{"id":2557,"date":"2020-08-12T11:00:00","date_gmt":"2020-08-12T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2557"},"modified":"2020-08-11T11:20:29","modified_gmt":"2020-08-11T02:20:29","slug":"cnormd-mv1","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cnormd-mv1","title":{"rendered":"\u6b63\u898f\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\u6b63\u898f\u5206\u5e03\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\u6b63\u898f\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u306f\uff1c\u6b63\u898f\u5206\u5e03\uff1e<\/a>\u306e\u8a18\u4e8b\u3092\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n\n

\n \n
\"\u6b63\u898f\u5206\u5e03\uff08\u30ac\u30a6\u30b9\u5206\u5e03\uff09\"<\/div>\n
\n
\u6b63\u898f\u5206\u5e03\uff08\u30ac\u30a6\u30b9\u5206\u5e03\uff09<\/div>\n
\u3000\u6b63\u898f\u5206\u5e03\uff08\u30ac\u30a6\u30b9\u5206\u5e03\u3001normal distribution\uff09\u306f\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u6b63\u898f\u5206\u5e03\u306f\u6570\u5b66\u7684\u306b\u3082\u4fbf\u5229\u306a\u6027\u8cea\u3092\u305f\u304f\u3055\u3093\u6301\u3063\u3066\u3044\u308b\u305f\u3081\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n

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

\u6b63\u898f\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563<\/h2>\n\n\n\n
\u671f\u5f85\u5024\u3068\u5206\u6563<\/strong><\/div>\n
\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\mu,\\sigma^{2}\\)\u306e\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim N(\\mu,\\sigma^{2})\\)\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]=\\mu,\\ \\ \\ \\mathrm{Var}[X]=\\sigma^{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

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

\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\mu,\\sigma^{2}\\)\u306e\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f

\\begin{align}
f(x)= \\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2\\sigma^{2}}(x-\\mu)^{2} \\right]
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f<\/a><\/a><\/a><\/a>\uff1c\u6b63\u898f\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] &= \\int_{-\\infty}^{\\infty}xf(x) dx\\\\
&= \\int_{-\\infty}^{\\infty}\\{(x-\\mu)+\\mu\\}f(x) dx\\\\
&= \\int_{-\\infty}^{\\infty}(x-\\mu)\\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2\\sigma^{2}}(x-\\mu)^{2} \\right] dx+\\int_{-\\infty}^{\\infty}\\mu f(x) dx\\\\
&= \\int_{-\\infty}^{\\infty}\\sigma\\left(\\frac{x-\\mu}{\\sigma}\\right)\\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^{2} \\right] dx+\\mu
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067\u3001\u5909\u6570\u5909\u63db\\(y=\\frac{x-\\mu}{\\sigma}\\)\u3068\u3059\u308b\u3068\u3001\\(dx=\\sigma dy\\)\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3001
\\begin{align}
\\mathrm{E}[X] &= \\int_{-\\infty}^{\\infty}\\sigma y\\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2}y^{2} \\right] \\sigma dy+\\mu \\\\
&= \\frac{ \\sigma }{ \\sqrt{2\\pi} }\\int_{-\\infty}^{\\infty} y\\exp\\left[ -\\frac{1}{2}y^{2} \\right] dy+\\mu \\\\
&= -\\frac{ \\sigma }{ \\sqrt{2\\pi} }\\int_{-\\infty}^{\\infty} \\left( -\\frac{1}{2}y^{2} \\right)^{\\prime}\\exp\\left[ -\\frac{1}{2}y^{2} \\right] dy+\\mu \\\\
&= -\\frac{ \\sigma }{ \\sqrt{2\\pi} } \\left[ \\exp\\left[ -\\frac{1}{2}y^{2} \\right] \\right]_{-\\infty}^{\\infty}+\\mu \\\\
&=\\mu
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u8a08\u7b97\u904e\u7a0b\u306f\u9577\u3044\u3067\u3059\u304c\u3001\u7f6e\u63db\u7a4d\u5206\u3092\u884c\u3046\u3060\u3051\u3067\u3059\u306e\u3067\u3001\u6bd4\u8f03\u7684\u5bb9\u6613\u306b\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u8a08\u7b97\u65b9\u6cd5\u306f\u899a\u3048\u3066\u304a\u304f\u3068\u3044\u3044\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002
\u3000\u6b21\u306b\u5206\u6563\u306b\u3064\u3044\u3066\u89e3\u3044\u3066\u3044\u304d\u307e\u3059\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}^{\\infty}\\{(x-\\mu)^{2}+2\\mu x-\\mu^{2}\\}f(x) dx\\\\
&= \\int_{-\\infty}^{\\infty}(x-\\mu)^{2}f(x)dx+2\\mu \\int_{-\\infty}^{\\infty}xf(x)dx-\\mu^{2}\\int_{-\\infty}^{\\infty}f(x) dx\\\\
&= \\int_{-\\infty}^{\\infty}(x-\\mu)^{2}\\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2\\sigma^{2}}(x-\\mu)^{2}\\right]dx+2\\mu \\mathrm{E}[X]-\\mu^{2}\\\\
&= \\frac{ \\sigma }{ \\sqrt{2\\pi} }\\int_{-\\infty}^{\\infty}\\left(\\frac{x-\\mu}{\\sigma}\\right)^{2}\\exp\\left[ -\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^{2}\\right]dx+2\\mu^{2}-\\mu^{2}\\\\
&= \\frac{ \\sigma^{2} }{ \\sqrt{2\\pi} }\\int_{-\\infty}^{\\infty}y^{2}\\exp\\left[ -\\frac{y^{2}}{2}\\right]dy+\\mu^{2}\\\\
&= \\frac{ \\sigma^{2} }{ \\sqrt{2\\pi} }\\int_{-\\infty}^{\\infty}-y\\left(\\exp\\left[ -\\frac{y^{2}}{2}\\right]\\right)^{\\prime}dy+\\mu^{2}\\\\
&= \\frac{ \\sigma^{2} }{ \\sqrt{2\\pi} }\\left\\{ \\left[ -y\\exp\\left[ -\\frac{y^{2}}{2} \\right] \\right]_{-\\infty}^{\\infty}+\\int_{-\\infty}^{\\infty}\\exp\\left[ -\\frac{y^{2}}{2} \\right]dy \\right\\}+\\mu^{2}\\\\
&= \\frac{ \\sigma^{2} }{ \\sqrt{2\\pi} }\\int_{-\\infty}^{\\infty}\\exp\\left[ -\\frac{y^{2}}{2} \\right]dy +\\mu^{2}\\\\
&= \\sigma^{2}+\\mu^{2}\\\\
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u305f\u3060\u3057\u3001\u6700\u5f8c\u306e\u5f0f\u5909\u5f62\u306f\u30ac\u30a6\u30b9\u7a4d\u5206
\\begin{align}
\\int_{-\\infty}^{\\infty}\\exp\\left[ -ax^{2} \\right]dx &= \\sqrt{\\frac{\\pi}{a}}
\\end{align}<\/div>
\u3092\u7528\u3044\u307e\u3057\u305f\u3002\u3053\u306e\u3053\u3068\u304b\u3089\u5206\u6563\u306f
\\begin{align}
\\mathrm{Var}[X] &= \\sigma^{2}+\\mu^{2}-\\mu^{2} \\\\
&= \\sigma^{2}
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u25a1<\/p>\n\n\n\n

\u3000\u3053\u3053\u307e\u3067\u3001\u8a08\u7b97\u3057\u3066\u3044\u3066\u3001\u3068\u3066\u3082\u9762\u5012\u3060\u3068\u611f\u3058\u307e\u3059\u3088\u306d\u3002
\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u307b\u304b\u306b\u6b6a\u5ea6\u30fb\u5c16\u5ea6\u3082\u6c42\u3081\u3088\u3046\u3068\u8003\u3048\u305f\u3068\u304d\u3001\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u4f7f\u3063\u3066\u6c42\u3081\u305f\u65b9\u304c\u697d\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u3089\u306b\u95a2\u3057\u3066\u306f\u3001\u4e0b\u306e\u30ea\u30f3\u30af\u304b\u3089\u304a\u9858\u3044\u3057\u307e\u3059\u3002<\/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\u6b63\u898f\u5206\u5e03\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 … <\/p>\n","protected":false},"author":1,"featured_media":2512,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[74,72],"tags":[31,73,115],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09\u03c31.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2557"}],"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=2557"}],"version-history":[{"count":18,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2557\/revisions"}],"predecessor-version":[{"id":2577,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2557\/revisions\/2577"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2512"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2557"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2557"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2557"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}