{"id":2597,"date":"2020-08-14T11:00:00","date_gmt":"2020-08-14T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2597"},"modified":"2020-07-29T20:28:46","modified_gmt":"2020-07-29T11:28:46","slug":"cnormd-sk","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cnormd-sk","title":{"rendered":"\u6b63\u898f\u5206\u5e03\u306e\u6b6a\u5ea6\u30fb\u5c16\u5ea6\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

\u3000\u3053\u306e\u8a18\u4e8b\u3067\u306f\u6b63\u898f\u5206\u5e03\u306e\u6b6a\u5ea6\u30fb\u5c16\u5ea6\u3092\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u6b6a\u5ea6\u30fb\u5c16\u5ea6\u306e\u6c42\u3081\u65b9\u304c\u5206\u304b\u3089\u306a\u3044\u65b9\u306f\u662f\u975e\u304a\u8aad\u307f\u304f\u3060\u3055\u3044\u3002\u6b6a\u5ea6\u30fb\u5c16\u5ea6\u304c\u308f\u304b\u3089\u306a\u3044\u65b9\u306f\uff1c\u6b6a\u5ea6\uff1e<\/a>\u30fb\uff1c\u5c16\u5ea6\uff1e<\/a>\u306e\u8a18\u4e8b\u3092\u3001\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
\"\u6b6a\u5ea6\u3068\u306f\u3010\u5b9a\u7fa9\u3068\u610f\u5473\u3092\u5fb9\u5e95\u89e3\u8aac\u3011\"<\/div>\n
\n
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\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2606\u2606\u2606\u2606 \u78ba\u7387\u5206\u5e03\u306e\u7279\u5fb4\u4ed8\u3051\u3068\u3057\u3066\u671f\u5f85\u5024\u3001\u5206\u6563\u306f\u3088\u304f\u4f7f\u7528\u3055\u308c\u308b\u6307\u6a19\u3067\u3059\u304c\u3001\uff12\u3064\u306b\u52a0\u3048\u3066\u6b6a\u5ea6\u30fb\u5c16\u5ea6\u304c\u4f7f\u7528\u3055\u308c\u308b\u3053\u3068\u304c\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n\n
\n \n
\"\u5c16\u5ea6\u3068\u306f\u3010\u5b9a\u7fa9\u3068\u610f\u5473\u3092\u5fb9\u5e95\u89e3\u8aac\u3011\"<\/div>\n
\n
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\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2606\u2606\u2606\u2606 \u78ba\u7387\u5206\u5e03\u306e\u7279\u5fb4\u4ed8\u3051\u3068\u3057\u3066\u671f\u5f85\u5024\u3001\u5206\u6563\u306f\u3088\u304f\u4f7f\u7528\u3055\u308c\u308b\u6307\u6a19\u3067\u3059\u304c\u3001\uff12\u3064\u306b\u52a0\u3048\u3066\u6b6a\u5ea6\u30fb\u5c16\u5ea6\u304c\u4f7f\u7528\u3055\u308c\u308b\u3053\u3068\u304c\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\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

\u6b63\u898f\u5206\u5e03\u306e\u6b6a\u5ea6\u30fb\u5c16\u5ea6<\/h2>\n\n\n
\u6b6a\u5ea6\u30fb\u5c16\u5ea6<\/strong><\/div>\n
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\u8a3c\u660e<\/h4>\n\n\n\n

\u3000\u6b63\u898f\u5206\u5e03\u306e\u6027\u8cea<\/a>\u304b\u3089\u3001\u6b6a\u5ea6\u30fb\u5c16\u5ea6\u306e\u5b9a\u7fa9\u3088\u308a\u671f\u5f85\u5024\u306b\u3088\u3063\u3066\u6b6a\u5ea6\u30fb\u5c16\u5ea6\u306e\u5024\u306f\u5909\u5316\u3057\u307e\u305b\u3093\u3002
\u203b\u3000\u3053\u308c\u306f\u78ba\u7387\u5909\u6570\\(X\\)\u3092\u671f\u5f85\u5024\u3067\u5f15\u3044\u305f\\(X-\\mu\\)\u306b\u3064\u3044\u3066\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u3068\u308b\u306e\u3067\u3001\u3069\u3093\u306a\u671f\u5f85\u3055\u308c\u308b\u3092\u6301\u3063\u3066\u3044\u305f\u3068\u3057\u3066\u3082\u3001\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u6c42\u3081\u308b\u969b\u306b\u306f\u671f\u5f85\u5024\u304c\u5fc5\u305a\\(0\\)\u306b\u306a\u308b\u304b\u3089\u3067\u3059\uff08\u6b63\u898f\u5206\u5e03\u306e\u7dda\u5f62\u95a2\u6570\u3088\u308a\u3053\u306e\u3053\u3068\u306f\u5206\u304b\u308a\u307e\u3059\uff09\u3002<\/p>\n\n\n\n

\u3088\u3063\u3066\u3001\u7c21\u5358\u5316\u306e\u305f\u3081\u671f\u5f85\u5024\u304c\\(0\\)\u3067\u3042\u308b\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim N(0,\\sigma^{2})\\)\u306b\u3064\u3044\u3066\u8003\u3048\u307e\u3059\u3002\u6b6a\u5ea6\u30fb\u5c16\u5ea6\u3092\u6c42\u3081\u308b\u969b\u306b\u306f\\(3\\)\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8\\(\\mathrm{E}[(X-\\mu)^{3}]\\)\u3068\\(4\\)\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8\\(\\mathrm{E}[(X-\\mu)^{4}]\\)\u306e\u8a08\u7b97\u304c\u5fc5\u8981\u306b\u306a\u308a\u307e\u3059\u3002\u5b9a\u7fa9\u901a\u308a\u8a08\u7b97\u3059\u308b\u3068\u9762\u5012\u306a\u306e\u3067\u3001\u7a4d\u7387\u6bcd\u95a2\u6570\u304b\u3089\u6c42\u3081\u3066\u3044\u304d\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u306b\u3064\u3044\u3066\u5206\u304b\u3089\u306a\u3044\u65b9\u306f\u4ee5\u4e0b\u306e\u30ea\u30f3\u30af\u304b\u3089\u304a\u9858\u3044\u3057\u307e\u3059\u3002<\/p>\n\n\n\n\n

\n \n
\"\u30e2\u30fc\u30e1\u30f3\u30c8\u3068\u306f\u3010\u5b9a\u7fa9\u304b\u3089\u89e3\u8aac\u3011\"<\/div>\n
\n
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\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u9ad8\u6821\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u5e73\u5747\u3084\u5206\u6563\u306a\u3069\u3092\u4e00\u822c\u5316\u3057\u305f\u3082\u306e\u304c\u30e2\u30fc\u30e1\u30f3\u30c8\uff08\u7a4d\u7387\uff09\u3067\u3059\u3002\u30e2\u30fc\u30e1\u30f3\u30c8\u306b\u306f[\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8]\u3068[\u5e73\u5747\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n

\u671f\u5f85\u5024\\(0\\)\u306e\u6b63\u898f\u5206\u5e03\\(N(0,\\sigma^{2})\\)\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u306f\\(M_{X}(t)=\\exp\\left[ \\frac{1}{2}\\sigma^{2}t^{2} \\right]\\)\u3068\u306a\u308a\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u304c\u3053\u306e\u3088\u3046\u306b\u306a\u308b\u3053\u3068\u306f\u4ee5\u4e0b\u306e\u30ea\u30f3\u30af\u304b\u3089\u304a\u9858\u3044\u3057\u307e\u3059\u3002<\/p>\n\n\n\n\n

\n \n
\"\u6b63\u898f\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3011\"<\/div>\n
\n
\u6b63\u898f\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3011<\/div>\n
\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u3000\u3053\u306e\u8a18\u4e8b\u3067\u306f\u6b63\u898f\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\u2026<\/div>\n <\/div>\n
<\/div>\n <\/a>\n <\/div>\n\n\n\n

\u3053\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u306e\u7b2c3,4\u6b21\u5c0e\u95a2\u6570\u3092\u6c42\u3081\u3066\u3044\u304f\u3068

\\begin{align}
M_{X}^{(1)}(t) &= \\sigma^{2} t\\exp\\left[ \\frac{1}{2}\\sigma^{2}t^{2} \\right], \\\\
M_{X}^{(2)}(t) &= \\sigma^{2} \\exp\\left[ \\frac{1}{2}\\sigma^{2}t^{2} \\right] +\\sigma^{4} t^{2}\\exp\\left[ \\frac{1}{2}\\sigma^{2}t^{2} \\right] \\\\
&= ( 1+\\sigma^{2} t^{2} )\\sigma^{2}\\exp\\left[ \\frac{1}{2}\\sigma^{2}t^{2} \\right],\\\\
M_{X}^{(3)}(t) &= 2\\sigma^{4} t\\exp\\left[ \\frac{1}{2}\\sigma^{2}t^{2} \\right]+( 1+\\sigma^{2} t^{2} )\\sigma^{4}t\\exp\\left[ \\frac{1}{2}\\sigma^{2}t^{2} \\right]\\\\
&= (3+\\sigma^{2} t^{2})\\sigma^{4} t\\exp\\left[ \\frac{1}{2}\\sigma^{2}t^{2} \\right],\\\\
M_{X}^{(4)}(t) &= (3\\sigma^{4}+3\\sigma^{6} t^{2}) \\exp\\left[ \\frac{1}{2}\\sigma^{2}t^{2} \\right]+(3+\\sigma^{2} t^{2})\\sigma^{6} t^{2}\\exp\\left[ \\frac{1}{2}\\sigma^{2}t^{2} \\right],
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u5b9f\u969b\u3001\\(1,2\\)\u6b21\u5c0e\u95a2\u6570\u3092\u7528\u3044\u308b\u3053\u3068\u3067\u671f\u5f85\u5024\u3001\u5206\u6563\u304c
\\begin{align}
\\mathrm{E}[X] &= \\left. M_{X}^{(1)}(t) \\right|_{t=0} = 0\\\\
\\mathrm{E}[X^{2}] &= \\mathrm{Var}[X] = \\left. M_{X}^{(2)}(t) \\right|_{t=0} = \\sigma^{2}
\\end{align}<\/div>
\u3068\u306a\u308b\u3053\u3068\u3082\u78ba\u8a8d\u3067\u304d\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u304b\u30893,4\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u6c42\u3081\u308b\u3068\u3001
\\begin{align}
\\mathrm{E}[X^{3}] &= \\left. M_{X}^{(3)}(t) \\right|_{t=0} = 0\\\\
\\mathrm{E}[X^{4}] &= \\left. M_{X}^{(4)}(t) \\right|_{t=0} = 3\\sigma^{4}
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u304b\u3089\u3001\u6c42\u3081\u305f\u3044\u6b6a\u5ea6\\(skew(X)\\)\u3068\u5c16\u5ea6\\(kurt(X)\\)\u306f
\\begin{align}
skew(X) &= \\frac{\\mathrm{E}[X^{3}]}{\\sigma^{3}} = 0 \\\\
kurt(X) &= \\frac{\\mathrm{E}[X^{4}]}{\\sigma^{4}} = 3
\\end{align}<\/div>
\u3068\u306a\u308a\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 \u3000\u3053\u306e\u8a18\u4e8b\u3067\u306f\u6b63\u898f\u5206\u5e03\u306e\u6b6a\u5ea6\u30fb\u5c16\u5ea6\u3092\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u6b6a\u5ea6\u30fb\u5c16\u5ea6\u306e\u6c42\u3081\u65b9\u304c\u5206\u304b\u3089\u306a\u3044\u65b9\u306f\u662f … <\/p>\n","protected":false},"author":1,"featured_media":2514,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[84],"tags":[86,115,85],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u6b63\u898f\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09\u03c3\uff1d1.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2597"}],"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=2597"}],"version-history":[{"count":13,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2597\/revisions"}],"predecessor-version":[{"id":2611,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2597\/revisions\/2611"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2514"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2597"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2597"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2597"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}