{"id":8685,"date":"2022-03-12T20:07:40","date_gmt":"2022-03-12T11:07:40","guid":{"rendered":"https:\/\/pandanote.info\/?p=8685"},"modified":"2022-08-07T13:06:34","modified_gmt":"2022-08-07T04:06:34","slug":"%e5%af%be%e6%95%b0%e6%ad%a3%e8%a6%8f%e5%88%86%e5%b8%83%e3%81%ae%e7%a2%ba%e7%8e%87%e5%af%86%e5%ba%a6%e9%96%a2%e6%95%b0%e3%82%92%e5%b0%8e%e5%87%ba%e3%81%97%e3%81%a6%e3%80%81%e3%81%a4%e3%81%84%e3%81%a7","status":"publish","type":"post","link":"https:\/\/pandanote.info\/?p=8685","title":{"rendered":"\u5bfe\u6570\u6b63\u898f\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u5c0e\u51fa\u3057\u3066\u3001\u3064\u3044\u3067\u306b\u671f\u5f85\u5024\u53ca\u3073\u5206\u6563\u3092\u8a08\u7b97\u3057\u3066\u307f\u305f\u3002"},"content":{"rendered":"

\u306f\u3058\u3081\u306b<\/h2>\n

\u5b9f\u9a13\u30c7\u30fc\u30bf\u3092\u306f\u3058\u3081\u3068\u3059\u308b\u5b9f\u4e16\u754c\u3092\u5bfe\u8c61\u3068\u3057\u305f\u30c7\u30fc\u30bf\u7fa4\u3092\u7d71\u8a08\u7684\u306b\u6271\u3046\u3068\u304d\u306b\u304b\u306a\u308a\u9ad8\u3044\u78ba\u7387\u3067\u767b\u5834\u3057\u3001\u304b\u3064\u4f55\u3068\u304b\u3059\u308b\u5fc5\u8981\u306b\u8feb\u3089\u308c\u308b\u306e\u304c\u3001\u300c\u5916\u308c\u5024\u300d\u3067\u3059\u3002<\/p>\n

\u4f8b\u3048\u3070\u3001\u5f97\u3089\u308c\u305f\u30c7\u30fc\u30bf\u304c $\\{ a_n \\} = \\{ 600,500,16000,1000,400,700,550,800,1200,900 \\}$\u306e\u3088\u3046\u306a\u3082\u306e\u3067\u3042\u3063\u305f\u308a\u3059\u308b\u3068\u300116000\u306f\u4ed6\u306e\u30c7\u30fc\u30bf\u5024\u3068\u6bd4\u8f03\u3059\u308b\u3068\u6841\u9055\u3044\u306b\u5927\u304d\u3044\u306e\u3067\u4f55\u3068\u306a\u304f\u9664\u5916\u3057\u305f\u304f\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u3057\u304b\u3057\u3001\u4e0a\u8a18\u306e\u30c7\u30fc\u30bf\u304c\u3001<\/p>\n

\u300c\u6771\u4eac\u306e\u6e2f\u533a\u5728\u4f4f\u306e\u65b9\u3092\u30e9\u30f3\u30c0\u30e0\u306b10\u4eba\u30d4\u30c3\u30af\u30a2\u30c3\u30d7\u3057\u305f\u3068\u304d\u306e\u3001\u5404\u4eba\u306e\u5e74\u53ce(\u5358\u4f4d:\u4e07\u5186)\u3002<\/strong>\u300d<\/p>\n

\u306e\u3088\u3046\u306a\u30c7\u30fc\u30bf\u3060\u3068\u3001\u5e74\u53ce16000\u4e07\u5186(=1.6\u5104\u5186)\u306e\u65b9\u304c(\u81ea\u5206\u306e\u5468\u56f2\u306b\u306f\u3044\u306a\u304b\u3063\u305f\u3068\u3057\u3066\u3082)\u5b9f\u969b\u306b\u306f\u3044\u3089\u3063\u3057\u3083\u308b\u53ef\u80fd\u6027\u304c\u3042\u308b\u3060\u3051\u306b\u3001\u300c16000\u300d\u3068\u3044\u3046\u30c7\u30fc\u30bf\u306f\u5916\u308c\u5024\u3068\u3057\u3066\u9664\u5916\u3057\u3066\u306f\u3044\u3051\u306a\u3044\u3082\u306e\u306a\u306e\u304b\u3082\u3001\u3068\u601d\u3044\u305f\u304f\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u3053\u3046\u3044\u3046\u3068\u304d\u306b\u306f\u30c7\u30fc\u30bf\u3092\u5bfe\u6570\u306b\u3059\u308b\u3068\u3001\u3082\u3057\u304b\u3059\u308b\u3068\u6b63\u898f\u5206\u5e03\u307f\u305f\u3044\u306a\u5206\u5e03\u306b\u306a\u3063\u305f\u308a\u3059\u308b\u306e\u304b\u3082\u3057\u308c\u307e\u305b\u3093(\u203b\u500b\u4eba\u306e\u3053\u3058\u3064\u3051\u3067\u3059)\u3002<\/p>\n

\u305d\u3053\u3067\u3001\u3042\u308b\u30c7\u30fc\u30bf\u5217 $\\{ a_k\\} (1 \\le k \\le n)$ \u306e\u5404\u30c7\u30fc\u30bf\u306b\u5bfe\u3057\u3066 $b_k = \\log a_k$ \u3092\u8003\u3048\u3001\u30c7\u30fc\u30bf\u5217 $\\{ b_k\\}$ \u304c\u6b63\u898f\u5206\u5e03\u306b\u306a\u308b\u3088\u3046\u306a\u5206\u5e03\u3092\u8003\u3048\u308b\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n

\u5bfe\u6570\u6b63\u898f\u5206\u5e03<\/h2>\n

\u78ba\u7387\u5909\u6570\u306e\u5909\u6570\u5909\u63db\u306e\u78ba\u8a8d<\/h3>\n

\u524d\u7bc0\u306e\u5206\u5e03\u306b\u3082\u3059\u3067\u306b\u308c\u3063\u304d\u3068\u3057\u305f\u65e5\u672c\u8a9e\u306e\u540d\u79f0\u304c\u3064\u3051\u3089\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n

\u300c\u5bfe\u6570\u6b63\u898f\u5206\u5e03<\/strong>\u300d\u3068\u3044\u3044\u307e\u3059\u3002<\/p>\n

\u524d\u7bc0\u3067\u306f\u30c7\u30fc\u30bf\u5217\u306b\u3064\u3044\u3066\u306f\u96e2\u6563\u7684\u306b\u6271\u3044\u307e\u3057\u305f\u304c\u3001\u3053\u3053\u304b\u3089\u306f\u8a08\u7b97\u306e\u90fd\u5408\u4e0a\u3001\u9023\u7d9a\u7684\u306a\u78ba\u7387\u5206\u5e03\u306b\u3064\u3044\u3066\u6271\u3046\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n

\u5bfe\u6570\u6b63\u898f\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u3068\u308a\u3042\u3048\u305a$q(y) (y$\u3092\u78ba\u7387\u5909\u6570\u3068\u3057\u307e\u3059\u3002$)$\u3068\u304a\u304d\u307e\u3059\u3002<\/p>\n

\u5c11\u3005\u898b\u6163\u308c\u306a\u3044\u5f62\u3067\u3059\u304c\u3001\u3057\u3070\u3089\u304f\u306e\u9593\u8f9b\u62b1\u9858\u3044\u307e\u3059\u3002<\/p>\n

$q(y)$\u306f\u4ee5\u4e0b\u306e(\\ref{eq:nddensity})\u5f0f\u3067\u8868\u3055\u308c\u308b\u6b63\u898f\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570$p(x)$<\/p>\n

\\begin{align}
\np(x) &= \\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}} \\label{eq:nddensity}
\n\\end{align}<\/p>\n

\u304b\u3089\u306e\u78ba\u7387\u5909\u6570\u306e\u5909\u63db\u3092\u884c\u3046\u3053\u3068\u3067\u5c0e\u51fa\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u304c\u3001\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306e\u5c0e\u51fa\u3092\u884c\u3046\u524d\u306b\u78ba\u7387\u5909\u6570\u306e\u5909\u63db\u306e\u65b9\u6cd5\u306b\u3064\u3044\u3066\u78ba\u8a8d\u3057\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n

(\\ref{eq:nddensity})\u5f0f\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570$X$\u304c\u3042\u3063\u305f\u3068\u304d\u306b\u3001$x \\le X \\le x+dx (dx > 0)$\u3068\u306a\u308b\u78ba\u7387\u306f\u8fd1\u4f3c\u7684\u306b$|p(x)dx|$\u3068\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u540c\u69d8\u306b\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570$q(y)$\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570$Y$\u304c\u3042\u3063\u305f\u3068\u304d\u306b$y \\le Y \\le y+dy (dy > 0)$\u3068\u306a\u308b\u78ba\u7387\u3082\u540c\u69d8\u306b$|q(y)dy|$\u3068\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u3053\u3053\u3067$p(x)$\u53ca\u3073$q(y)$\u304c\u9023\u7d9a\u95a2\u6570\u3067\u3042\u308b\u3068\u3059\u308b\u3068\u3001$dx$\u53ca\u3073$dy$\u306b\u3064\u3044\u3066\u306f(\\ref{eq:pxqy})\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3088\u3046\u306b\u3068\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

\\begin{align}
\n|q(y)dy| &= |p(x)dx| \\label{eq:pxqy}
\n\\end{align}<\/p>\n

\u3055\u3089\u306b(\\ref{eq:pxqy})\u5f0f\u306e\u4e21\u8fba\u3092\u5f62\u5f0f\u7684\u306b$|dy|$\u3067\u5272\u308a\u307e\u3059\u3002<\/p>\n

\u3059\u308b\u3068\u2026<\/p>\n

\\begin{align}
\n|q(y)| &= \\left|p(x)\\frac{dx}{dy}\\right| \\label{eq:dxqy}
\n\\end{align}<\/p>\n

\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

\u5bfe\u6570\u6b63\u898f\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306e\u5c0e\u51fa<\/h3>\n

\u4e00\u65b9\u3067\u3001\u78ba\u7387\u5909\u6570$Y$\u306e\u5bfe\u6570\u3067\u3042\u308b$X$\u304c\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u3067\u3042\u308b\u305f\u3081\u3001$x$\u53ca\u3073$y$\u306e\u95a2\u4fc2\u306f(\\ref{eq:xlogy})\u5f0f\u306e\u95a2\u4fc2\u5f0f\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

\\begin{align}
\nx &= \\log y \\label{eq:xlogy}
\n\\end{align}<\/p>\n

\u305d\u3053\u3067\u3001(\\ref{eq:xlogy})\u5f0f\u306e\u4e21\u8fba\u3092$y$\u3067\u5fae\u5206\u3057\u305f\u3082\u306e\u3092(\\ref{eq:dxqy})\u5f0f\u306b\u4ee3\u5165\u3057\u3064\u3064(\\ref{eq:nddensity})\u5f0f\u3092\u7528\u3044\u308b\u3068\u3001(\\ref{eq:logndfirst})\u5f0f\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u307e\u3059\u3002<\/p>\n

\\begin{align}
\n |q(y)| &= \\left|p(\\log y)\\frac{d}{dy}(\\log y)\\right| \\nonumber \\cr
\n &= \\left|p(\\log y)\\frac{1}{y}\\right| \\nonumber \\cr
\n &= \\left|\\frac{1}{y}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(\\log y-\\mu)^2}{2\\sigma^2}}\\right| \\nonumber \\cr
\n &= \\left|\\frac{1}{\\sqrt{2\\pi\\sigma^2}y}e^{-\\frac{(\\log y-\\mu)^2}{2\\sigma^2}}\\right| \\label{eq:logndfirst}
\n\\end{align}<\/p>\n

$q(y)$\u306f\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3067\u3042\u308b\u305f\u3081$q(y) \\gt 0$\u3068\u306a\u308a\u307e\u3059\u3002\u307e\u305f\u3001(\\ref{eq:logndfirst})\u5f0f\u306e\u53f3\u8fba\u3082$\\left( -\\infty, \\infty \\right)$\u3067\u8ca0\u3067\u306a\u3044\u5024\u3092\u3068\u308b\u305f\u3081\u3001\u4e21\u8fba\u3068\u3082\u306b\u7d76\u5bfe\u5024\u8a18\u53f7\u3092\u5916\u3059\u3053\u3068\u304c\u3067\u304d\u3066\u2026<\/p>\n

\\begin{align}
\nq(y) &= \\frac{1}{\\sqrt{2\\pi\\sigma^2}y}e^{-\\frac{(\\log y-\\mu)^2}{2\\sigma^2}} \\label{eq:logndsecond}
\n\\end{align}<\/p>\n

\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u307e\u3067\u306f$q(y)$\u304c\u9023\u7d9a\u95a2\u6570\u3067\u3042\u308b\u3068\u4eee\u5b9a\u3057\u3066\u8a08\u7b97\u3057\u307e\u3057\u305f\u304c\u3001(\\ref{eq:logndsecond})\u5f0f\u3088\u308a\u6c42\u307e\u3063\u305f$q(y)$\u306f\u9023\u7d9a\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

\u306a\u304a\u3001(\\ref{eq:logndsecond})\u5f0f\u306e\u5909\u6570$y$\u53ca\u3073\u95a2\u6570\u3092\u793a\u3059\u8a18\u53f7$q$\u3068\u3044\u3046\u6587\u5b57\u81ea\u4f53\u306f\u7279\u5225\u306a\u610f\u5473\u3092\u6301\u3061\u307e\u305b\u3093\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u305d\u308c\u305e\u308c$x$\u53ca\u3073$p$\u3068\u66f8\u304d\u63db\u3048\u3066\u3082\u826f\u3044\u306e\u3067\u2026<\/p>\n

\\begin{align}
\n p(x) &= \\frac{1}{\\sqrt{2\\pi\\sigma^2}x}e^{-\\frac{(\\log x-\\mu)^2}{2\\sigma^2}} \\nonumber \\cr
\n &= \\frac{1}{x\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{(\\log x-\\mu)^2}{2\\sigma^2}\\right] \\label{eq:logndfinal}
\n\\end{align}<\/p>\n

\u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002$\\blacksquare$<\/p>\n

\u306a\u304a\u3001(\\ref{eq:logndfinal})\u5f0f\u306e$p(x)$\u306e\u5b9a\u7fa9\u57df\u306f$(0,\\infty)$\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

Wikipedia\u306e\u5bfe\u6570\u6b63\u898f\u5206\u5e03\u306e\u30da\u30fc\u30b8[1<\/a>]\u306b\u306f$\\mu=0$\u3067\u56fa\u5b9a\u3057\u3066$\\sigma$\u3092\u5909\u5316\u3055\u305b\u305f\u3068\u304d\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306e\u30b0\u30e9\u30d5\u304c\u63b2\u8f09\u3055\u308c\u3066\u3044\u307e\u3059(\u203b\u3053\u306e\u8a18\u4e8b\u3092\u6700\u521d\u306b\u66f8\u3044\u305f\u6642\u70b9(2022\u5e743\u6708)\u306e\u60c5\u5831\u3067\u3059\u3002)\u306e\u3067\u3001$\\sigma=\\displaystyle\\frac{1}{4}$\u3067\u56fa\u5b9a\u3057\u3066$\\mu$\u3092\u5909\u5316\u3055\u305b\u305f\u3068\u304d\u306e\u30b0\u30e9\u30d5\u3092Inkscape(1.1.2)\u3067\u4f5c\u3063\u3066\u307f\u307e\u3057\u305f\u2193<\/p>\n

\"\"<\/a><\/p>\n

(\u5bfe\u6570\u3067\u306a\u3044\u4e00\u822c\u7684\u306a)\u6b63\u898f\u5206\u5e03\u306e\u5834\u5408\u306b\u306f$\\mu$\u3092\u5909\u5316\u3055\u305b\u308b\u3068\u30b0\u30e9\u30d5\u306e\u5f62\u306f\u5909\u5316\u305b\u305a\u306b$x$\u8ef8\u65b9\u5411\u306b$\\mu$\u3060\u3051\u79fb\u52d5\u3057\u307e\u3059\u304c\u3001\u5bfe\u6570\u6b63\u898f\u5206\u5e03\u306e\u5834\u5408\u306b\u306f\u5c71\u306e\u9802\u304c\u79fb\u52d5\u3059\u308b\u3060\u3051\u3067\u306a\u304f\u3001\u30b0\u30e9\u30d5\u306e\u5f62\u3082\u5909\u5316\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

\u5bfe\u6570\u6b63\u898f\u5206\u5e03\u306e\u671f\u5f85\u5024<\/h2>\n

\u6b21\u306b\u3001(\\ref{eq:logndfinal})\u5f0f\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570$X$\u306e\u671f\u5f85\u5024$E[X]$\u306b\u3064\u3044\u3066\u8003\u3048\u307e\u3059\u3002<\/p>\n

\u4f8b\u306b\u3088\u3063\u3066\u3001\u30b5\u30af\u30b5\u30af\u3068\u8a08\u7b97\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n

\\begin{align}
\n E[X] &= \\int_{0}^{\\infty} xp(x)dx \\nonumber \\cr
\n &= \\int_{0}^{\\infty} x \\frac{1}{x\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{(\\log x-\\mu)^2}{2\\sigma^2}\\right]dx \\nonumber \\cr
\n &= \\int_{0}^{\\infty} \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{(\\log x-\\mu)^2}{2\\sigma^2}\\right]dx \\label{eq:expectionfirst}
\n\\end{align}<\/p>\n

\u3068\u3053\u3053\u307e\u3067\u306f\u5206\u5b50\u306b\u3042\u3063\u305f$x$\u304c\u6d88\u3048\u3066\u304f\u308c\u3066\u3044\u3044\u611f\u3058\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u304c\u3001$u = \\log x$\u3068\u7f6e\u304f\u3068\u3059\u3053\u3057\u69d8\u5b50\u304c\u5909\u308f\u3063\u3066\u304d\u307e\u3059\u3002<\/p>\n

$x$\u304c$0 \\to \\infty$\u3068\u5909\u5316\u3059\u308b\u3068$u$\u306f$-\\infty \\to \\infty$\u3068\u5909\u5316\u3059\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3064\u3064\u8a08\u7b97\u3092\u9032\u3081\u3001(\\ref{eq:expectationsecond})\u5f0f\u306e\u3088\u3046\u306b\u5909\u5f62\u3057\u307e\u3059\u3002<\/p>\n

\\begin{align}
\n E[X] &= \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{(u-\\mu)^2}{2\\sigma^2}\\right]\\frac{dx}{du}du \\nonumber \\cr
\n &= \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{(u-\\mu)^2}{2\\sigma^2}\\right]\\frac{d(e^u)}{du}du \\nonumber \\cr
\n &= \\int_{-\\infty}^{\\infty} \\frac{e^u}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{(u-\\mu)^2}{2\\sigma^2}\\right]du \\cr
\n &= \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[u-\\frac{(u-\\mu)^2}{2\\sigma^2}\\right]du \\label{eq:expectationsecond}
\n\\end{align}<\/p>\n

(\\ref{eq:expectationsecond})\u5f0f\u306e$\\exp[\\cdot]$\u306e$\\cdot$\u306e\u90e8\u5206\u306f\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u307e\u3059($u$\u3068$\\mu$\u306f\u6587\u5b57\u306e\u5f62\u304c\u4f3c\u3066\u3044\u307e\u3059\u304c\u3001\u3081\u3052\u305a\u306b\u8a08\u7b97\u3057\u307e\u3059)\u3002<\/p>\n

\\begin{align}
\n u-\\frac{(u-\\mu)^2}{2\\sigma^2} &= -\\frac{(u-\\mu)^2-2u\\sigma^2}{2\\sigma^2} \\nonumber \\cr
\n &= -\\frac{u^2-2u\\mu+\\mu^2-2u\\sigma^2}{2\\sigma^2} \\nonumber \\cr
\n &= -\\frac{u^2-2u(\\mu+\\sigma^2)+\\mu^2}{2\\sigma^2} \\nonumber \\cr
\n &= -\\frac{[u-(\\mu+\\sigma^2)]^2-(\\mu+\\sigma^2)^2+\\mu^2}{2\\sigma^2} \\nonumber \\cr
\n &= -\\frac{[u-(\\mu+\\sigma^2)]^2-\\mu^2-2\\mu\\sigma^2-\\sigma^4+\\mu^2}{2\\sigma^2} \\nonumber \\cr
\n &= -\\frac{[u-(\\mu+\\sigma^2)]^2-2\\mu\\sigma^2-\\sigma^4}{2\\sigma^2} \\nonumber \\cr
\n &= -\\frac{[u-(\\mu+\\sigma^2)]^2}{2\\sigma^2}+\\frac{2\\mu\\sigma^2+\\sigma^4}{2\\sigma^2} \\nonumber \\cr
\n &= -\\frac{[u-(\\mu+\\sigma^2)]^2}{2\\sigma^2}+\\frac{2\\mu+\\sigma^2}{2} \\label{eq:expectationthird}
\n\\end{align}<\/p>\n

(\\ref{eq:expectationthird})\u5f0f\u3092(\\ref{eq:expectationsecond})\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068$\\mu$\u53ca\u3073$\\sigma$\u306f\u5b9a\u6570\u3067\u3059\u306e\u3067\u2026<\/p>\n

\\begin{align}
\n E[X] &= \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{[u-(\\mu+\\sigma^2)]^2}{2\\sigma^2}+\\frac{2\\mu+\\sigma^2}{2}\\right]du \\nonumber \\cr
\n &= \\exp\\left(\\frac{2\\mu+\\sigma^2}{2}\\right) \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{[u-(\\mu+\\sigma^2)]^2}{2\\sigma^2}\\right]du \\label{eq:expectationfourth}
\n\\end{align}<\/p>\n

\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067(\\ref{eq:expectationfourth})\u5f0f\u306e\u53f3\u8fba\u306e\u88ab\u7a4d\u5206\u95a2\u6570\u306b\u7740\u76ee\u3059\u308b\u3068\u3001\u5e73\u5747$(\\mu+\\sigma^2)$\u3067\u5206\u6563$\\sigma^2$\u306e\u6b63\u898f\u5206\u5e03\u3092\u8868\u3057\u3066\u3044\u3066\u3001\u304b\u3064\u305d\u308c\u3092$(-\\infty, \\infty)$\u306e\u7bc4\u56f2\u3067\u7a4d\u5206\u3059\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u304c\u3001\u3053\u306e\u8a08\u7b97\u7d50\u679c\u306f1\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u3088\u3063\u3066\u3001<\/p>\n

\\begin{align}
\nE[X] &= \\exp\\left(\\frac{2\\mu+\\sigma^2}{2}\\right) \\label{eq:expectationfinal}
\n\\end{align}<\/p>\n

\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002$\\blacksquare$<\/p>\n

\u306a\u304a\u3001(\\ref{eq:expectationfinal})\u5f0f\u306e$\\mu$\u53ca\u3073$\\sigma$\u306f\u5bfe\u6570\u6b63\u898f\u5206\u5e03\u306e\u5e73\u5747\u5024\u53ca\u3073\u6a19\u6e96\u504f\u5dee\u3092\u8868\u3059\u3082\u306e\u3067\u306f\u306a\u3044\u3053\u3068\u306b\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3059\u3002<\/p>\n

\u5bfe\u6570\u6b63\u898f\u5206\u5e03\u306e\u5206\u6563<\/h2>\n

\u6b21\u306b\u3001(\\ref{eq:logndfinal})\u5f0f\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570$X$\u306e\u5206\u6563$V[X]$\u306b\u3064\u3044\u3066\u8003\u3048\u307e\u3059\u3002<\/p>\n

\u306a\u304a\u3001$V[X] = E[X^2]-(E[X])^2$\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u3066\u3001$E[X]$\u306b\u3064\u3044\u3066\u306f\u524d\u7bc0\u3067\u8a08\u7b97\u6e08\u307f\u3067\u3059\u306e\u3067\u3001$E[X^2]$\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u6ce8\u529b\u3057\u307e\u3059\u3002<\/p>\n

$E[X^2]$\u306f(\\ref{eq:somfirst})\u5f0f\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

\\begin{align}
\nE[X^2] &= \\int_{0}^{\\infty} x^2p(x)dx \\nonumber \\cr
\n &= \\int_{0}^{\\infty} x^2 \\frac{1}{x\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{(\\log x-\\mu)^2}{2\\sigma^2}\\right]dx \\nonumber \\cr
\n &= \\int_{0}^{\\infty} \\frac{x}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{(\\log x-\\mu)^2}{2\\sigma^2}\\right]dx \\label{eq:somfirst}
\n\\end{align}<\/p>\n

\u524d\u7bc0\u306e\u8b70\u8ad6\u3068\u540c\u69d8\u306b\u3001$u = \\log x$\u3068\u304a\u3044\u3066\u3001$x$\u304c$0 \\to \\infty$\u3068\u5909\u5316\u3059\u308b\u3068$u$\u306f$-\\infty \\to \\infty$\u3068\u5909\u5316\u3059\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3064\u3064\u8a08\u7b97\u3092\u9032\u3081\u307e\u3059\u3002<\/p>\n

\\begin{align}
\n E[X^2] &= \\int_{-\\infty}^{\\infty} \\frac{e^u}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{(u-\\mu)^2}{2\\sigma^2}\\right]\\frac{dx}{du}du \\nonumber \\cr
\n &= \\int_{-\\infty}^{\\infty} \\frac{e^u}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{(u-\\mu)^2}{2\\sigma^2}\\right]\\frac{d(e^u)}{du}du \\nonumber \\cr
\n &= \\int_{-\\infty}^{\\infty} \\frac{e^{2u}}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{(u-\\mu)^2}{2\\sigma^2}\\right]du \\cr
\n &= \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[2u-\\frac{(u-\\mu)^2}{2\\sigma^2}\\right]du \\label{eq:somsecond}
\n\\end{align}<\/p>\n

(\\ref{eq:somsecond})\u5f0f\u306e$\\exp[\\cdot]$\u306e$\\cdot$\u306e\u90e8\u5206\u306f\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u307e\u3059(\u671f\u5f85\u5024\u3092\u8a08\u7b97\u3057\u305f\u969b\u306e\u8a08\u7b97\u3068\u4f3c\u305f\u3088\u3046\u306a\u611f\u3058\u306e\u8a08\u7b97\u304c\u7d9a\u304d\u307e\u3059\u3002)\u3002<\/p>\n

\\begin{align}
\n 2u-\\frac{(u-\\mu)^2}{2\\sigma^2} &= -\\frac{(u-\\mu)^2-4u\\sigma^2}{2\\sigma^2} \\nonumber \\cr
\n &= -\\frac{u^2-2u\\mu+\\mu^2-4u\\sigma^2}{2\\sigma^2} \\nonumber \\cr
\n &= -\\frac{u^2-2u(\\mu+2\\sigma^2)+\\mu^2}{2\\sigma^2} \\nonumber \\cr
\n &= -\\frac{[u-(\\mu+2\\sigma^2)]^2-(\\mu+2\\sigma^2)^2+\\mu^2}{2\\sigma^2} \\nonumber \\cr
\n &= -\\frac{[u-(\\mu+2\\sigma^2)]^2-\\mu^2-4\\mu\\sigma^2-4\\sigma^4+\\mu^2}{2\\sigma^2} \\nonumber \\cr
\n &= -\\frac{[u-(\\mu+2\\sigma^2)]^2-4\\mu\\sigma^2-4\\sigma^4}{2\\sigma^2} \\nonumber \\cr
\n &= -\\frac{[u-(\\mu+2\\sigma^2)]^2}{2\\sigma^2}+\\frac{4\\mu\\sigma^2+4\\sigma^4}{2\\sigma^2} \\nonumber \\cr
\n &= -\\frac{[u-(\\mu+2\\sigma^2)]^2}{2\\sigma^2}+2\\mu+2\\sigma^2 \\label{eq:somthird}
\n\\end{align}<\/p>\n

(\\ref{eq:somthird})\u5f0f\u3092(\\ref{eq:somsecond})\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068$\\mu$\u53ca\u3073$\\sigma$\u306f\u5b9a\u6570\u3067\u3059\u306e\u3067\u2026<\/p>\n

\\begin{align}
\n E[X^2] &= \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{[u-(\\mu+2\\sigma^2)]^2}{2\\sigma^2}+2\\mu+2\\sigma^2\\right]du \\nonumber \\cr
\n &= \\exp\\left(2\\mu+2\\sigma^2\\right) \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{[u-(\\mu+2\\sigma^2)]^2}{2\\sigma^2}\\right]du \\label{eq:somfourth}
\n\\end{align}<\/p>\n

\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067(\\ref{eq:somfourth})\u5f0f\u306e\u53f3\u8fba\u306e\u88ab\u7a4d\u5206\u95a2\u6570\u306b\u7740\u76ee\u3059\u308b\u3068\u3001\u5e73\u5747$(\\mu+2\\sigma^2)$\u3067\u5206\u6563$\\sigma^2$\u306e\u6b63\u898f\u5206\u5e03\u3092\u8868\u3057\u3066\u3044\u3066\u3001\u304b\u3064\u305d\u308c\u3092$(-\\infty, \\infty)$\u306e\u7bc4\u56f2\u3067\u7a4d\u5206\u3059\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u304c\u3001\u3053\u306e\u8a08\u7b97\u7d50\u679c\u306f1\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u3088\u3063\u3066\u3001<\/p>\n

\\begin{align}
\nE[X^2] &= \\exp\\left(2\\mu+2\\sigma^2\\right) \\label{eq:somfinal}
\n\\end{align}<\/p>\n

\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

(\\ref{eq:expectationfinal})\u5f0f\u53ca\u3073(\\ref{eq:somfinal})\u5f0f\u306e\u7d50\u679c\u3088\u308a\u3001$V[X]$\u306f<\/p>\n

\\begin{align}
\n V[X] &= E[X^2]-(E[X])^2 \\nonumber \\cr
\n &= \\exp\\left(2\\mu+2\\sigma^2\\right) – \\left[\\exp\\left(\\frac{2\\mu+\\sigma^2}{2}\\right)\\right]^2 \\nonumber \\cr
\n &= \\exp\\left(2\\mu+2\\sigma^2\\right) – \\exp\\left(2\\mu+\\sigma^2\\right) \\nonumber \\cr
\n &= \\exp\\left(2\\mu+\\sigma^2\\right)\\left[\\exp(\\sigma^2)-1\\right] \\label{eq:variancefinal}
\n\\end{align}<\/p>\n

\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002$\\blacksquare$<\/p>\n

\u307e\u3068\u3081<\/h2>\n

(\\ref{eq:logndfinal})\u5f0f\u306f\u5909\u63db\u5143\u306e\u6b63\u898f\u5206\u5e03\u306e\u5f0f((\\ref{eq:nddensity})\u5f0f)\u306e$x$\u3092\u5358\u7d14\u306b$\\log x$\u3067\u7f6e\u304d\u63db\u3048\u305f\u3060\u3051\u3067\u306a\u304f\u3001\u5168\u4f53\u3092$x$\u3067\u5272\u3063\u305f\u3082\u306e\u306b\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3059\u3002<\/p>\n

\u672c\u8a18\u4e8b\u306e\u6700\u521d\u306b\u6319\u3052\u305f\u500b\u4eba\u306e\u5e74\u53ce\u306e\u5206\u5e03\u306e\u3088\u3046\u306b\u5916\u308c\u5024\u306b\u3082\u4f55\u3089\u304b\u306e\u610f\u5473\u304c\u3042\u308b\u3068\u8003\u3048\u3089\u308c\u308b\u6642\u306b\u306f\u6b63\u898f\u5206\u5e03\u306b\u7121\u7406\u306b\u5f53\u3066\u306f\u3081\u308b\u3088\u308a\u306f\u5bfe\u6570\u6b63\u898f\u5206\u5e03\u7b49\u306e\u4ed6\u306e\u5206\u5e03\u95a2\u6570\u3092\u5f53\u3066\u306f\u3081\u305f\u65b9\u304c\u826f\u3044\u5834\u5408\u304c\u3042\u308a\u305d\u3046\u3067\u3059\u3002<\/p>\n

\u78ba\u7387\u5909\u6570\u306e\u5909\u6570\u5909\u63db\u306f\u8a08\u7b97\u304c\u3084\u3084\u3053\u3057\u304b\u3063\u305f\u308a\u9593\u9055\u3048\u3084\u3059\u3044\u8a08\u7b97\u306b\u306a\u308b\u3053\u3068\u304c\u591a\u3044\u306e\u3067\u3001\u3064\u3044\u3067\u306b\u5fa9\u7fd2\u3059\u308b\u3053\u3068\u306b\u3057\u307e\u3057\u305f\u3002\u4f55\u304b\u306e\u3054\u53c2\u8003\u306b\u3057\u3066\u3044\u305f\u3060\u3051\u308b\u3068\u5e78\u3044\u3067\u3059\u3002<\/p>\n

\u3053\u306e\u8a18\u4e8b\u306f\u4ee5\u4e0a\u3067\u3059\u3002<\/p>\n

References \/ \u53c2\u8003\u6587\u732e<\/h2>\n
    \n
  1. Wikipedia\u306e\u5bfe\u6570\u6b63\u898f\u5206\u5e03\u306e\u30da\u30fc\u30b8<\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    \u306f\u3058\u3081\u306b \u5b9f\u9a13\u30c7\u30fc\u30bf\u3092\u306f\u3058\u3081\u3068\u3059\u308b\u5b9f\u4e16\u754c\u3092\u5bfe\u8c61\u3068\u3057\u305f\u30c7\u30fc\u30bf\u7fa4\u3092\u7d71\u8a08\u7684\u306b\u6271\u3046\u3068\u304d\u306b\u304b\u306a\u308a\u9ad8\u3044\u78ba\u7387\u3067\u767b\u5834\u3057\u3001\u304b\u3064\u4f55\u3068\u304b\u3059\u308b\u5fc5\u8981\u306b\u8feb\u3089\u308c\u308b\u306e\u304c\u3001\u300c\u5916\u308c\u5024\u300d\u3067\u3059\u3002 \u4f8b\u3048\u3070\u3001\u5f97\u3089\u308c\u305f\u30c7\u30fc\u30bf\u304c $\\{ a_n \\} = \\{ 600,500,16000,1000,400,700,550,800,1200,900 \\}$\u306e\u3088\u3046\u306a\u3082\u306e\u3067\u3042\u3063\u305f\u308a\u3059\u308b\u3068\u300116000\u306f\u4ed6\u306e\u30c7\u30fc\u30bf\u5024\u3068\u6bd4\u8f03\u3059\u308b\u3068\u6841\u9055\u3044\u306b\u5927\u304d\u3044\u306e\u3067\u4f55\u3068\u306a\u304f\u2026 Read More »<\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":8688,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[31,13],"tags":[],"class_list":["post-8685","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-inkscape","category-13"],"_links":{"self":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/8685","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=8685"}],"version-history":[{"count":8,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/8685\/revisions"}],"predecessor-version":[{"id":9405,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/8685\/revisions\/9405"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/media\/8688"}],"wp:attachment":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8685"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8685"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8685"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}