{"id":2112,"date":"2020-07-25T11:00:00","date_gmt":"2020-07-25T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2112"},"modified":"2020-08-11T11:15:37","modified_gmt":"2020-08-11T02:15:37","slug":"dpoid-mc","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/dpoid-mc","title":{"rendered":"\u30dd\u30a2\u30bd\u30f3\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\u30dd\u30a2\u30bd\u30f3\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\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u306f\uff1c\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff1e<\/a>\u306e\u8a18\u4e8b\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n\n

\n \n
\"\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\"<\/div>\n
\n
\u30dd\u30a2\u30bd\u30f3\u5206\u5e03<\/div>\n
\u3000\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08Poisson distribution\uff09\u306f\u96e2\u6563\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306f\u4e00\u5b9a\u9593\u9694\u306e\u6642\u9593\u3067\u767a\u751f\u3059\u308b\u4e8b\u8c61\u306e\u6570\u3092\u8868\u3057\u307e\u3059\u3002\u3082\u3046\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n

\u30dd\u30a2\u30bd\u30f3\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
\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\lambda\\)\u306e\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim Po(\\lambda)\\)\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\n
\\begin{align}\nM_{X}(t)=\\exp\\left[ \\lambda\\left\\{ \\exp[t]-1 \\right\\} \\right],\\ \\ \\ \\phi_{X}(t)=\\exp\\left[ \\lambda\\left\\{ \\exp[it]-1 \\right\\} \\right]\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

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

\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\lambda\\)\u306e\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u78ba\u7387\u95a2\u6570\u306f

\\begin{align}
f(x)= \\frac{\\lambda^{x}e^{-\\lambda}}{x!}
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f\uff1c\u30dd\u30a2\u30bd\u30f3\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\u7a4d\u7387\u6bcd\u95a2\u6570\u306e\u5b9a\u7fa9\u304b\u3089
\\begin{align}
M_{X}(t) &= \\mathrm{E}\\left[ \\exp[Xt] \\right] \\\\
&= \\sum_{x=0}^{\\infty} e^{xt} \\frac{\\lambda^{x}e^{-\\lambda}}{x!} \\\\
&= e^{-\\lambda}\\sum_{x=0}^{\\infty} \\frac{(\\lambda e^{t})^{x}}{x!} \\\\
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067\u3001\\(e^{x}\\)\u306b\u3064\u3044\u3066\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3092\u3057\u3066\u307f\u307e\u3059\u3002\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u306f\u95a2\u6570\\(f(x)\\)\u306b\u5bfe\u3057\u3066
\\begin{align}
f(x) = \\sum_{k=0}^{\\infty} \\frac{f^{(k)}(0)}{k!}x^{k}
\\end{align}<\/div>
\u3068\u3044\u3046\u5f62\u306b\u5909\u5f62\u3059\u308b\u3053\u3068\u3067\u3059\u3002\u3053\u3053\u3067\\(f^{(i)}(x)\\)\u306f\\(f(x)\\)\u3092\\(i\\)\u56de\u5fae\u5206\u3057\u305f\u3082\u306e\u3092\u8868\u3057\u307e\u3059\u3002\\(f(x)=e^{x}\\)\u306b\u3064\u3044\u3066\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3092\u8003\u3048\u308b\u3068\u304d\u3001
\\begin{align}
f^{(1)}(0) = 1,\\ \\ f^{(2)}(0) = 1,\\ \\ f^{(3)} = 1,\\ \\ \\cdots\\ \\ ,f^{(n)}(0) = 1
\\end{align}<\/div>
\u3068\u306a\u308b\u3053\u3068\u3092\u7528\u3044\u308b\u3068\u3001
\\begin{align}
e^{x} &= \\sum_{k=0}^{\\infty} \\frac{1}{k!}x^{k} = \\sum_{k=0}^{\\infty} \\frac{x^{k}}{k!}
\\end{align}<\/div>
\u304c\u6210\u7acb\u3057\u307e\u3059\u3002\u3044\u307e\\(x=\\lambda e^{t}\\)\u3068\u3059\u308b\u3068
\\begin{align}
e^{\\lambda e^{t}} &= \\sum_{k=0}^{\\infty} \\frac{(\\lambda e^{t})^{k}}{k!}
\\end{align}<\/div>
\u3068\u306a\u308b\u306e\u3067
\\begin{align}
M_{X}(t) &= e^{-\\lambda}\\cdot e^{\\lambda e^{t}} \\\\
&= \\exp\\left[ \\lambda\\left\\{ \\exp[t]-1 \\right\\} \\right]
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002
\u3000\u540c\u69d8\u306b\u3057\u3066\u7279\u6027\u95a2\u6570\u3092\u6c42\u3081\u307e\u3059\u3002\u7279\u6027\u95a2\u6570\u306e\u5b9a\u7fa9\u304b\u3089
\\begin{align}
\\phi_{X}(t) &= \\mathrm{E}\\left[ \\exp[iXt] \\right] \\\\
&= \\sum_{x=0}^{\\infty} e^{ixt} \\frac{\\lambda^{x}e^{-\\lambda}}{x!} \\\\
&= e^{-\\lambda}\\sum_{x=0}^{\\infty} \\frac{(\\lambda e^{it})^{x}}{x!} \\\\
&= e^{-\\lambda}\\cdot e^{\\lambda e^{it}} \\\\
&= \\exp\\left[ \\lambda\\left\\{ \\exp[it]-1 \\right\\} \\right]
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u9014\u4e2d\u306e\u5f0f\u5909\u5f62\u3067\u306f\\(e^{\\lambda e^{it}}\\)\u306b\u3064\u3044\u3066\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b
\\begin{align}
e^{\\lambda e^{it}} &= \\sum_{k=0}^{\\infty} \\frac{(\\lambda e^{it})^{k}}{k!}
\\end{align}<\/div>
\u306e\u7d50\u679c\u3092\u7528\u3044\u307e\u3057\u305f\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\u30dd\u30a2\u30bd\u30f3\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 … <\/p>\n","protected":false},"author":1,"featured_media":2095,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[76],"tags":[108,80,79],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2112"}],"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=2112"}],"version-history":[{"count":7,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2112\/revisions"}],"predecessor-version":[{"id":2122,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2112\/revisions\/2122"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2095"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2112"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2112"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2112"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}