{"id":1631,"date":"2020-07-02T11:00:00","date_gmt":"2020-07-02T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1631"},"modified":"2020-08-11T11:10:05","modified_gmt":"2020-08-11T02:10:05","slug":"cud-mv","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cud-mv","title":{"rendered":"\u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\uff09\u306e\u671f\u5f85\u5024\u3068\u5206\u6563\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\u2606\u2606\u2606\u2606<\/span><\/p>\n\n\n\n\n\n\n\n

\u3053\u306e\u8a18\u4e8b\u3067\u306f\u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\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\u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\uff09\u306e\u57fa\u672c\u60c5\u5831\u306f\uff1c\u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\uff09\uff1e<\/a>\u306e\u8a18\u4e8b\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\uff09\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563<\/h3>\n\n\n\n
\u671f\u5f85\u5024\u3068\u5206\u6563<\/strong><\/div>\n
\u4e00\u69d8\u5206\u5e03\\(X\\sim U(\\alpha,\\ \\beta)\\)\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
\\begin{align}\n\\mathrm{E}[X]=\\frac{\\alpha+\\beta}{2},\\ \\ \\ \\mathrm{Var}[X]=\\frac{(\\beta-\\alpha)^{2}}{12}\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

\u78ba\u7387\u5909\u6570\u304c\\(X\\sim U(\\alpha,\\ \\beta)\\)\u306b\u5f93\u3063\u3066\u3044\u308b\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\\(X\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f

\\begin{align}
f(x)=\\left\\{
\\begin{array}{ccc}
& \\displaystyle\\frac{1}{\\beta-\\alpha} &\uff08\\alpha \\leq x \\leq \\beta\uff09\\\\
& 0 &\uff08\u305d\u306e\u4ed6\uff09
\\end{array}\\right.
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f\uff1c\u4e00\u69d8\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
\\begin{align}
\\mathrm{E}[X] &= \\int_{-\\infty}^{\\infty}xf(x)dx \\\\
&= \\int_{-\\infty}^{\\alpha}xf(x)dx+\\int_{\\alpha}^{\\beta}xf(x)dx+\\int_{\\beta}^{\\infty}xf(x)dx \\\\
&= \\int_{\\alpha}^{\\beta}\\frac{x}{\\beta-\\alpha}dx \\\\
&= \\left[ \\frac{x^{2}}{2(\\beta-\\alpha)} \\right]_{\\alpha}^{\\beta} \\\\
&= \\frac{\\beta^{2}-\\alpha^{2}}{2(\\beta-\\alpha)} \\\\
&=\\frac{\\beta+\\alpha}{2}
\\end{align}<\/div>
\u304c\u6210\u7acb\u3057\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}] &= \\int_{-\\infty}^{\\infty}x^{2}f(x)dx \\\\
&= \\int_{-\\infty}^{\\alpha}x^{2}f(x)dx+\\int_{\\alpha}^{\\beta}x^{2}f(x)dx+\\int_{\\beta}^{\\infty}x^{2}f(x)dx \\\\
&= \\int_{\\alpha}^{\\beta}\\frac{x^{2}}{\\beta-\\alpha}dx \\\\
&= \\left[ \\frac{x^{3}}{3(\\beta-\\alpha)} \\right]_{\\alpha}^{\\beta} \\\\
&= \\frac{\\beta^{3}-\\alpha^{3}}{3(\\beta-\\alpha)} \\\\
&=\\frac{\\alpha^{2}+\\alpha\\beta+\\beta^{2}}{3}
\\end{align}<\/div>
\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3001\u4e00\u69d8\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\u5206\u6563\u306f
\\begin{align}
\\mathrm{Var}[X] = \\frac{(\\beta-\\alpha)^{2}}{12}
\\end{align}<\/div>
\u304c\u6210\u7acb\u3057\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\u2606\u2606\u2606\u2606 \u3053\u306e\u8a18\u4e8b\u3067\u306f\u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\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 … <\/p>\n","protected":false},"author":1,"featured_media":1604,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[74,72],"tags":[96,31,73],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\u4e00\u69d8\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1631"}],"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=1631"}],"version-history":[{"count":11,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1631\/revisions"}],"predecessor-version":[{"id":1643,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1631\/revisions\/1643"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/1604"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=1631"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=1631"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=1631"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}