{"id":1974,"date":"2020-07-17T11:00:00","date_gmt":"2020-07-17T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1974"},"modified":"2020-08-11T11:13:38","modified_gmt":"2020-08-11T02:13:38","slug":"nbiod-mv","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/nbiod-mv","title":{"rendered":"\u8ca0\u306e\uff12\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\u8ca0\u306e\uff12\u9805\u5206\u5e03\uff08\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff09\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\u8ca0\u306e\uff12\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u306f\uff1c\u8ca0\u306e\uff12\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
\"\u8ca0\u306e\uff12\u9805\u5206\u5e03\uff08\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff09\"<\/div>\n
\n
\u8ca0\u306e\uff12\u9805\u5206\u5e03\uff08\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff09<\/div>\n
\u8ca0\u306e\uff12\u9805\u5206\u5e03\uff08negative binomial distribution\uff09\u306f\u96e2\u6563\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3001\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\\(k\\)\u56de\u6210\u529f\u3059\u308b\u307e\u3067\u8a66\u884c\u3092\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n

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

\u8ca0\u306e\uff12\u9805\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563<\/h2>\n\n\n\n
\u671f\u5f85\u5024\u3068\u5206\u6563<\/strong><\/div>\n
\u8ca0\u306e\uff12\u9805\u5206\u5e03\uff08\\(p\\):\u6210\u529f\u78ba\u7387\u3001\\(k\\)\uff1a\u6210\u529f\u56de\u6570\uff09\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\)\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]=\\frac{k(1-p)}{p},\\ \\ \\ \\mathrm{Var}[X]=\\frac{ k(1-p) }{ p^{2} }\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

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

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

\\(X\\)\u306e\u78ba\u7387\u95a2\u6570\u306f

\\begin{align}
f(x)= _{k+x-1}C_{k-1}p^{k}(1-p)^{x}
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f<\/a><\/a><\/a>\uff1c\u8ca0\u306e\uff12\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] &= \\sum_{x=0}^{\\infty}xf(x) \\\\
&= \\sum_{x=1}^{\\infty}x\\cdot\\ _{k+x-1}C_{k-1}p^{k}(1-p)^{x}\\\\
&= \\sum_{x=1}^{n}x\\cdot\\ \\frac{(k+x-1)!}{(k-1)!x!}p^{k}(1-p)^{x} \\\\
&= kp^{k}(1-p) \\sum_{x=0}^{n}\\ \\frac{(k+x)!}{k!x!}(1-p)^{x} \\\\
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067\u3001
\\begin{align}
\\frac{1}{(1-x)^{k+1}}
\\end{align}<\/div>
\u306b\u3064\u3044\u3066\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3092\u884c\u3044\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)=1\/\\{(1-x)^{k+1}\\}\\)\u306b\u3064\u3044\u3066\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3092\u8003\u3048\u308b\u3068\u304d\u3001
\\begin{align}
f^{(1)}(0) = k+1\\ \\ f^{(2)}(0) = (k+1)(k+2),\\ \\ \\cdots\\ \\ ,f^{(n)}(0) = (k+1)(k+2)\\cdots(k+n)
\\end{align}<\/div>
\u3068\u306a\u308b\u3053\u3068\u3092\u7528\u3044\u308b\u3068\u3001
\\begin{align}
\\frac{1}{(1-x)^{k+1}} &= \\sum_{n=0}^{\\infty}\\ _{n+k}C_{n}x^{n}
\\end{align}<\/div>
\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\u3001\\(x=n,\\ \\ (1-p)=x\\)\u3067\u66f8\u304d\u63db\u3048\u308b\u3068
\\begin{align}
\\sum_{x=0}^{\\infty}\\ _{x+k}C_{x}(1-p)^{x} &= \\frac{1}{\\{1-(1-p)\\}^{k+1}} \\\\
&= p^{-k-1}
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3088\u3063\u3066\u6c42\u3081\u305f\u3044\u671f\u5f85\u5024\u306f
\\begin{align}
\\mathrm{E}[X] = kp^{k}(1-p)p^{-k-1} = \\frac{k(1-p)}{p}
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002
\u3000\u3042\u3068\u306f\u5206\u6563\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002\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^{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^{2}] &= \\sum_{x=0}^{\\infty}x^{2}f(x) \\\\
&= \\sum_{x=0}^{\\infty}x^{2}\\cdot\\ _{k+x-1}C_{k-1}p^{k}(1-p)^{x}\\\\
&= \\sum_{x=1}^{\\infty}\\{x(x-1)+x\\}\\cdot\\ \\frac{(k+x-1)!}{(k-1)!x!}p^{k}(1-p)^{x} \\\\
&= \\sum_{x=2}^{\\infty}x(x-1)\\cdot\\ \\frac{(k+x-1)!}{(k-1)!x!}p^{k}(1-p)^{x} + \\frac{k(1-p)}{p}\\\\
&= p^{k}(1-p)^{2}(k+1)k\\sum_{x=0}^{\\infty}\\frac{(k+x+1)!}{x!(k+1)!}(1-p)^{x} + \\frac{k(1-p)}{p}\\\\
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u5148\u307b\u3069\u306e\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u306e\u7d50\u679c\u3092\u7528\u3044\u308b\u3068
\\begin{align}
\\sum_{x=0}^{\\infty}\\ _{x+k+1}C_{x}(1-p)^{x} &= \\frac{1}{\\{1-(1-p)\\}^{k+2}} \\\\
&= p^{-k-2}
\\end{align}<\/div>
\u3068\u306a\u308b\u306e\u3067
\\begin{align}
\\mathrm{E}[X^{2}] &= p^{k}(1-p)^{2}(k+1)kp^{-k-2} + \\frac{k(1-p)}{p}\\\\
&= (k+1)k\\frac{(1-p)^{2}}{p^{2}} + \\frac{k(1-p)}{p}
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u3092\u7528\u3044\u308c\u3070\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\u8ca0\u306e\uff12\u9805\u5206\u5e03\uff08\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff09\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 … <\/p>\n","protected":false},"author":1,"featured_media":1960,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[74,72],"tags":[31,73,105],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff08p0.4\uff09.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1974"}],"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=1974"}],"version-history":[{"count":14,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1974\/revisions"}],"predecessor-version":[{"id":1995,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1974\/revisions\/1995"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/1960"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=1974"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=1974"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=1974"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}