{"id":4543,"date":"2024-08-14T18:05:30","date_gmt":"2024-08-14T09:05:30","guid":{"rendered":"https:\/\/aiknot.jp\/media-top\/?p=4543"},"modified":"2024-08-16T11:26:07","modified_gmt":"2024-08-16T02:26:07","slug":"%e3%83%a2%e3%83%bc%e3%83%a1%e3%83%b3%e3%83%88%e3%81%a8%e3%81%af%ef%bc%9f%ef%bc%94%e6%ac%a1%e3%83%a2%e3%83%bc%e3%83%a1%e3%83%b3%e3%83%88%e3%81%be%e3%81%a7%e6%b1%82%e3%82%81%e3%81%a6%e3%81%bf%e3%82%8b","status":"publish","type":"post","link":"https:\/\/aiknot.jp\/media-top\/?p=4543","title":{"rendered":"\u30e2\u30fc\u30e1\u30f3\u30c8\u3068\u306f\uff1f\uff14\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8$E(X^4)$\u306e\u7b97\u51fa\u3068\u516c\u5f0f\u306e\u5c0e\u51fa\u65b9\u6cd5\u3092\u89e3\u8aac\uff01"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">\u30e2\u30fc\u30e1\u30f3\u30c8\u3068\u306f\uff1f<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">\u78ba\u7387\u5206\u5e03\u3092\u628a\u63e1\u3059\u308b\u305f\u3081\u306e\u5fc5\u8981\u306a\u8981\u7d20<\/h3>\n\n\n\n<p>\u78ba\u7387\u5206\u5e03\u306f\u6b63\u898f\u5206\u5e03\u306e\u3088\u3046\u306a\u5f62\u72b6\u3092\u3068\u3063\u3066\u3044\u306a\u3044\u3053\u3068\u3082\u591a\u304f\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u6642\u3001\u5f62\u72b6\u306e\u7279\u5fb4\u3092\u628a\u63e1\u3059\u308b\u305f\u3081\u306b\u6c42\u3081\u308b\u306e\u304c\u30e2\u30fc\u30e1\u30f3\u30c8\u3067\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u30e2\u30fc\u30e1\u30f3\u30c8\u306e\u5b9a\u7fa9<\/h3>\n\n\n\n<p>\u539f\u70b9\u4ed8\u8fd1\u306en\u6b21\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u306f\u6b21\u306e\u901a\u308a\u3067\u3001\u7a4d\u7387\u3068\u547c\u79f0\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p>\\[\u03bc_n=E(X^n)\\]<\/p>\n\n\n\n<p>\u5e73\u5747\u5468\u308a\u306en\u6b21\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u306f\u6b21\u306e\u901a\u308a\u3067\u3059\u3002<\/p>\n\n\n\n<p>\\[\u03bc\u00b4_n=E\\{(X-\u03bc)^n\\}\\]<\/p>\n\n\n\n<p>\u78ba\u7387\u5909\u6570\u3092\uff11\u56de\u4e57\u3058\u3066\u7b97\u51fa\u3059\u308b\u671f\u5f85\u5024\u306f\u3001\uff11\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8$\u03bc_1$\u3067\u3042\u308b\u3068\u8a00\u3048\u307e\u3059\u3057\u3001\u5206\u6563\u306f\uff12\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8$\u03bc\u00b4_2$\u3067\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u6c42\u3081\u308b<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570<\/h3>\n\n\n\n<p>\u6b63\u898f\u5206\u5e03\u306f\u6b21\u306e\u5f0f\u3067\u8868\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p>\\[f(x)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(x-\u03bc)^2}{2\\sigma^2}}\\]<\/p>\n\n\n\n<p>\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306f\u5e73\u57470\u3001\u5206\u65631\u306e\u6b63\u898f\u5206\u5e03\u306a\u306e\u3067\u3001$\u03bc=0$\u3001$\\sigma^2=1$\u3092\u4ee3\u5165\u3059\u308b\u3068\u6b21\u5f0f\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\\[f(x)=\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{x^2}{2}}\\]<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"764\" height=\"472\" src=\"https:\/\/aiknot.jp\/media-top\/wp-content\/uploads\/2024\/08\/N01.png\" alt=\"\" class=\"wp-image-4552\" style=\"width:453px;height:auto\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306e\uff11\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8<\/h3>\n\n\n\n<p>\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306b\u78ba\u7387\u5909\u6570\u3092\uff11\u56de\u639b\u3051\u305f\u3082\u306e\u304c\u3001\uff11\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8\uff08\u671f\u5f85\u5024\uff09\u306e\u5b9a\u7fa9\u5f0f\u3067\u3059\u3002<\/p>\n\n\n\n<p>\\[E(X^1)=\\displaystyle \\int_{-\\infty}^{\\infty}x^1\u00d7\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{x^2}{2}}dx\\]<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"764\" height=\"472\" src=\"https:\/\/aiknot.jp\/media-top\/wp-content\/uploads\/2024\/08\/1\u6b21-.png\" alt=\"\" class=\"wp-image-4572\" style=\"width:496px;height:auto\"\/><\/figure>\n\n\n\n<p>\u30bc\u30ed\u3092\u5883\u754c\u306b\u3057\u3066\u6b63\u8ca0\u304c\u53cd\u8ee2\u3057\u305f\u5de6\u53f3\u5bfe\u79f0\u306b\u306a\u3063\u3066\u3044\u308b\u305f\u3081\u3001\uff11\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8\uff08\u671f\u5f85\u5024\uff09\u3092\u8a08\u7b97\u3059\u308c\u3070\u30bc\u30ed\u306b\u306a\u308b\u4e8b\u304c\u5206\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306e\uff12\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8<\/h3>\n\n\n\n<p>\\[E(X^2)=\\displaystyle \\int_{-\\infty}^{\\infty}x^2\u00d7\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{x^2}{2}}dx\\]<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"764\" height=\"472\" src=\"https:\/\/aiknot.jp\/media-top\/wp-content\/uploads\/2024\/08\/2\u6b21.png\" alt=\"\" class=\"wp-image-4574\" style=\"width:504px;height:auto\"\/><\/figure>\n\n\n\n<p>\u5f8c\u8ff0\u306e\u300c\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306e\uff14\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8\u300d\u3067\u8a73\u7d30\u306a\u8a08\u7b97\u904e\u7a0b\u3092\u8a18\u8ff0\u3057\u3066\u3044\u307e\u3059\u3002\u540c\u69d8\u306e\u8a08\u7b97\u65b9\u6cd5\u3067\uff12\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8\u306f\uff11\u3068\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306e\uff13\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8<\/h3>\n\n\n\n<p>\\[E(X^3)=\\displaystyle \\int_{-\\infty}^{\\infty}x^3\u00d7\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{x^2}{2}}dx\\]<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"765\" height=\"472\" src=\"https:\/\/aiknot.jp\/media-top\/wp-content\/uploads\/2024\/08\/3\u6b21.png\" alt=\"\" class=\"wp-image-4575\" style=\"width:513px;height:auto\"\/><\/figure>\n\n\n\n<p>\u30b0\u30e9\u30d5\u306b\u3057\u3066\u30d7\u30ed\u30c3\u30c8\u3059\u308b\u3068\u3001\u5947\u6570\u30e2\u30fc\u30e1\u30f3\u30c8\u306f\u30bc\u30ed\u306b\u306a\u308b\u4e8b\u304c\u660e\u767d\u3067\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306e\uff14\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8<\/h3>\n\n\n\n<p>\\[E(X^4)=\\displaystyle \\int_{-\\infty}^{\\infty}x^4\u00d7\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{x^2}{2}}dx\\]<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"765\" height=\"473\" src=\"https:\/\/aiknot.jp\/media-top\/wp-content\/uploads\/2024\/08\/4\u6b21.png\" alt=\"\" class=\"wp-image-4577\" style=\"width:520px;height:auto\"\/><\/figure>\n\n\n\n<p>$=\\frac{1}{\\sqrt{2\\pi}}\\displaystyle \\int_{-\\infty}^{\\infty}x^4\u00d7e^{-\\frac{x^2}{2}}$<\/p>\n\n\n\n<p>\u90e8\u5206\u7a4d\u5206\u306e\u516c\u5f0f$\\displaystyle \\int_{-\\infty}^{\\infty}f(x)g(x)dx=f(x)G(x)-\\displaystyle \\int_{-\\infty}^{\\infty}f\u00b4(x)G(x)dx$\u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>$f(x)=x^3$<br>$g(x)=x\u00d7e^{-\\frac{x^2}{2}}$<\/p>\n\n\n\n<p>$g(x)$\u306f\u7a4d\u5206\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u306e\u3067\u3001\u305d\u306e\u969b\u5f8c\u306e$(3)$\u306e\u8a08\u7b97\u304c\u7c21\u5358\u306b\u306a\u308b\u3088\u3046\u8abf\u6574\u3057$x$\u3092\u3064\u3051\u307e\u3059\u3002<br>\u5148\u306b$G(x)$\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$G(x)=\\displaystyle \\int x\u00d7e^{-\\frac{x^2}{2}}dx$<\/p>\n\n\n\n<p>\u7f6e\u63db\u7a4d\u5206\u3092\u4f7f\u7528\u3057\u3001$t={-\\frac{x^2}{2}}$\u3068\u7f6e\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>$G(x)=\\displaystyle \\int x\u00d7e^tdx$$\u3000\u2026(1)$<\/p>\n\n\n\n<p>\u3053\u306e\u307e\u307e\u3067\u306f\u89e3\u3051\u306a\u3044\u305f\u3081\u3001t\u3092\u5fae\u5206\u3057\u3066\u4ee3\u5165\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$t={-\\frac{x^2}{2}}$<br>$t=-\\frac{1}{2}x^2$<br>$\\frac{dt}{dx}=-x$<br>$dx=-\\frac{1}{x}dt$$\u3000\u2026(2)$<\/p>\n\n\n\n<p>(2)\u3092(1)\u306b\u4ee3\u5165\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$G(x)=\\displaystyle \\int x\u00d7e^t\u00d7-\\frac{1}{x}dt$$\u3000\u2026(3)$<br>$=\\displaystyle \\int -e^tdt=-e^t$<\/p>\n\n\n\n<p>\u30cd\u30a4\u30d4\u30a2\u6570\u306a\u306e\u3067\u7a4d\u5206\u3057\u3066\u3082\u5909\u308f\u3089\u306a\u3044\u3002$t$\u3092$x$\u306b\u623b\u3059\u3002<\/p>\n\n\n\n<p>$G(x)=-e^{-\\frac{x^2}{2}}$$\u3000\u2026(4)$<\/p>\n\n\n\n<p>\u90e8\u5206\u7a4d\u5206\u306e\u516c\u5f0f$\\displaystyle \\int_{-\\infty}^{\\infty}f(x)g(x)dx=f(x)G(x)-\\displaystyle \\int_{-\\infty}^{\\infty}f\u00b4(x)G(x)dx$\u306b\u4ee3\u5165\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>$\\frac{1}{\\sqrt{2\\pi}}\\displaystyle \\int_{-\\infty}^{\\infty}x^3\u30fbx\u00d7e^{-\\frac{x^2}{2}}dx=\\frac{1}{\\sqrt{2\\pi}}\\{x^3\u30fb-e^{-\\frac{x^2}{2}}-\\displaystyle \\int_{-\\infty}^{\\infty}3x^2\u30fb-e^{-\\frac{x^2}{2}}dx\\}$<br>$=-\\frac{1}{\\sqrt{2\\pi}}\\{x^3\u30fbe^{-\\frac{x^2}{2}}\\}+3\u30fb\\frac{1}{\\sqrt{2\\pi}}\\{\\displaystyle \\int_{-\\infty}^{\\infty}x^2\u30fbe^{-\\frac{x^2}{2}}dx\\}$<\/p>\n\n\n\n<p>$\\{x^3\u30fbe^{-\\frac{x^2}{2}}\\}$\u3068\u3044\u3046\u95a2\u6570\u306b\u7740\u76ee\u3059\u308b\u3068\u3001$x$\u304c\u6975\u9650\u306b\u8fd1\u3065\u3044\u305f\u3068\u304d\u3001$e^{-\\frac{x^2}{2}}$\u304c\u5c0f\u3055\u304f\u306a\u308b\u30b9\u30d4\u30fc\u30c9\u304c$x^3$\u304c\u5927\u304d\u304f\u306a\u308b\u30b9\u30d4\u30fc\u30c9\u3088\u308a\u901f\u3044\u305f\u3081\u7121\u9650\u9060\u3067\u30bc\u30ed\u306b\u53ce\u675f\u3057\u3066\u6b21\u306e\u30b0\u30e9\u30d5\u3068\u306a\u308a\u3001\u7a4d\u5206\u5168\u4f53\u306b\u5bfe\u3059\u308b\u5f71\u97ff\u306f\u7121\u8996\u3067\u304d\u308b\u307b\u3069\u5c0f\u3055\u3044\u3068\u8a55\u4fa1\u3067\u304d\u308b\u305f\u3081\u3001\u7b2c\uff11\u9805\u306f\u30bc\u30ed\u3068\u306a\u308b\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized is-style-border\"><img decoding=\"async\" width=\"750\" height=\"449\" src=\"https:\/\/aiknot.jp\/media-top\/wp-content\/uploads\/2024\/08\/\u5408\u6210\u95a2\u6570.png\" alt=\"\" class=\"wp-image-4660\" style=\"width:538px;height:auto\"\/><\/figure>\n\n\n\n<p>$=3\u30fb\\frac{1}{\\sqrt{2\\pi}}\\{\\displaystyle \\int_{-\\infty}^{\\infty}x^2\u30fbe^{-\\frac{x^2}{2}}dx\\}$$\u2026(5)$<\/p>\n\n\n\n<p>(5)\u306f\u3001\u7a4d\u5206\u304c\u305d\u306e\u307e\u307e\u3067\u306f\u89e3\u3051\u306a\u3044\u306e\u3067\u3001\u518d\u5ea6\u90e8\u5206\u7a4d\u5206\u3092\u4f7f\u7528\u3059\u308b\u3002<\/p>\n\n\n\n<p>$f(x)=x$<br>$g(x)=x\u00d7e^{-\\frac{x^2}{2}}$<\/p>\n\n\n\n<p>\u524d\u56de\u306e\u7f6e\u63db\u7a4d\u5206\u3068$g(x)$\u306e\u5b9a\u7fa9\u304c\u540c\u3058\u3067\u3042\u308b\u305f\u3081\u3001(4)\u3092\u4f7f\u7528\u3057\u3066\u3053\u306e\u90e8\u5206\u7a4d\u5206\u3092\u89e3\u304f\u3068\u3001<\/p>\n\n\n\n<p>$\\displaystyle \\int_{-\\infty}^{\\infty}x^2\u30fbe^{-\\frac{x^2}{2}}dx=x\u30fb-e^{-\\frac{x^2}{2}}-\\displaystyle \\int_{-\\infty}^{\\infty}1\u30fb-e^{-\\frac{x^2}{2}}dx$<br>$=x\u30fb-e^{-\\frac{x^2}{2}}+\\displaystyle \\int_{-\\infty}^{\\infty}e^{-\\frac{x^2}{2}}dx$<\/p>\n\n\n\n<p>\u3084\u306f\u308a\u3001\u7b2c\uff11\u9805\u306f\u30bc\u30ed\u306b\u306a\u308b\u305f\u3081\u3001<\/p>\n\n\n\n<p>$=\\displaystyle \\int_{-\\infty}^{\\infty}e^{-\\frac{x^2}{2}}dx$$\u3000\u2026(6)$<\/p>\n\n\n\n<p>\u30ac\u30a6\u30b9\u7a4d\u5206\u306e\u95a2\u9023\u516c\u5f0f$\\displaystyle \\int_{-\\infty}^{\\infty}e^{-ax^2}=\\sqrt{\\frac{\\pi}{a}}$\u3092\u4f7f\u7528\u3057\u3066(6)\u3092\u89e3\u304f\u3002<\/p>\n\n\n\n<p>$=\\sqrt{\\frac{\\pi}{\\frac{1}{2}}}$<br>$=\\sqrt{2\\pi}$$\u3000\u2026(7)$<\/p>\n\n\n\n<p>(7)\u3092(5)\u306e\u53f3\u5074\u306b\u623b\u3059\u3068\u3001<\/p>\n\n\n\n<p>$=3\u30fb\\frac{1}{\\sqrt{2\\pi}}\\{\\sqrt{2\\pi}\\}$<br>$=3$<\/p>\n\n\n\n<p>\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306e\uff14\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8\u306f\uff13\u306b\u306a\u308b\u4e8b\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u30e2\u30fc\u30e1\u30f3\u30c8\u306e\u6027\u8cea<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">\u5947\u6570\u30e2\u30fc\u30e1\u30f3\u30c8<\/h3>\n\n\n\n<p>\u5947\u6570\u30e2\u30fc\u30e1\u30f3\u30c8\u306f\u30bc\u30ed\u3092\u5883\u754c\u306b\u6b63\u8ca0\u304c\u53cd\u8ee2\u3057\u3066\u3044\u308b\u5f62\u72b6\u3067\u3042\u308b\u305f\u3081\u3001\u7a4d\u5206\u3059\u308b\u3068\u30bc\u30ed\u306b\u306a\u308b\u4e8b\u304c\u660e\u3089\u304b\u3067\u3057\u305f\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u5076\u6570\u30e2\u30fc\u30e1\u30f3\u30c8<\/h3>\n\n\n\n<p>\u5076\u6570\u30e2\u30fc\u30e1\u30f3\u30c8\u306f\u3001\u4e57\u6570\u304c\u5897\u3048\u308b\u3068\u90e8\u5206\u7a4d\u5206\u306e\u8a08\u7b97\u5de5\u7a0b\u304c\u305d\u306e\u5206\u7e70\u308a\u8fd4\u3055\u308c\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002\u3088\u3063\u3066\u3001\u4e8c\u91cd\u968e\u4e57\u306b\u3088\u3063\u3066\u7f6e\u304d\u63db\u3048\u308b\u3053\u3068\u304c\u53ef\u80fd\u3067\u3042\u308a\u3001\u5076\u6570\u30e2\u30fc\u30e1\u30f3\u30c8\u306f\u6b21\u5f0f\u3067\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>\\[E(X^n)=\\frac{n!}{2^k\u30fbk!}\u3000n=2k,k=0,1,2,3,4,5,,,,\\]<\/p>\n\n\n\n<p>\u3053\u306e\u516c\u5f0f\u306f\u3001\u30d6\u30e9\u30c3\u30af\u30b7\u30e7\u30fc\u30eb\u30ba\u65b9\u7a0b\u5f0f\u306e\u5c0e\u51fa\u904e\u7a0b\u3067\u3001\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u304b\u3089\u4f0a\u85e4\u306e\u30ec\u30f3\u30de\u3092\u5c0e\u304f\u305f\u3081\u306b\u5229\u7528\u3057\u307e\u3059\u3002\u4ee5\u4e0b\u306b\u5f53\u8a72\u516c\u5f0f\u306e\u5c0e\u51fa\u904e\u7a0b\u3092\u793a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\\[E(X^n)=\\displaystyle \\int_{-\\infty}^{\\infty}x^n\u00d7\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{x^2}{2}}dx\\]<br>$=\\frac{1}{\\sqrt{2\\pi}}\\displaystyle \\int_{-\\infty}^{\\infty}x^n\u00d7e^{-\\frac{x^2}{2}}$<\/p>\n\n\n\n<p>\uff14\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u6c42\u3081\u305f\u6642\u3068\u540c\u69d8\u306b\u3001\u90e8\u5206\u7a4d\u5206\u3092\u4f7f\u7528\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$f(x)=x^{n-1}$<br>$g(x)=x\u00d7e^{-\\frac{x^2}{2}}$<br>$f\u00b4(x)=(n-1)x^{n-2}$<br>$G(x)=-e^{-\\frac{x^2}{2}}$<\/p>\n\n\n\n<p>$\\frac{1}{\\sqrt{2\\pi}}\\displaystyle \\int_{-\\infty}^{\\infty}x^{n-1}\u30fbx\u00d7e^{-\\frac{x^2}{2}}dx=\\frac{1}{\\sqrt{2\\pi}}\\{x^{n-1}\u30fb-e^{-\\frac{x^2}{2}}-\\displaystyle \\int_{-\\infty}^{\\infty}(n-1)x^{n-2}\u30fb-e^{-\\frac{x^2}{2}}dx\\}$<br>$=-\\frac{1}{\\sqrt{2\\pi}}\\{x^{n-1}\u30fbe^{-\\frac{x^2}{2}}\\}+(n-1)\u30fb\\frac{1}{\\sqrt{2\\pi}}\\{\\displaystyle \\int_{-\\infty}^{\\infty}x^{n-2}\u30fbe^{-\\frac{x^2}{2}}dx\\}$<br>$=(n-1)\u30fb\\frac{1}{\\sqrt{2\\pi}}\\{\\displaystyle \\int_{-\\infty}^{\\infty}x^{n-2}\u30fbe^{-\\frac{x^2}{2}}dx\\}$$\u3000\u2026(8)$<\/p>\n\n\n\n<p>\u3053\u3053\u3067\u3001$(8)$\u306e$(n-1)$\u3092\u9664\u304f\u53f3\u5074\u306f\u3001$E(X^{n-2})$\u306e\u5b9a\u7fa9\u5f0f\u306b\u4ed6\u306a\u3089\u306a\u3044\u305f\u3081\u3001<\/p>\n\n\n\n<p>$E(X^n)=(n-1)E(X^{n-2})$$\u3000\u2026(9)$<\/p>\n\n\n\n<p>\u3068\u7f6e\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u540c\u69d8\u306e\u8a08\u7b97\u3092\u7e70\u308a\u8fd4\u3059\u3068\u3001<\/p>\n\n\n\n<p>$E(X^{n-2})=(n-3)E(X^{n-4})$<br>$E(X^{n-4})=(n-5)E(X^{n-6})$<\/p>\n\n\n\n<p>\u3068\u306a\u308b\u305f\u3081\u3001\u3053\u308c\u3089\u3092$(9)$\u306b\u4ee3\u5165\u3059\u308b\u3002<\/p>\n\n\n\n<p>$E(X^n)=(n-1)(n-3)(n-5)\u2026E(X^2)$<\/p>\n\n\n\n<p>n\u306f\u5076\u6570\u3067\u3042\u308b\u305f\u3081\u3001\u6700\u7d42\u7684\u306bE(X^2)\u306b\u884c\u304d\u7740\u304f\u3002<br>\uff12\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8$E(X^2)$\u306f\u8a08\u7b97\u3059\u308b\u3068\uff11\u3067\u3042\u308b\u305f\u3081\u3001\u6b21\u306e\u901a\u308a\uff12\u91cd\u968e\u4e57\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>$E(X^n)=(n-1)(n-3)(n-5)\u20261$<\/p>\n\n\n\n<p>\u3053\u306e\u5f0f\u306f$n$\u304c\u5076\u6570\u306e\u6642\u3057\u304b\u6210\u308a\u7acb\u305f\u306a\u3044\u305f\u3081\u3001\u6b63\u3057\u304f\u8a18\u8f09\u3059\u308b\u3068\u521d\u3081\u306b\u793a\u3057\u305f\u516c\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\\[E(X^n)=\\frac{n!}{2^k\u30fbk!}\u3000n=2k,k=0,1,2,3,4,5,,,,\\]<\/p>\n\n\n\n<p><br><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u30e2\u30fc\u30e1\u30f3\u30c8\u3068\u306f\uff1f \u78ba\u7387\u5206\u5e03\u3092\u628a\u63e1\u3059\u308b\u305f\u3081\u306e\u5fc5\u8981\u306a\u8981\u7d20 \u78ba\u7387\u5206\u5e03\u306f\u6b63\u898f\u5206\u5e03\u306e\u3088\u3046\u306a\u5f62\u72b6\u3092\u3068\u3063\u3066\u3044\u306a\u3044\u3053\u3068\u3082\u591a\u304f\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u6642\u3001\u5f62\u72b6\u306e\u7279\u5fb4\u3092\u628a\u63e1\u3059\u308b\u305f\u3081\u306b\u6c42\u3081\u308b\u306e\u304c\u30e2\u30fc\u30e1\u30f3\u30c8\u3067\u3059\u3002 \u30e2\u30fc\u30e1\u30f3\u30c8\u306e\u5b9a\u7fa9 \u539f\u70b9\u4ed8\u8fd1\u306en\u6b21\u306e\u30e2\u30fc\u30e1 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":20,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"swell_btn_cv_data":"","footnotes":""},"categories":[27],"tags":[],"class_list":["post-4543","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-27"],"_links":{"self":[{"href":"https:\/\/aiknot.jp\/media-top\/index.php?rest_route=\/wp\/v2\/posts\/4543","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/aiknot.jp\/media-top\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/aiknot.jp\/media-top\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/aiknot.jp\/media-top\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/aiknot.jp\/media-top\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4543"}],"version-history":[{"count":142,"href":"https:\/\/aiknot.jp\/media-top\/index.php?rest_route=\/wp\/v2\/posts\/4543\/revisions"}],"predecessor-version":[{"id":4715,"href":"https:\/\/aiknot.jp\/media-top\/index.php?rest_route=\/wp\/v2\/posts\/4543\/revisions\/4715"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/aiknot.jp\/media-top\/index.php?rest_route=\/wp\/v2\/media\/20"}],"wp:attachment":[{"href":"https:\/\/aiknot.jp\/media-top\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4543"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/aiknot.jp\/media-top\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4543"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/aiknot.jp\/media-top\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4543"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}