{"id":1767,"date":"2020-07-09T11:00:00","date_gmt":"2020-07-09T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1767"},"modified":"2020-08-11T11:11:55","modified_gmt":"2020-08-11T02:11:55","slug":"dbd-mv","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/dbd-mv","title":{"rendered":"\uff12\u9805\u5206\u5e03\uff08\u4e8c\u9805\u5206\u5e03\uff09\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011"},"content":{"rendered":"\n<p class=\"has-text-align-right\"><span style=\"text-decoration: underline;\">\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<p>\u3053\u306e\u8a18\u4e8b\u3067\u306f\uff12\u9805\u5206\u5e03\uff08\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\uff12\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/binomial-distribution\">\uff1c\uff12\u9805\u5206\u5e03\uff1e<\/a>\u306e\u8a18\u4e8b\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<!-- \u6a2a\u9577 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1927349886186426\" data-ad-slot=\"8656035513\" data-ad-format=\"horizontal\" data-full-width-responsive=\"false\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2>\uff12\u9805\u5206\u5e03\uff08\u4e8c\u9805\u5206\u5e03\uff09\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563<\/h2>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u671f\u5f85\u5024\u3068\u5206\u6563<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\uff12\u9805\u5206\u5e03\\(X\\sim Bin(n,\\ p)\\)\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\n<div style=\"overflow-x: auto;\">\\begin{align}\n\\mathrm{E}[X]=np,\\ \\ \\ \\mathrm{Var}[X]=np(1-p)\n\\end{align}<\/div>\n<\/div>\n\n\n\n<p>\u671f\u5f85\u5024\u30fb\u5206\u6563\u3092\u6c42\u3081\u308b\u969b\u306b\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mean-definition\">\uff1c\u671f\u5f85\u5024\u306e\u5b9a\u7fa9\uff1e<\/a>\u304a\u3088\u3073<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/variance\">\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<p>\u3000<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h4>\u8a3c\u660e<\/h4>\n\n\n\n<p>\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<br><div style=\"overflow-x: auto;\">\\begin{align}<br>f(x)=_{n}C_{x}p^{x}(1-p)^{n-x}<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/binomial-distribution\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/binomial-distribution\">\uff1c\uff12\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\uff1e<\/a><\/a>\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<br>\u3000\u307e\u305a\u3001\u671f\u5f85\u5024\u3092\u6c42\u3081\u3066\u3044\u304d\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mathrm{E}[X] &amp;= \\sum_{x=0}^{n}xf(x) \\\\<br>&amp;= \\sum_{x=0}^{n}x\\cdot\\ _{n}C_{x}p^{x}(1-p)^{n-x} \\\\<br>&amp;= \\sum_{x=1}^{n}x\\cdot\\ \\frac{n!}{(n-x)!x!}p^{x}(1-p)^{n-x} \\\\<br>&amp;=\\sum_{x=1}^{n} \\frac{n!}{(n-x)!(x-1)!}p^{x}(1-p)^{n-x} \\\\<br>&amp;= np\\sum_{x=1}^{n} \\frac{(n-1)!}{(n-x)!(x-1)!}p^{x-1}(1-p)^{n-x} \\\\<br>&amp;= np\\sum_{k=0}^{n-1} \\frac{(n-1)!}{\\{(n-1)-k\\}!k!}p^{k}(1-p)^{(n-1)-k} \\\\<br>&amp;= np\\sum_{k=0}^{n-1}\\  _{n-1}C_{k}p^{k}(1-p)^{(n-1)-k} \\\\<br>&amp;= np<br>\\end{align}<\/div><br>\u304c\u6210\u7acb\u3057\u307e\u3059\u3002\u305f\u3060\u3057\\(k=x-1\\)\u3067\u3059\u3002\u3053\u3053\u3067\u3001\u6700\u5f8c\u306e\u884c\u306e\u8a08\u7b97\u306f\u4e8c\u9805\u95a2\u4fc2\u3092\u7528\u3044\u307e\u3059\u3002\u898b\u65b9\u306b\u3088\u3063\u3066\u306f\u78ba\u7387\\(p\\)\u306e\u8a66\u884c\u56de\u6570\\(n-1\\)\u306e\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u3057\u305f\uff12\u9805\u5206\u5e03\u306e\u78ba\u7387\u95a2\u6570\u306e\u7dcf\u548c\u306b\u306a\u3063\u3066\u3044\u308b\u306e\u3067\\(1\\)\u3068\u3059\u308b\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002<br>\u3000\u3042\u3068\u306f\u5206\u6563\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/variance\">\uff1c\u5206\u6563\u306e\u5b9a\u7fa9\uff1e<\/a>\u306e\u8a18\u4e8b\u304b\u3089\u5206\u6563\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mathrm{Var}[X] &amp;= \\mathrm{E}[X^{2}]-\\mathrm{E}[X]^{2}<br>\\end{align}<\/div><br>\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<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mathrm{E}[X^{2}] &amp;= \\sum_{x=0}^{n}x^{2}f(x) \\\\<br>&amp;= \\sum_{x=0}^{n}x^{2}\\cdot\\ _{n}C_{x}p^{x}(1-p)^{n-x} \\\\<br>&amp;= \\sum_{x=1}^{n}x^{2}\\cdot\\ \\frac{n!}{(n-x)!x!}p^{x}(1-p)^{n-x} \\\\<br>&amp;= \\sum_{x=1}^{n}\\{x(x-1)+x\\}\\cdot\\ \\frac{n!}{(n-x)!x!}p^{x}(1-p)^{n-x} \\\\<br>&amp;= \\sum_{x=1}^{n}\\{x(x-1)+x\\}\\cdot\\ \\frac{n!}{(n-x)!x!}p^{x}(1-p)^{n-x} \\\\<br>&amp;= \\sum_{x=1}^{n}x(x-1)\\cdot\\ \\frac{n!}{(n-x)!x!}p^{x}(1-p)^{n-x}+\\sum_{x=1}^{n}x\\cdot\\ \\frac{n!}{(n-x)!x!}p^{x}(1-p)^{n-x} \\\\<br>&amp;= \\sum_{x=1}^{n}\\cdot\\ \\frac{n!}{(n-x)!(x-2)!}p^{x}(1-p)^{n-x}+\\mathrm{E}[X] \\\\<br>&amp;= n(n-1)p^{2}\\sum_{k=0}^{n-2}\\cdot\\ \\frac{(n-2)!}{\\{(n-2)-k\\}!k!}p^{k}(1-p)^{n-x}+np \\\\<br>&amp;= n(n-1)p^{2}+np<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002\u305f\u3060\u3057\\(k=x-2\\)\u3067\u3059\u3002\u3053\u308c\u3089\u304b\u3089\u3001\uff12\u9805\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\u5206\u6563\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mathrm{Var}[X] = np(1-p)<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"has-text-align-right\">\u25a1<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\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\uff12\u9805\u5206\u5e03\uff08\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 &#8230; <\/p>\n","protected":false},"author":1,"featured_media":1749,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[74,72],"tags":[99,31,73],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\uff12\u9805\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570-p0.4\uff09-1.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1767"}],"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=1767"}],"version-history":[{"count":26,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1767\/revisions"}],"predecessor-version":[{"id":1875,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1767\/revisions\/1875"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/1749"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=1767"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=1767"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=1767"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}