{"id":2537,"date":"2018-08-22T23:27:42","date_gmt":"2018-08-22T14:27:42","guid":{"rendered":"https:\/\/pandanote.info\/?p=2537"},"modified":"2022-08-07T00:27:44","modified_gmt":"2022-08-06T15:27:44","slug":"%e3%81%a1%e3%82%87%e3%81%84%e3%81%a8%e9%87%8e%e6%9a%ae%e7%94%a8%e3%81%a7%e6%ad%a3%e8%a6%8f%e5%88%86%e5%b8%83%e3%81%ab%e5%be%93%e3%81%86%e7%8b%ac%e7%ab%8b%e3%81%aa%e7%a2%ba%e7%8e%87%e5%a4%89%e6%95%b0","status":"publish","type":"post","link":"https:\/\/pandanote.info\/?p=2537","title":{"rendered":"\u3061\u3087\u3044\u3068\u91ce\u66ae\u7528\u3067\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u72ec\u7acb\u306a\u78ba\u7387\u5909\u6570\u306e\u548c\u3068\u5dee\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u8a08\u7b97\u3057\u3066\u307f\u305f\u306e\u3067\u3001\u30e1\u30e2\u66f8\u304d\u3057\u3066\u307f\u305f\u3002"},"content":{"rendered":"

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

\u3061\u3087\u3044\u3068\u91ce\u66ae\u306a\u7528\u4e8b\u3067\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u72ec\u7acb\u306a\u78ba\u7387\u5909\u6570\u306e\u548c\u3068\u5dee\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u8a08\u7b97\u3057\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u304f\u306a\u308a\u305d\u3046\u306a\u306e\u3067\u3001\u305d\u3082\u305d\u3082\u3057\u3063\u304b\u308a\u52c9\u5f37\u3057\u3066\u3044\u305f\u306e\u304b\u3069\u3046\u304b\u3082\u601d\u3044\u51fa\u305b\u307e\u305b\u3093\u304c\u3001<\/del>\u601d\u3044\u51fa\u3059\u3053\u3068\u306b\u3057\u307e\u3057\u305f\u3002<\/p>\n

\u306a\u304a\u3001\u672c\u8a18\u4e8b\u3067\u306f\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5909\u6570\u53ca\u3073\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306e\u307f\u306b\u3064\u3044\u3066\u8003\u3048\u308b\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n

\u307e\u305a\u306f\u548c\u306e\u516c\u5f0f\u304b\u3089\u3002<\/h2>\n

\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306b\u9650\u3089\u305a\u3001\u4e00\u822c\u306b\u72ec\u7acb\u306a\u78ba\u7387\u5909\u6570$X$,$Y$\u3068\u305d\u308c\u3089\u306b\u5bfe\u5fdc\u3059\u308b\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570$p_1(x),p_2(x) (x \\in \\mathbb{R})$\u306b\u3064\u3044\u3066\u3001\u78ba\u7387\u5909\u6570\u306e\u548c$X+Y$\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570$p(x)$\u306f\u4ee5\u4e0b\u306e\u5f0f\u3067\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n

\\begin{eqnarray}
\np(x) &=& \\int^{\\infty}_{-\\infty}\\,p_1(x-y)p_2(y)dy \\label{eq:probsum}
\n\\end{eqnarray}<\/p>\n

(\\ref{eq:probsum})\u5f0f\u306e\u5c0e\u51fa\u306f\u81ea\u529b\u3067\u3084\u308d\u3046\u3068\u3057\u305f\u3082\u306e\u306e\u3001\u529b\u5c3d\u304d\u307e\u3057\u305f\u3002\uff3f|\uffe3|\u25cb<\/p>\n

\u81ea\u5206\u306b\u3068\u3063\u3066\u308f\u304b\u308a\u306b\u304f\u304b\u3063\u305f\u306e\u306f\u3001\u4ee5\u4e0b\u306e\u30dd\u30a4\u30f3\u30c8\u3060\u3063\u305f\u306e\u3067\u3001\u5f8c\u3067\u601d\u3044\u51fa\u305b\u308b\u3088\u3046\u306b\u30e1\u30e2\u3057\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n

    \n
  1. \u6700\u7d42\u76ee\u7684\u306f\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u6c42\u3081\u308b\u306e\u3067\u3059\u304c\u3001\u307e\u305a\u3001\u78ba\u7387\u3092\u8a08\u7b97\u3057\u3001\u305d\u308c\u304b\u3089\u5fae\u5206\u3059\u308b\u3053\u3068\u3002<\/li>\n
  2. \u8db3\u3057\u5408\u308f\u305b\u308b2\u500b\u306e\u78ba\u7387\u5909\u6570\u304c\u72ec\u7acb\u3067\u3042\u308b\u304b\u3069\u3046\u304b\u78ba\u8a8d\u3059\u308b\u3053\u3068\u3002\u72ec\u7acb\u3067\u306a\u3044\u5834\u5408\u306f\u5225\u306e\u5c0e\u51fa\u65b9\u91dd\u3092\u8003\u3048\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059(\u53c2\u8003\u6587\u732e[1])\u3002<\/li>\n
  3. \u7a4d\u5206\u9806\u5e8f\u306e\u4ea4\u63db\u304c\u53ef\u80fd\u306a\u6761\u4ef6\u306b\u3064\u3044\u3066\u306f\u7a4d\u5206\u306e\u524d\u306b\u5fc5\u305a\u78ba\u8a8d\u3059\u308b\u3053\u3068\u3002<\/li>\n<\/ol>\n

    \u7573\u307f\u8fbc\u307f\u7a4d\u5206\u306e\u8a08\u7b97\u3002<\/h2>\n

    (\\ref{eq:probsum})\u5f0f\u304c\u7406\u89e3\u3067\u304d\u305f\u3068\u3053\u308d\u3067\u3001$p(x)$\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n

    $p_1(x)$\u304c\u5e73\u5747$\\mu_1$,\u5206\u6563$\\sigma_1^2$\u306e\u6b63\u898f\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3067\u3001$p_2(x)$\u304c\u5e73\u5747$\\mu_2$,\u5206\u6563$\\sigma_2^2$\u306e\u6b63\u898f\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3068\u3059\u308b\u3068\u3001$p_1(x)$\u53ca\u3073$p_2(x)$\u306f(\\ref{eq:probden1})\u53ca\u3073(\\ref{eq:probden2})\u5f0f\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

    \\begin{eqnarray}
    \np_1(x) &=& \\frac{1}{\\sqrt{2\\pi}\\sigma_1}e^{-\\frac{(x-\\mu_1)^2}{2\\sigma_1^2}} \\label{eq:probden1} \\\\
    \np_2(x) &=& \\frac{1}{\\sqrt{2\\pi}\\sigma_2}e^{-\\frac{(x-\\mu_2)^2}{2\\sigma_2^2}} \\label{eq:probden2}
    \n\\end{eqnarray}<\/p>\n

    (\\ref{eq:probden1})\u53ca\u3073(\\ref{eq:probden2})\u5f0f\u3092(\\ref{eq:probsum})\u5f0f\u306b\u4ee3\u5165\u3057\u3066\u8a08\u7b97\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n

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

    \\begin{eqnarray}
    \np(x) &=& \\frac{1}{2\\pi\\sigma_1\\sigma_2}\\int^{\\infty}_{-\\infty}\\exp\\left\\{-\\frac{(x-y-\\mu_1)^2}{2\\sigma_1^2}-\\frac{(y-\\mu_2)^2}{2\\sigma_2^2}\\right\\}dy \\label{eq:probden}
    \n\\end{eqnarray}
    \n\u3068\u306a\u308a\u307e\u3059\u3002\u6b21\u306f\u3001
    \n\\begin{eqnarray}
    \n\\int^{\\infty}_{-\\infty}e^{\\frac{(x-\\mu)^2}{b}}dx &=& \\sqrt{b\\pi} \\label{eq:gaussformula}
    \n\\end{eqnarray}
    \n\u304c\u9069\u7528\u3067\u304d\u308b\u5f62\u306b\u6301\u3061\u8fbc\u307f\u305f\u3044\u306e\u3067\u3001(\\ref{eq:probden})\u5f0f\u53f3\u8fba\u306e$\\exp\\left\\{\\cdot\\right\\}$\u3092$\\exp\\left\\{-\\displaystyle\\frac{1}{2}(\\cdot)\\right\\}$\u3068\u5909\u5f62\u3057\u305f\u4e0a\u3067\u3001$(\\cdot)$\u306e\u90e8\u5206\u3092$y$\u306b\u3064\u3044\u3066\u5e73\u65b9\u5b8c\u6210\u3055\u305b\u307e\u3059\u3002\u3053\u308c\u306f\u3001
    \n\\begin{eqnarray}
    \n\\def\\xmumu{\\displaystyle\\frac{x-\\mu_1}{\\sigma_1^2}+\\displaystyle\\frac{\\mu_2}{\\sigma_2^2}}
    \n\\def\\sigmasigma{\\displaystyle\\frac{1}{\\sigma_1^2}+\\displaystyle\\frac{1}{\\sigma_2^2}}
    \n&{}& \\frac{(x-y-\\mu_1)^2}{\\sigma_1^2}+\\frac{(y-\\mu_2)^2}{\\sigma_2^2} \\nonumber \\\\
    \n&=& y^2\\left(\\frac{1}{\\sigma_1}+\\frac{1}{\\sigma_2}\\right)-2y\\left(\\xmumu\\right)+\\frac{(x-\\mu_1)^2}{\\sigma_1^2}+\\frac{\\mu_2^2}{\\sigma_2^2} \\nonumber \\\\
    \n&=& \\left(\\sigmasigma\\right)\\left\\{y-\\frac{\\xmumu}{\\sigmasigma}\\right\\}^2 + \\frac{(x-\\mu_1)^2}{\\sigma_1^2}+\\frac{\\mu_2^2}{\\sigma_2^2}-\\frac{\\left(\\xmumu\\right)^2}{\\sigmasigma} \\label{eq:sq}
    \n\\end{eqnarray}
    \n\u3068\u5e73\u65b9\u5b8c\u6210\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

    \u672c\u5f53\u306b\u5e73\u65b9\u5b8c\u6210\u304c\u3067\u304d\u308b\u306e\u304b\u3061\u3087\u3063\u3068\u5148\u884c\u304d\u306b\u4e0d\u5b89\u3092\u899a\u3048\u308b(\u3069\u306e\u304f\u3089\u3044\u4e0d\u5b89\u304b\u3068\u3044\u3046\u3068\u3001\u3053\u306e\u8a18\u4e8b<\/a>\u306e\u8a08\u7b97\u304f\u3089\u3044\u4e0d\u5b89\u3067\u3059\u3002)\u3068\u3053\u308d\u3067\u3059\u304c\u3001\u3053\u3053\u306f\u5c11\u3057\u6211\u6162\u3057\u3066\u3001(\\ref{eq:sq})\u306e\u53f3\u8fba\u306e\u5f0f\u306e\u3046\u3061\u3001\u5e73\u65b9\u5b8c\u6210\u304b\u3089\u5916\u308c\u305f\u90e8\u5206\u3092\u3082\u3046\u5c11\u3057\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n

    \u3059\u308b\u3068\u2026
    \n\\begin{eqnarray}
    \n\\def\\xmumu{\\displaystyle\\frac{x-\\mu_1}{\\sigma_1^2}+\\displaystyle\\frac{\\mu_2}{\\sigma_2^2}}
    \n\\def\\sigmasigma{\\displaystyle\\frac{1}{\\sigma_1^2}+\\displaystyle\\frac{1}{\\sigma_2^2}}
    \n\\def\\sigmasigmap{\\sigma_1^2+\\sigma_2^2}
    \n&{}& \\frac{(x-\\mu_1)^2}{\\sigma_1^2}+\\frac{\\mu_2^2}{\\sigma_2^2}-\\frac{\\left(\\xmumu\\right)^2}{\\sigmasigma} \\nonumber \\\\
    \n&=& \\frac{(x-\\mu_1)^2}{\\sigma_1^2}+\\frac{\\mu_2^2}{\\sigma_2^2}-\\frac{\\displaystyle\\frac{(x-\\mu_1)^2}{\\sigma_1^4} + \\displaystyle\\frac{2(x-\\mu_1)\\mu_2}{\\sigma_1^2\\sigma_2^2} + \\displaystyle\\frac{\\mu^2}{\\sigma_2^4}}{\\sigmasigma} \\nonumber \\\\
    \n&=& \\frac{(x-\\mu_1)^2}{\\sigma_1^2}+\\frac{\\mu_2^2}{\\sigma_2^2}-\\frac{1}{\\sigma_1^2+\\sigma_2^2}\\left\\{\\frac{\\sigma_2^2}{\\sigma_1^2}(x-\\mu_1)^2+2(x-\\mu_1)\\mu_2+\\frac{\\sigma_1^2}{\\sigma_2^2}\\mu_2^2\\right\\} \\nonumber \\\\
    \n&=& \\left\\{\\frac{1}{\\sigma_1^2}-\\frac{\\sigma_2^2}{\\sigma_1^2(\\sigmasigmap)}\\right\\}(x-\\mu_1)^2 – \\frac{1}{\\sigmasigmap}\\cdot 2(x-\\mu_1)\\mu_2+ \\nonumber \\\\
    \n&{}& \\left\\{\\frac{1}{\\sigma_2^2}-\\frac{\\sigma_1^2}{\\sigma_2^2(\\sigmasigmap)}\\right\\}\\mu_2^2 \\nonumber \\\\
    \n&=& \\frac{1}{\\sigmasigmap}\\left\\{(x-\\mu_1)^2 – 2(x-\\mu_1)\\mu_2 + \\mu_2^2 \\right\\} \\nonumber \\\\
    \n&=& \\frac{(x – \\mu_1 – \\mu_2)^2}{\\sigmasigmap} \\label{eq:residualerror}
    \n\\end{eqnarray}<\/p>\n

    (\\ref{eq:sq})\u53ca\u3073(\\ref{eq:residualerror})\u5f0f\u3092\u5229\u7528\u3057\u3066\u3001(\\ref{eq:probden})\u5f0f\u306e\u53f3\u8fba\u3092\u66f8\u304d\u63db\u3048\u308b\u3068\u3001
    \n\\begin{eqnarray}
    \n\\def\\xmumu{\\displaystyle\\frac{x-\\mu_1}{\\sigma_1^2}+\\displaystyle\\frac{\\mu_2}{\\sigma_2^2}}
    \n\\def\\sigmasigma{\\displaystyle\\frac{1}{\\sigma_1^2}+\\displaystyle\\frac{1}{\\sigma_2^2}}
    \n\\def\\sigmasigmap{\\sigma_1^2+\\sigma_2^2}
    \np(x) &=& \\frac{1}{2\\pi\\sigma_1\\sigma_2}{\\Huge\\int}_{-\\infty}^{\\infty}\\!\\!\\!\\exp\\left[-\\frac{1}{2}\\left\\{\\left(\\sigmasigma\\right)\\left(y-\\frac{\\xmumu}{\\sigmasigma}\\right)^2\\right.\\right. \\nonumber \\\\
    \n&{}&\\left.\\left. +\\frac{1}{\\sigmasigmap}(x – \\mu_1 – \\mu_2)^2\\right\\}\\right]dy \\nonumber \\\\
    \n&=& \\frac{1}{2\\pi\\sigma_1\\sigma_2}e^{-\\frac{(x – \\mu_1 – \\mu_2)^2}{2(\\sigmasigmap)}} \\nonumber \\\\
    \n&{}& {\\Huge\\int}_{-\\infty}^{\\infty}\\!\\!\\!\\exp\\left[-\\frac{1}{2}\\left\\{\\left(\\sigmasigma\\right)\\left(y-\\frac{\\xmumu}{\\sigmasigma}\\right)^2\\right\\}\\right]dy \\label{eq:probdensqres}
    \n\\end{eqnarray}
    \n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

    (\\ref{eq:probdensqres})\u5f0f\u53f3\u8fba\u306e\u88ab\u7a4d\u5206\u95a2\u6570\u306f$b = \\displaystyle\\frac{2}{\\displaystyle\\frac{1}{\\sigma_1^2}+\\displaystyle\\frac{1}{\\sigma_2^2}}, \\mu = \\frac{\\displaystyle\\frac{x-\\mu_1}{\\sigma_1^2}+\\displaystyle\\frac{\\mu_2}{\\sigma_2^2}}{\\displaystyle\\frac{1}{\\sigma_1^2}+\\displaystyle\\frac{1}{\\sigma_2^2}}$\u3068\u7f6e\u304f\u3068\u3001(\\ref{eq:gaussformula})\u5f0f\u3088\u308a\u3001
    \n\\begin{eqnarray}
    \n\\def\\xmumu{\\displaystyle\\frac{x-\\mu_1}{\\sigma_1^2}+\\displaystyle\\frac{\\mu_2}{\\sigma_2^2}}
    \n\\def\\sigmasigma{\\displaystyle\\frac{1}{\\sigma_1^2}+\\displaystyle\\frac{1}{\\sigma_2^2}}
    \n\\def\\sigmasigmap{\\sigma_1^2+\\sigma_2^2}
    \n{\\Huge\\int}_{-\\infty}^{\\infty}\\!\\!\\!\\exp\\left[-\\frac{1}{2}\\left\\{\\left(\\sigmasigma\\right)\\left(y-\\frac{\\xmumu}{\\sigmasigma}\\right)^2\\right\\}\\right]dy &=& \\sqrt{\\frac{2\\pi}{\\sigmasigma}} \\label{eq:insideofint}
    \n\\end{eqnarray}
    \n\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002\u305d\u3053\u3067\u3001\u3053\u308c\u3092(\\ref{eq:probdensqres})\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\u3001
    \n\\begin{eqnarray}
    \n\\def\\xmumu{\\displaystyle\\frac{x-\\mu_1}{\\sigma_1^2}+\\displaystyle\\frac{\\mu_2}{\\sigma_2^2}}
    \n\\def\\sigmasigma{\\displaystyle\\frac{1}{\\sigma_1^2}+\\displaystyle\\frac{1}{\\sigma_2^2}}
    \n\\def\\sigmasigmap{\\sigma_1^2+\\sigma_2^2}
    \np(x) &=& \\frac{1}{2\\pi\\sigma_1\\sigma_2}\\sqrt{\\frac{2\\pi}{\\sigmasigma}}e^{-\\frac{(x – \\mu_1 – \\mu_2)^2}{2(\\sigmasigmap)}} \\nonumber \\\\
    \n&=& \\frac{1}{\\sqrt{2\\pi(\\sigmasigmap)}}e^{-\\frac{(x – \\mu_1 – \\mu_2)^2}{2(\\sigmasigmap)}} \\label{eq:results}
    \n\\end{eqnarray}
    \n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

    ((\\ref{eq:probden1})\u53ca\u3073(\\ref{eq:probden2})\u5f0f\u3068(\\ref{eq:results})\u5f0f\u3092\u3088\u304f\u898b\u6bd4\u3079\u308b\u3068\u3001$p(x)$\u306f\u5e73\u5747$\\mu_1+\\mu_2$,\u5206\u6563$\\sigma_1^2+\\sigma_2^2$\u306e\u6b63\u898f\u5206\u5e03\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u3088\u3063\u3066\u3001\u78ba\u7387\u5909\u6570\u306e\u548c$X+Y$\u306f\u3053\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570$p(x)$\u306b\u5f93\u3046\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

    \u5dee\u306e\u516c\u5f0f\u306e\u8a08\u7b97\u3067\u3059\u3002<\/h2>\n

    \u524d\u7bc0\u306e\u8a08\u7b97\u3067\u78ba\u7387\u5909\u6570\u306e\u548c\u304c\u5f93\u3046\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304c\u8a08\u7b97\u3067\u304d\u305f\u306e\u3067\u3001\u6b21\u306f\u78ba\u7387\u5909\u6570\u306e\u5dee$X-Y$\u304c\u5f93\u3046\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570$q(x)$\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n

    \u72ec\u7acb\u306a\u78ba\u7387\u5909\u6570$X$,$Y$\u3068\u305d\u308c\u3089\u306b\u5bfe\u5fdc\u3059\u308b\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570$q_1(x),q_2(x) (x \\in \\mathbb{R})$\u306b\u3064\u3044\u3066\u3001\u78ba\u7387\u5909\u6570\u306e\u5dee$X-Y$\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570$q(x)$\u306f\u4ee5\u4e0b\u306e\u5f0f\u3067\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n

    \\begin{eqnarray}
    \nq(x) &=& \\int^{\\infty}_{-\\infty}\\,q_1(x+y)q_2(y)dy \\label{eq:probsub}
    \n\\end{eqnarray}<\/p>\n

    \u548c\u306e\u5834\u5408\u3068\u540c\u69d8\u306b\u3001(\\ref{eq:probsub})\u5f0f\u306b(\\ref{eq:probden1})\u53ca\u3073(\\ref{eq:probden2})\u5f0f\u3092\u4ee3\u5165\u3057\u3066\u8a08\u7b97\u3057\u3066\u3044\u304d\u307e\u3059\u3002$q_1(x+y)$\u306b\u95a2\u9023\u3059\u308b\u9805\u306b\u3064\u3044\u3066\u306f\u7b26\u53f7\u304c\u5909\u308f\u308b\u3068\u3053\u308d\u304c\u3042\u308a\u307e\u3059\u304c\u3001(\\ref{eq:insideofint})\u5f0f\u306b\u76f8\u5f53\u3059\u308b\u7a4d\u5206\u306b\u3064\u3044\u3066\u306f\u548c\u306e\u5834\u5408\u306b\u8a08\u7b97\u3057\u305f\u88ab\u7a4d\u5206\u95a2\u6570\u304c$x$\u65b9\u5411\u306b\u5e73\u884c\u79fb\u52d5\u3057\u305f\u3082\u306e\u3092$(-\\infty,\\infty)$\u3067\u7a4d\u5206\u3059\u308b\u306e\u3067\u3001\u8a08\u7b97\u7d50\u679c\u306f\u548c\u306e\u5834\u5408\u3068\u7b49\u3057\u304f\u306a\u308a\u307e\u3059\u3002\u307e\u305f\u3001(\\ref{eq:residualerror})\u5f0f\u306b\u76f8\u5f53\u3059\u308b$y$\u306b\u3064\u3044\u3066\u306e\u5b9a\u6570\u9805\u306f(\u7d30\u304b\u3044\u8a08\u7b97\u306f\u7701\u7565\u3057\u307e\u3059\u304c)\u3001
    \n\\begin{eqnarray}
    \n\\def\\xmumup{\\displaystyle\\frac{\\mu_2}{\\sigma_2^2}-\\displaystyle\\frac{x-\\mu_1}{\\sigma_1^2}}
    \n\\def\\sigmasigma{\\displaystyle\\frac{1}{\\sigma_1^2}+\\displaystyle\\frac{1}{\\sigma_2^2}}
    \n\\def\\sigmasigmap{\\sigma_1^2+\\sigma_2^2}
    \n\\frac{(x-\\mu_1)^2}{\\sigma_1^2}+\\frac{\\mu_2^2}{\\sigma_2^2}-\\frac{\\left(\\xmumup\\right)^2}{\\sigmasigma} &=& \\frac{1}{\\sigmasigmap}(x-\\mu_1+\\mu_2)^2 \\label{eq:residualerrorsub}
    \n\\end{eqnarray}
    \n\u306b\u306a\u308a\u307e\u3059\u3002\u3088\u3063\u3066\u3001(\\ref{eq:insideofint})\u53ca\u3073(\\ref{eq:residualerrorsub})\u5f0f\u3088\u308a
    \n\\begin{eqnarray}
    \n\\def\\xmumu{\\displaystyle\\frac{x-\\mu_1}{\\sigma_1^2}+\\displaystyle\\frac{\\mu_2}{\\sigma_2^2}}
    \n\\def\\sigmasigma{\\displaystyle\\frac{1}{\\sigma_1^2}+\\displaystyle\\frac{1}{\\sigma_2^2}}
    \n\\def\\sigmasigmap{\\sigma_1^2+\\sigma_2^2}
    \nq(x) &=& \\frac{1}{2\\pi\\sigma_1\\sigma_2}\\sqrt{\\frac{2\\pi}{\\sigmasigma}}e^{-\\frac{(x – \\mu_1 + \\mu_2)^2}{2(\\sigmasigmap)}} \\nonumber \\\\
    \n&=& \\frac{1}{\\sqrt{2\\pi(\\sigmasigmap)}}e^{-\\frac{(x – \\mu_1 + \\mu_2)^2}{2(\\sigmasigmap)}} \\label{eq:resultssub}
    \n\\end{eqnarray}
    \n\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n

    (\\ref{eq:probden1})\u53ca\u3073(\\ref{eq:probden2})\u5f0f\u3068(\\ref{eq:resultssub})\u5f0f\u306e\u7d50\u679c\u3088\u308a\u3001$q(x)$\u306f\u5e73\u5747$\\mu_1-\\mu_2$,\u5206\u6563$\\sigma_1^2+\\sigma_2^2$\u306e\u6b63\u898f\u5206\u5e03\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u3088\u3063\u3066\u3001\u78ba\u7387\u5909\u6570\u306e\u5dee$X-Y$\u306f\u3053\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570$q(x)$\u306b\u5f93\u3046\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

    \u307e\u3068\u3081\u306e\u3088\u3046\u306a\u3082\u306e<\/h2>\n

    \u3053\u3053\u307e\u3067\u306e\u8a08\u7b97\u3067\u3001\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304c\u6b63\u898f\u5206\u5e03\u306b\u306a\u308b\u72ec\u7acb\u306a\u78ba\u7387\u5909\u6570\u306e\u548c\u3068\u5dee\u304c\u5f93\u3046\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306b\u3064\u3044\u3066\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002\u8a08\u7b97\u306e\u7d50\u679c\u5f97\u3089\u308c\u305f\u7d50\u8ad6\u306f\u3044\u305f\u3063\u3066\u30b7\u30f3\u30d7\u30eb\u304b\u3064\u57fa\u672c\u7684\u306a\u3082\u306e\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u304c\u3001\u7d50\u8ad6\u3060\u3051\u3092\u6697\u8a18\u3057\u3066\u3044\u308b\u3060\u3051\u3060\u3068\u3001\u3044\u3056\u3068\u3044\u3046\u3068\u304d\u306b\u4f7f\u3048\u306a\u3044\u6c17\u304c\u3057\u305f\u306e\u3067\u3001\u601d\u3044\u51fa\u3057\u3064\u3044\u3066\u306b\u8a08\u7b97\u3057\u3066\u307f\u305f\u6b21\u7b2c\u3067\u3059\u3002<\/p>\n

    \u306a\u304a\u3001\u4eca\u307e\u3067\u6570\u5b66\u7684\u306a\u8a71\u984c\u304c\u542b\u307e\u308c\u3066\u3044\u308b\u8a18\u4e8b\u306b\u3064\u3044\u3066\u306f\u3001\u300c\u6570\u5b66\u7684\u30c1\u30e9\u30b7\u306e\u88cf\u300d\u3068\u3044\u3046\u30ab\u30c6\u30b4\u30ea\u30fc\u3092\u8a2d\u5b9a\u3057\u3066\u3044\u307e\u3057\u305f\u304c\u3001\u30ab\u30c6\u30b4\u30ea\u30fc\u540d\u3068\u3057\u3066\u3044\u308d\u3044\u308d\u3068\u30a2\u30ec\u306a\u611f\u3058\u3082\u3057\u3066\u3044\u305f\u306e\u3067\u3001\u300cpanda\u306e\u5927\u8a08\u7b97\u7528\u7d19\u300d\u3068\u3044\u3046\u30ab\u30c6\u30b4\u30ea\u30fc\u540d\u306b\u5909\u66f4\u3057\u307e\u3057\u305f\u3002\u3053\u306e\u65b9\u304c\u672c\u30b5\u30a4\u30c8\u306e\u30bf\u30a4\u30c8\u30eb\u3068\u306e\u6574\u5408\u6027\u3082\u53d6\u308c\u3066\u3001\u306a\u304b\u306a\u304b\u3044\u3044\u611f\u3058\u3060\u3068\u601d\u3044\u307e\u3059(\u203b\u500b\u4eba\u306e\u611f\u60f3\u3067\u3059)\u3002<\/p>\n

    \u307e\u305f\u3001\u3064\u3044\u3067\u3068\u8a00\u3063\u3066\u306f\u4f55\u3067\u3059\u304c\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304c\u6a19\u6e96<\/strong>\u6b63\u898f\u5206\u5e03\u306b\u306a\u308b\u72ec\u7acb\u306a\u78ba\u7387\u5909\u6570\u306e\u7a4d\u304c\u5f93\u3046\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3082\u8a08\u7b97\u3057\u3066\u307f\u307e\u3057\u305f\u3002\u5909\u5f62Bessel\u95a2\u6570\u304c\u767b\u5834\u3057\u307e\u3059\u3002\u8a73\u3057\u304f\u306f\u3053\u3061\u3089\u3092\u53c2\u7167<\/a>\u3067\u3059\u3002<\/p>\n

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

    \u53c2\u8003\u6587\u732e<\/h2>\n