{"id":1882,"date":"2020-07-13T11:00:00","date_gmt":"2020-07-13T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1882"},"modified":"2020-08-11T11:13:02","modified_gmt":"2020-08-11T02:13:02","slug":"dgeod-mc","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/dgeod-mc","title":{"rendered":"\u5e7e\u4f55\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\u5e7e\u4f55\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\u5e7e\u4f55\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u306f\uff1c\u5e7e\u4f55\u5206\u5e03\uff1e<\/a>\u306e\u8a18\u4e8b\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\u5e7e\u4f55\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
\u5e7e\u4f55\u5206\u5e03\\(X\\sim Geo(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)=\\frac{pe^{t}}{1-(1-p)e^{t}},\\ \\ \\ \\phi_{X}(t)=\\frac{pe^{it}}{1-(1-p)e^{it}}\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 Geo(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)= p(1-p)^{x-1}
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f\uff1c\u5e7e\u4f55\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}^{\\infty} e^{tx}p(1-p)^{x-1} \\\\
&= pe^{t}\\sum_{x=0}^{\\infty} \\left\\{ e^{t}(1-p) \\right\\}^{x-1} \\\\
&= pe^{t}\\cdot\\frac{ 1-\\left\\{e^{t}(1-p)\\right\\}^{\\infty} }{ 1-e^{t}(1-p)} \\\\
&= \\frac{pe^{t}}{1-e^{t}(1-p)}
\\end{align}<\/div>
\u304c\u6210\u7acb\u3057\u307e\u3059\u3002\u5f0f\u5909\u5f62\u306b\u306f\u7b49\u6bd4\u6570\u5217\u306e\u548c\u306e\u516c\u5f0f\u3092\u7528\u3044\u307e\u3057\u305f\u3002
\u3000\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}^{\\infty} e^{itx}p(1-p)^{x-1} \\\\
&= pe^{it}\\sum_{x=0}^{\\infty} \\left\\{ e^{it}(1-p) \\right\\}^{x-1} \\\\
&= pe^{it}\\cdot\\frac{ 1-\\left\\{e^{it}(1-p)\\right\\}^{\\infty} }{ 1-e^{it}(1-p)} \\\\
&= \\frac{pe^{it}}{1-e^{it}(1-p)}
\\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\u5e7e\u4f55\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 … <\/p>\n","protected":false},"author":1,"featured_media":1868,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[76],"tags":[101,80,79],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u5e7e\u4f55\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\/1882"}],"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=1882"}],"version-history":[{"count":8,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1882\/revisions"}],"predecessor-version":[{"id":1893,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1882\/revisions\/1893"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/1868"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=1882"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=1882"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=1882"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}