\u30ac\u30a6\u30b9\u95a2\u6570<\/h3>\n
\u4e00\u822c\u306b\u30ac\u30a6\u30b9\u95a2\u6570\u3068\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5f62\uff1a
\n$$ a \\exp\\left\\{ -\\frac{(x-b)^2}{2 c^2} \\right\\} $$<\/p>\n
\u6b63\u898f\u5206\u5e03\u95a2\u6570<\/h3>\n
\u7279\u306b\u6b63\u898f\u5206\u5e03\u95a2\u6570\u3068\u306f
\n$$ N(\\mu, \\sigma^2) \\equiv \\frac{1}{\\sqrt{2\\pi} \\sigma} \\exp\\left\\{ – \\frac{(x-\\mu)^2}{2\\sigma^2}\\right\\}$$ \u3067\u3042\u308a\uff0c\u78ba\u7387\u7d71\u8a08\u3067\u6700\u91cd\u8981\u304b\u3064\u4e2d\u5fc3\u7684\u5f79\u5272\u3092\u679c\u305f\u3059\u3002<\/p>\n
\u7279\u306b\uff0c
\n$$\\int_{-\\infty}^{\\infty} N(\\mu=0, \\sigma^2 = 1\/2)\\, dx = \\frac{1}{\\sqrt{\\pi}} \\int_{-\\infty}^{\\infty} e^{-x^2} dx = 1$$ \u3068\u306a\u308a\uff0c\u30ac\u30a6\u30b9\u7a4d\u5206\u3002<\/p>\n
\\(\\mu, \\ \\sigma^2 \\) \u304c\u4e00\u822c\u306e\u5834\u5408\u3067\u3082\uff0c\\(\\displaystyle y \\equiv \\frac{x-\\mu}{\\sqrt{2}\\sigma}, \\ dy = \\frac{dx}{\\sqrt{2}\\sigma} \\) \u3068\u3044\u3046\u5909\u6570\u5909\u63db\u3092\u884c\u3046\u3068\uff0c
\n$$\\int_{-\\infty}^{\\infty}\\frac{1}{\\sqrt{2\\pi} \\sigma} \\exp\\left\\{ – \\frac{(x-\\mu)^2}{2\\sigma^2}\\right\\} dx =
\n\\frac{1}{\\sqrt{\\pi}} \\int_{-\\infty}^{\\infty} \\exp\\left\\{ -y^2\\right\\} dy = 1$$ \u3068\u306a\u308b\u3002\u5168\u533a\u9593\u3067\u306e\u7a4d\u5206\u304c1\u306b\u898f\u683c\u5316\u3055\u308c\u3066\u3044\u308b\u3053\u3068\u304b\u3089\uff0c\u3053\u306e\u6b63\u898f\u5206\u5e03\u95a2\u6570\u3092\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3068\u304b\uff0c\u78ba\u7387\u5206\u5e03\u95a2\u6570\u3068\u3044\u3046\u3053\u3068\u3082\u3042\u308b\u3002\u3059\u3079\u3066\u306e\u4e8b\u8c61\u306e\u8d77\u3053\u308a\u3046\u308b\u78ba\u7387\u3092\u8db3\u3057\u4e0a\u3052\u308b\u3068\uff0c\u5168\u78ba\u7387\u306f1\u306b\u306a\u308b\u3067\u3057\u3087\u3002<\/p>\n
\u671f\u5f85\u5024<\/h4>\n
\uff08\u4ee5\u524d\u306f\u52e2\u3044\u3067\u300c\u5e73\u5747\u5024\u300d\u3068\u66f8\u3044\u3066\u3044\u305f\u3051\u3069\uff0c\u305d\u3057\u3066 Wikipedia \u3067\u3082 $\\mu$ \u306f\u3057\u3070\u3057\u3070\u300c\u5e73\u5747\u300d\u3068\u66f8\u304b\u308c\u3066\u3044\u305f\u308a\u3059\u308b\u3057… \u3068\u601d\u3063\u3066\u3044\u305f\u304c\uff0c\u4e16\u306e\u4e2d\u3067\u306f\u300c\u5e73\u5747\u5024\u300d\u3068\u300c\u671f\u5f85\u5024\u300d\u3092\u533a\u5225\u3059\u308b\u3088\u3046\u306a\u306e\u3067\uff0c\u300c\u671f\u5f85\u5024\u300d\u3068\u4fee\u6b63\u3057\u3066\u307f\u305f\u3002\u78ba\u7387\u5909\u6570\u306e\u300c\u671f\u5f85\u5024\u300d\u3068\u306f\uff0c\u5909\u6570\u306e\u5b9f\u73fe\u5024\u306b\u78ba\u7387\u306e\u91cd\u307f\u3092\u3064\u3051\u305f\u52a0\u91cd\u5e73\u5747\u3067\u3042\u308b<\/a>\uff0c\u3068\u3044\u3046\u3053\u3068\u3067\u3044\u3044\u3067\u3057\u3087\u3046\u304b\u3002\uff09<\/p>\n \u78ba\u7387\u5909\u6570 \\(x\\) \u304c\u6b63\u898f\u5206\u5e03 \\(N(\\mu, \\sigma^2)\\) \u306b\u5f93\u3046\u3068\u304d\uff0c\\( x\\) \u306e\u671f\u5f85\u5024 \\(\\langle x \\rangle\\) \u306f \uff08 \\(\\displaystyle y \\equiv \\frac{x-\\mu}{\\sqrt{2}\\sigma}, \\ dy = \\frac{dx}{\\sqrt{2}\\sigma} \\) \u3068\u3044\u3046\u5909\u6570\u5909\u63db\u3092\u4f7f\u3063\u3066\uff09 \u307e\u305f\uff0c\\( (x-\\langle x \\rangle)^2\\) \u306e\u671f\u5f85\u5024\u3067\u3042\u308b\u5206\u6563\u306f \u3055\u3089\u306b\u306f\uff0c\u3053\u306e\u5206\u6563\u306e\u5e73\u65b9\u6839 \\(\\sigma\\) \u3092\u6a19\u6e96\u504f\u5dee\u3068\u3044\u3044\uff0c\u3053\u308c\u3089\u304b\u3089\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u300c\u504f\u5dee\u5024\u300d\u3068\u547c\u3070\u308c\u308b\u5024\u304c\u5b9a\u7fa9\u3055\u308c\u308b\u3002 \u305f\u3068\u3048\u3070\uff0c\u504f\u5dee\u5024 $70$ \u3068\u3044\u3048\u3070\uff0c<\/p>\n \\begin{eqnarray} \u3068\u306a\u308b\u3002\u4e0b\u306e\u30b0\u30e9\u30d5\u304b\u3089\uff0c$\\mu -2 \\sigma \\leq x \\leq \\mu + 2 \\sigma$ \u306f $95.45\\%$ \u306a\u306e\u3067\uff0c\u6b8b\u308a $100 – 95.45 = 4.55$ \u3092 $2$ \u3067\u5272\u308b\u3068\uff0c$4.55\/2 = 2.275 \\simeq 2.28$ \u3068\u306a\u308a\uff0c\u307e\u3041\uff0c\u4e0a\u4f4d $2.28\\%$ \u306b\u5165\u308b\u306e\u304c\u504f\u5dee\u5024 $70$ \u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n <\/p>\n
\n\\begin{eqnarray}
\n\\langle x \\rangle &=& \\int_{-\\infty}^{\\infty} x N(\\mu, \\sigma^2) dx \\\\
\n&=& \\int_{-\\infty}^{\\infty}\\bigl( (x – \\mu) + \\mu\\bigr) N(\\mu, \\sigma^2) dx \\\\
\n&=& \\frac{\\sqrt{2}\\sigma}{\\sqrt{\\pi}} \\int_{-\\infty}^{\\infty} y e^{-y^2} dy + \\frac{\\mu}{\\sqrt{\\pi}} \\int_{-\\infty}^{\\infty} e^{-y^2} dy \\\\
\n&=& \\mu
\n\\end{eqnarray}
\n\u3064\u307e\u308a\uff0c\\(\\mu\\) \u306f\u78ba\u7387\u5909\u6570 \\(x\\) \u306e\u671f\u5f85\u5024\u3092\u8868\u3059\u3002<\/p>\n\u5206\u6563<\/h4>\n
\n$$\\langle (x-\\langle x \\rangle)^2 \\rangle \\equiv \\int_{-\\infty}^{\\infty} (x-\\langle x \\rangle)^2 N(\\mu, \\sigma^2) dx = \\cdots = \\sigma^2$$ \u3067\u3042\u308b\u3053\u3068\u3082\u7c21\u5358\u306b\u308f\u304b\u308b\u3002\u3064\u307e\u308a\uff0c\\(\\sigma^2\\) \u306f\u78ba\u7387\u5909\u6570 \\(x\\) \u306e\u5206\u6563\u3092\u8868\u3059\u3002<\/p>\n\u504f\u5dee\u5024<\/h4>\n
\n$$ T = \\frac{10\\times(x-\\mu)}{\\sigma} + 50$$ \u6614\u306e\u53d7\u9a13\u751f\uff08\u4eca\u3082\uff1f\uff09\u306f\u3053\u306e\u300c\u504f\u5dee\u5024\u300d\u3068\u3044\u3046\u5024\u306b\u4e00\u559c\u4e00\u6182\u3057\u306a\u304c\u3089\uff0c\u53d7\u9a13\u52c9\u5f37\u3092\u3057\u3066\u3044\u305f\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n\u504f\u5dee\u5024 70 \u306f\u4e0a\u4f4d\u4f55%\uff1f<\/h5>\n
\nT = 70 &=& \\frac{10\\times(x-\\mu)}{\\sigma} + 50 \\\\
\n\\therefore\\ \\ x &=& \\mu + 2 \\sigma
\n\\end{eqnarray}<\/p>\n\u6b63\u898f\u5206\u5e03\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/h3>\n
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