{"id":2580,"date":"2020-08-13T11:00:00","date_gmt":"2020-08-13T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=2580"},"modified":"2020-07-29T17:41:12","modified_gmt":"2020-07-29T08:41:12","slug":"cnormd-mc","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cnormd-mc","title":{"rendered":"\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"},"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\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\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

\u6b63\u898f\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570<\/h2>\n\n\n
\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570<\/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\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002
\\begin{align} M_{X}(t)=\\exp\\left[ \\mu t+\\frac{1}{2}\\sigma^{2}t^{2} \\right],\\ \\ \\ \\phi_{X}(t)=\\exp\\left[ i\\mu t-\\frac{1}{2}\\sigma^{2}t^{2} \\right] \\end{align}<\/div><\/div>\n\n\n

\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u3092\u6c42\u3081\u308b\u969b\u306b\u306f\uff1c\u7a4d\u7387\u6bcd\u95a2\u6570\u306e\u5b9a\u7fa9\uff1e<\/a>\u304a\u3088\u3073\uff1c\u7279\u6027\u95a2\u6570\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\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\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u6c42\u3081\u3066\u3044\u304d\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u306e\u5b9a\u7fa9\u304b\u3089\u76f4\u63a5\u8a08\u7b97\u3059\u308b\u3068\u3001
\\begin{align}
M_{X}(t) &= \\mathrm{E}\\left[ \\exp[tX] \\right] \\\\
&= \\int_{-\\infty}^{\\infty} \\exp[tx] \\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2\\sigma^{2}}(x-\\mu)^{2}\\right]\u00a0 dx \\\\
&= \\int_{-\\infty}^{\\infty} \\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2\\sigma^{2}}(x-\\mu)^{2}+tx\\right]\u00a0 dx \\\\
&= \\int_{-\\infty}^{\\infty} \\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2\\sigma^{2}}(x^{2}-2\\mu x+\\mu^{2}-2\\sigma^{2}tx)\\right]\u00a0 dx \\\\
&= \\int_{-\\infty}^{\\infty} \\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2\\sigma^{2}}\\left\\{ (x-(\\mu+\\sigma^{2}t))^{2}-2\\mu\\sigma^{2} t-\\sigma^{4}t^{2} \\right\\}\\right]\u00a0 dx \\\\
&= \\exp\\left[ \\mu t+\\frac{1}{2}\\sigma^{2}t^{2} \\right]\\int_{-\\infty}^{\\infty} \\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2\\sigma^{2}}\\left\\{ x-(\\mu+\\sigma^{2}t)\\right\\}^{2} \\right]\u00a0 dx \\\\
&= \\exp\\left[ \\mu t+\\frac{1}{2}\\sigma^{2}t^{2} \\right]
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u6700\u5f8c\u306e\u5f0f\u5909\u5f62\u306f
\\begin{align}
\\int_{-\\infty}^{\\infty} \\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2\\sigma^{2}}\\left\\{ x-(\\mu+\\sigma^{2}t)\\right\\}^{2} \\right]\u00a0 dx
\\end{align}<\/div>
\u306b\u3064\u3044\u3066\u898b\u3066\u307f\u308b\u3068\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\mu+\\sigma^{2}t,\\ \\sigma^{2}\\)\u306e\u6b63\u898f\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306e\u7a4d\u5206\u306b\u306a\u3063\u3066\u3044\u308b\u306e\u3067\u3001\\(1\\)\u3068\u306a\u308a\u307e\u3059\u3002
\u3000\u540c\u69d8\u306b\u3057\u3066\u7279\u6027\u95a2\u6570\u3092\u6c42\u3081\u307e\u3059\u3002\u7279\u6027\u95a2\u6570\u306e\u5b9a\u7fa9\u304b\u3089
\\begin{align}
\\phi_{X}(t) &= \\mathrm{E}\\left[ \\exp[itX] \\right] \\\\
&= \\int_{-\\infty}^{\\infty} \\exp[itx] \\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2\\sigma^{2}}(x-\\mu)^{2}\\right]\u00a0 dx \\\\
&= \\int_{-\\infty}^{\\infty} \\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2\\sigma^{2}}(x-\\mu)^{2}+itx\\right]\u00a0 dx \\\\
&= \\int_{-\\infty}^{\\infty} \\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2\\sigma^{2}}(x^{2}-2\\mu x+\\mu^{2}-2i\\sigma^{2}tx)\\right]\u00a0 dx \\\\
&= \\int_{-\\infty}^{\\infty} \\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2\\sigma^{2}}\\left\\{ (x-(\\mu+i\\sigma^{2}t))^{2}-2i\\mu\\sigma^{2} t+\\sigma^{4}t^{2} \\right\\}\\right]\u00a0 dx \\\\
&= \\exp\\left[ i\\mu t-\\frac{1}{2}\\sigma^{2}t^{2} \\right]\\int_{-\\infty}^{\\infty} \\frac{ 1 }{ \\sqrt{2\\pi}\\sigma }\\exp\\left[ -\\frac{1}{2\\sigma^{2}}\\left\\{ x-(\\mu+i\\sigma^{2}t)\\right\\}^{2} \\right]\u00a0 dx \\\\
&= \\exp\\left[ i\\mu t-\\frac{1}{2}\\sigma^{2}t^{2} \\right]
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3061\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\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 … <\/p>\n","protected":false},"author":1,"featured_media":2513,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[76,84],"tags":[115,80,79],"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\u03bc0.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2580"}],"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=2580"}],"version-history":[{"count":15,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2580\/revisions"}],"predecessor-version":[{"id":2595,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/2580\/revisions\/2595"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/2513"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=2580"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=2580"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=2580"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}