dbd-mc","title":{"rendered":"\uff12\u9805\u5206\u5e03\uff08\u4e8c\u9805\u5206\u5e03\uff09\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\uff12\u9805\u5206\u5e03\uff08\u4e8c\u9805\u5206\u5e03\uff09\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\uff12\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u306f\uff1c\uff12\u9805\u5206\u5e03\uff1e<\/a>\u306e\u8a18\u4e8b\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\uff12\u9805\u5206\u5e03\uff08\u4e8c\u9805\u5206\u5e03\uff09\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570<\/h3>\n\n\n\n
\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570<\/strong><\/div>\n
\u4e8c\u9805\u5206\u5e03\\(X\\sim Bin(n,\\ p)\\)\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)=\\left\\{ pe^{t}+(1-p) \\right\\}^{n},\\ \\ \\ \\phi_{X}(t)=\\left\\{ pe^{it}+(1-p) \\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

\u78ba\u7387\u5909\u6570\u304c\\(X\\sim Bin(n,\\ p)\\)\u306b\u5f93\u3063\u3066\u3044\u308b\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\\(X\\)\u306e\u78ba\u7387\u95a2\u6570\u306f

\\begin{align}
f(x)= _{n}C_{x}p^{x}(1-p)^{n-x}
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f\uff1c\uff12\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\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]] \\\\
&= \\sum_{x=0}^{n} e^{tx}\\ _{n}C_{x}p^{x}(1-p)^{n-x} \\\\
&= \\sum_{x=0}^{n}\\ _{n}C_{x}(pe^{t})^{x}(1-p)^{n-x} \\\\
&= \\left\\{ pe^{t}+(1-p) \\right\\}^{n} \\\\
\\end{align}<\/div>
\u304c\u6210\u7acb\u3057\u307e\u3059\u3002\u6700\u5f8c\u306e\u5f0f\u5909\u5f62\u306f\u4e8c\u9805\u95a2\u4fc2\u3092\u7528\u3044\u307e\u3059\u3002\u4e8c\u9805\u95a2\u4fc2\u3068\u306f
\\begin{align}
(p+q)^{n} = \\sum_{k=0}^{n}\\ _{n}C_{k}p^{k}q^{n-k}
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3064\u3068\u3044\u3046\u95a2\u4fc2\u3067\u3059\u3002
\u3053\u306e\u95a2\u4fc2\u306f\u7279\u6027\u95a2\u6570\u306e\u5834\u5408\u306b\u3082\u7528\u3044\u307e\u3059\u3002\u7279\u6027\u95a2\u6570\u306e\u5b9a\u7fa9\u304b\u3089
\\begin{align}
\\phi_{X}(t) &= \\mathrm{E}[\\exp[itX]] \\\\
&= \\sum_{x=0}^{n} e^{itx}\\ _{n}C_{x}p^{x}(1-p)^{n-x} \\\\
&= \\sum_{x=0}^{n}\\ _{n}C_{x}(pe^{it})^{x}(1-p)^{n-x} \\\\
&= \\left\\{ pe^{it}+(1-p) \\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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