{"id":1255,"date":"2020-06-21T11:00:00","date_gmt":"2020-06-21T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1255"},"modified":"2020-06-30T21:19:16","modified_gmt":"2020-06-30T12:19:16","slug":"cf","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cf","title":{"rendered":"\u7279\u6027\u95a2\u6570\u3068\u306f\u3010\u5b9a\u7fa9\u3068\u610f\u5473\u3092\u89e3\u8aac\u3011"},"content":{"rendered":"\n<p class=\"has-text-align-right\"><span style=\"text-decoration: underline;\">\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<p><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mgf\">\u7a4d\u7387\u6bcd\u95a2\u6570<\/a>\u3092\u4e00\u822c\u5316\u3057\u305f\u3082\u306e\u306b\u7279\u6027\u95a2\u6570\u304c\u3042\u308a\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u306f\u671f\u5f85\u5024\u304c\u5b58\u5728\u3059\u308b\u5206\u5e03\u3067\u306a\u3044\u3068\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u305b\u3093\uff08\u30b3\u30fc\u30b7\u30fc\u5206\u5e03\u306a\u3069\uff09\u3002\u3057\u304b\u3057\u3001\u7279\u6027\u95a2\u6570\u306f\u3069\u306e\u3088\u3046\u306a\u5206\u5e03\u306b\u5bfe\u3057\u3066\u3082\u5fc5\u305a\u5b58\u5728\u3057\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u304c\u3082\u3064\u6027\u8cea\u3092\u7279\u6027\u95a2\u6570\u3082\u6301\u3063\u3066\u3044\u308b\u306e\u3067\u3001\u304b\u306a\u308a\u4fbf\u5229\u306a\u95a2\u6570\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<div id=\"toc_container\" class=\"toc_light_blue no_bullets\"><p class=\"toc_title\">\u76ee\u6b21<\/p><ul class=\"toc_list\"><li><a href=\"#i\"><span class=\"toc_number toc_depth_1\">1<\/span> \u7279\u6027\u95a2\u6570\u306e\u5b9a\u7fa9<\/a><ul><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_2\">1.1<\/span> \u7279\u6027\u95a2\u6570\u3068\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8<\/a><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_2\">1.2<\/span> \u7279\u6027\u95a2\u6570\u3068\u5909\u6570\u5909\u63db<\/a><\/li><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_2\">1.3<\/span> \u7279\u6027\u95a2\u6570\u3068\u78ba\u7387\u5909\u6570\u306e\u548c<\/a><\/li><li><a href=\"#i-5\"><span class=\"toc_number toc_depth_2\">1.4<\/span> \u7279\u6027\u95a2\u6570\u3068\u78ba\u7387\u5206\u5e03<\/a><\/li><\/ul><\/li><li><a href=\"#i-6\"><span class=\"toc_number toc_depth_1\">2<\/span> \u53cd\u8ee2\u5b9a\u7406<\/a><\/li><li><a href=\"#i-7\"><span class=\"toc_number toc_depth_1\">3<\/span> \u30e2\u30fc\u30e1\u30f3\u30c8\u306b\u95a2\u3059\u308b\u95a2\u9023\u8a18\u4e8b<\/a><\/li><\/ul><\/div>\n<h2><span id=\"i\">\u7279\u6027\u95a2\u6570\u306e\u5b9a\u7fa9<\/span><\/h2>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u7279\u6027\u95a2\u6570(characteristic function(cf))<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\n\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u7279\u6027\u95a2\u6570\u306f\u8a18\u53f7\\(\\phi_{X}(t)\\)\u3067\u8868\u3057\u3001\u4e0b\u5f0f\u3067\u5b9a\u7fa9\u3057\u307e\u3059\u3002$$\\phi_{X}(t) = \\mathrm{E}\\left[ e^{itX} \\right] $$\u3053\u3053\u3067\\(i^{2}=-1\\)\u3067\u3059\u3002\n<\/div>\n<p>&nbsp;<\/p>\n\n\n\n<p>\u7279\u6027\u95a2\u6570\u304c\u5fc5\u305a\u5b58\u5728\u3059\u308b\u7406\u7531\u306a\u3093\u3067\u3059\u304c\u3001\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u516c\u5f0f\u304b\u3089$$e^{itX}=\\cos(tX)+i\\sin(tX)$$\u3092\u7528\u3044\u308c\u3070$$\\left| e^{itX} \\right|=\\cos^{2}(tX)+\\sin^{2}(tX)=1$$\u3068\u306a\u308b\u3053\u3068\u3092\u7528\u3044\u308b\u3068\u3001$$\\left| \\phi_{X}(t) \\right|&lt;1$$\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3068\u304d\\(\\phi_{X}(t)\\)\u304c\u7d76\u5bfe\u53ce\u675f\u3059\u308c\u3070\\(\\phi_{X}(t)\\)\u3082\u53ce\u675f\u3059\u308b\u3053\u3068\u304b\u3089\u7279\u6027\u95a2\u6570\u306f\u5fc5\u305a\u5b58\u5728\u3059\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"wp-block-liquid-speech-balloon liquid-speech-balloon-wrap liquid-speech-balloon-00  liquid-speech-balloon-short\"><div class=\"liquid-speech-balloon-avatar\"><\/div><div class=\"liquid-speech-balloon-text\" style=\"border-color:#d6ebf8;background-color:#d6ebf8;color:#000000\"><p>\u7279\u6027\u95a2\u6570\u306b\u95a2\u3057\u3066\u306f\u6b21\u306e\u3088\u3046\u306a\u6027\u8cea\u304c\u3042\u308a\u307e\u3059\u3002<\/p><div class=\"liquid-speech-balloon-arrow\" style=\"border-color:transparent #d6ebf8 transparent transparent\"><\/div><\/div><\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<!-- \u78ba\u7387\u6bcd\u95a2\u6570 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1927349886186426\" data-ad-slot=\"8307158523\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<h3><span id=\"i-2\">\u7279\u6027\u95a2\u6570\u3068\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8<\/span><\/h3>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u7279\u6027\u95a2\u6570\u3068\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\n\u7279\u6027\u95a2\u6570\\(M_{X}(t)\\)\u3092\u7528\u3044\u3066\u3001\\(k\\)\u6b21\u306e\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\\(\\mu_{k}^{\\prime}\\)\u3092\u6b21\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002$$\\mu_{k}^{\\prime}= \\left.(-i)^{k}\\frac{d^{k}}{dt^{k}}\\phi_{X}(t)\\right|_{t=0}$$\n<\/div>\n<p>&nbsp;<\/p>\n\n\n\n<p>\u203b\u3000\u30e2\u30fc\u30e1\u30f3\u30c8\u306b\u3064\u3044\u3066\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/moment\">\uff1c\u30e2\u30fc\u30e1\u30f3\u30c8\u306e\u8a18\u4e8b\uff1e<\/a>\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<p>\u3053\u306e\u5f0f\u306e\u8a08\u7b97\u65b9\u6cd5\u3067\u3059\u304c\u3001\u307e\u305a\\(\\phi_{X}(t)\\)\u3092\\(t\\)\u3067\\(k\\)\u56de\u5fae\u5206\u3057\u305f\u5f8c\u306b\\(t\\)\u306b\\(0\\)\u3092\u4ee3\u5165\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"wp-block-liquid-speech-balloon liquid-speech-balloon-wrap liquid-speech-balloon-00  liquid-speech-balloon-short\"><div class=\"liquid-speech-balloon-avatar\"><\/div><div class=\"liquid-speech-balloon-text\" style=\"border-color:#d6ebf8;background-color:#d6ebf8;color:#000000\"><p>\u8a3c\u660e\u81ea\u4f53\u306f\u3068\u3063\u3066\u3082\u7c21\u5358\u3067\u3059\u304c\u3001\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u306e\u77e5\u8b58\u304c\u5fc5\u8981\u306b\u306a\u308a\u307e\u3059\u3002<\/p><div class=\"liquid-speech-balloon-arrow\" style=\"border-color:transparent #d6ebf8 transparent transparent\"><\/div><\/div><\/div>\n\n\n\n<p>\\(e^{itX}\\)\u3092\\(t\\)\u306b\u3064\u3044\u3066\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3057\u3066\u307f\u308b\u3068<br><div style=\"overflow-x: auto;\">\\begin{align}<br>e^{itX} = 1+(it)X+\\frac{(it)^{2}}{2!}X^{2}+\\frac{(it)^{3}}{3!}X^{3}+\\cdots<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u7d50\u679c\u3092\u7528\u3044\u3066\u3001\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u66f8\u304d\u63db\u3048\u308b\u3068<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\phi_{X}(t) &amp;= \\mathrm{E}\\left[ 1+(it)X+\\frac{(it)^{2}}{2!}X^{2}+\\frac{(it)^{3}}{3!}X^{3}+\\cdots \\right] \\\\<br>&amp;= 1+it\\mathrm{E}\\left[ X \\right]+\\frac{(it)^{2}}{2!}\\mathrm{E}\\left[ X^{2} \\right] + \\frac{(it)^{3}}{3!}\\mathrm{E}\\left[ X^{3} \\right]+\\cdots\\\\<br>&amp;= 1+it\\mu_{1}^{\\prime}+\\frac{(it)^{2}}{2!}\\mu_{2}^{\\prime} + \\frac{(it)^{3}}{3!}\\mu_{3}^{\\prime}+\\cdots<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002\\(t\\)\u3067\\(k\\)\u56de\u5fae\u5206\u3059\u308b\u3068<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\frac{d^{k}}{dt^{k}}M_{X}(t) &amp;= i^{k}\\mu_{k}^{\\prime} +i^{k+1}t\\mu_{k+1}^{\\prime}+\\cdots<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u5f0f\u306b\\(t=0\\)\u3092\u4ee3\u5165\u3057\u4e21\u5074\u306b\\((-i)^{k}\\)\u3092\u304b\u3051\u308b\u3068<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\left.(-i)^{k}\\frac{d^{k}}{dt^{k}}M_{X}(t)\\right|_{t=0} = \\mu_{k}^{\\prime}<br>\\end{align}<\/div><br>\u304c\u6210\u7acb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n\n\n<h3><span id=\"i-3\">\u7279\u6027\u95a2\u6570\u3068\u5909\u6570\u5909\u63db<\/span><\/h3>\n\n\n\n<p>\u7279\u6027\u95a2\u6570\u306f\u7a4d\u7387\u6bcd\u95a2\u6570\u3068\u540c\u3058\u3088\u3046\u306a\u6027\u8cea\u3092\u6301\u3063\u3066\u3044\u307e\u3059\u3002\u4ee5\u964d\u306f\u7279\u6027\u95a2\u6570\u7248\u306e\u6027\u8cea\u3092\u7d39\u4ecb\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u7279\u6027\u95a2\u6570\u3068\u5909\u6570\u5909\u63db<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\n\u78ba\u7387\u5909\u6570\\(X\\)\u3092\u5b9a\u6570\\(a,b\\)\u3092\u7528\u3044\u3066\u5909\u6570\u5909\u63db\u3057\u305f\\(Y=a+bX\\)\u306e\u7279\u6027\u95a2\u6570\\(\\phi_{Y}(t)\\)\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002$$\\phi_{Y}(t) = e^{iat}\\phi_{X}(bt)$$\n<\/div>\n<p>&nbsp;<\/p>\n\n\n\n<p>\u5b9a\u7fa9\u3069\u304a\u308a\u306b\u8a08\u7b97\u3059\u308b\u3068\u6c42\u3081\u3089\u308c\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\phi_{Y}(t) &amp;= \\mathrm{E}\\left[ e^{itY} \\right] \\\\<br>&amp;= \\mathrm{E}\\left[ e^{it(a+bX)} \\right] \\\\<br>&amp;= e^{iat}\\mathrm{E}\\left[ e^{ibtX} \\right] \\\\<br>&amp;= e^{iat}\\phi_{X}(bt)<br>\\end{align}<\/div><br>\u3053\u308c\u3067\u3001\u5909\u6570\u5909\u63db\u3057\u305f\u78ba\u7387\u5909\u6570\u306e\u7279\u6027\u95a2\u6570\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h3><span id=\"i-4\">\u7279\u6027\u95a2\u6570\u3068\u78ba\u7387\u5909\u6570\u306e\u548c<\/span><\/h3>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u7279\u6027\u95a2\u6570\u3068\u78ba\u7387\u5909\u6570\u306e\u548c<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\n\u4e92\u3044\u306b\u72ec\u7acb\u306a\u78ba\u7387\u5909\u6570\\(X_{1},X_{2},\\cdots,X_{n}\\)\u306e\u7279\u6027\u95a2\u6570\u3092\u305d\u308c\u305e\u308c\\(\\phi_{X_{1}}(t),\\phi_{X_{2}}(t),\\cdots,\\phi_{X_{n}}(t)\\)\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\\(X=X_{1}+X_{2}+\\cdots+X_{n}\\)\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\\(\\phi_{X}(t)\\)\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002$$\\phi_{X}(t) = \\prod_{i=1}^{n}\\phi_{X_{i}}(t)$$\u7279\u306b\u3001\\(X_{1},X_{2},\\cdots,X_{n}\\)\u304c\u3059\u3079\u3066\u540c\u3058\u7279\u6027\u95a2\u6570\\(\\phi_{X^{\\prime}}(t)\\)\u3092\u6301\u3064\u3068\u304d$$\\phi_{X}(t) = \\left[\\phi_{X^{\\prime}}(t)\\right]^{n}$$\u3068\u306a\u308a\u307e\u3059\u3002\n<\/div>\n<p>&nbsp;<\/p>\n\n\n\n<p>\u3053\u306e\u5f0f\u306e\u8a3c\u660e\u3082\u5b9a\u7fa9\u3069\u304a\u308a\u306b\u8a08\u7b97\u3059\u308c\u3070\u3067\u3066\u304d\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\phi_{X}(t) &amp;= \\mathrm{E}\\left[ e^{itX} \\right] \\\\<br>&amp;= \\mathrm{E}\\left[ e^{it(X_{1}+X_{2}+\\cdots+X_{n})} \\right] \\\\<br>&amp;= \\mathrm{E}\\left[ e^{itX_{1}} \\right]\\mathrm{E}\\left[ e^{itX_{2}} \\right]\\cdots\\mathrm{E}\\left[ e^{itX_{n}} \\right] \\\\<br>&amp;= \\prod_{i=1}^{n}\\phi_{X_{i}}(t)<br>\\end{align}<\/div><\/p>\n\n\n\n\n\n<h3><span id=\"i-5\">\u7279\u6027\u95a2\u6570\u3068\u78ba\u7387\u5206\u5e03<\/span><\/h3>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u7279\u6027\u95a2\u6570\u3068\u78ba\u7387\u5206\u5e03<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\n\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u7279\u6027\u95a2\u6570\\(\\phi_{X}(t)\\)\u3068\u3001\u78ba\u7387\u5909\u6570\\(Y\\)\u306e\u7279\u6027\u95a2\u6570\\(\\phi_{Y}(t)\\)\u304c\\(t=0\\)\u306e\u8fd1\u508d\u3067\u7b49\u3057\u3051\u308c\u3070\u3001\\(X\\)\u3068\\(Y\\)\u306f\u540c\u3058\u5206\u5e03\u3092\u6301\u3061\u307e\u3059\u3002\n<\/div>\n<p>&nbsp;<\/p>\n\n\n\n<p>\u8a3c\u660e\u306f\u3084\u3084\u9762\u5012\u306a\u306e\u3067\u3001\u3053\u3053\u3067\u306f\u7701\u7565\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"wp-block-liquid-speech-balloon liquid-speech-balloon-wrap liquid-speech-balloon-00  liquid-speech-balloon-short\"><div class=\"liquid-speech-balloon-avatar\"><\/div><div class=\"liquid-speech-balloon-text\" style=\"border-color:#d6ebf8;background-color:#d6ebf8;color:#000000\"><p>\u3053\u306e\u6027\u8cea\u306f<strong>\u7279\u6027\u95a2\u6570\u304c\u7b49\u3057\u3051\u308c\u3070\u78ba\u7387\u5206\u5e03\u3082\u7b49\u3057\u3044\u3053\u3068\u3092\u4fdd\u8a3c\u3057\u3066\u3044\u307e\u3059\u3002<\/strong>\u7a4d\u7387\u6bcd\u95a2\u6570\u3067\u3082\u540c\u3058\u3053\u3068\u304c\u8a00\u3048\u307e\u3057\u305f\u304c\u3001\u7279\u6027\u95a2\u6570\u3067\u3082\u540c\u3058\u3053\u3068\u304c\u3067\u304d\u308b\u3053\u3068\u304c\u91cd\u8981\u3067\u3059\uff01<\/p><div class=\"liquid-speech-balloon-arrow\" style=\"border-color:transparent #d6ebf8 transparent transparent\"><\/div><\/div><\/div>\n\n\n\n<h2><span id=\"i-6\">\u53cd\u8ee2\u5b9a\u7406<\/span><\/h2>\n\n\n\n<p>\u7279\u6027\u95a2\u6570\u304b\u3089<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/probability-function\">\u78ba\u7387\u95a2\u6570\u30fb\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570<\/a>\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u53cd\u8ee2\u5b9a\u7406\u3068\u3044\u3046\u3082\u306e\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u7bc0\u3067\u306f\u53cd\u8ee2\u5b9a\u7406\u306b\u3064\u3044\u3066\u7d39\u4ecb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<div style=\"display: inline-block; background: #33cc33; padding: 5px 10px; color: #ffffff; border-radius: 5px 5px 0 0;\"><strong>\u53cd\u8ee2\u5b9a\u7406(inversion theorem)<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\n\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u7279\u6027\u95a2\u6570\\(\\phi_{X}(t)\\)\u306b\u3064\u3044\u3066$$\\int_{-\\infty}^{\\infty}\\phi_{X}(t)dt,\\ \\ \\ \\ \\int_{-\\infty}^{\\infty}\\left|\\phi_{X}(t)\\right|dt$$\u304c\u5b58\u5728\u3059\u308b\u3068\u304d\u3092\u8003\u3048\u307e\u3059\uff08\u3053\u306e\u3053\u3068\u3092\u7d76\u5bfe\u53ef\u7a4d\u5206\u3068\u3044\u3044\u307e\u3059\uff09\u3002<br>\n\u3000\u3053\u306e\u3068\u304d\u78ba\u7387\u95a2\u6570\u30fb\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f\u305d\u308c\u305e\u308c\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<br>\n\u30fb\u96e2\u6563\u578b\u306e\u78ba\u7387\u5909\u6570\u306e\u5834\u5408<br>\n\u3000\\(X\\)\u304c\u6574\u6570\u5024\u306e\u307f\u3092\u3068\u308b\u96e2\u6563\u578b\u306e\u78ba\u7387\u5909\u6570\u306e\u5834\u5408\u3001\u78ba\u7387\u95a2\u6570\\(p(x)\\)\u306f\u6b21\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002$$p(x) = \\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi}e^{-itx}\\phi_{X}(t)dt$$\n\u30fb\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5909\u6570\u306e\u5834\u5408<br>\n\u3000\\(X\\)\u304c\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5909\u6570\u306e\u5834\u5408\u3001\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\\(f(x)\\)\u306f\u6b21\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002$$f(x) = \\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}e^{-itx}\\phi_{X}(t)dt$$\n<\/div>\n<p>&nbsp;<\/p>\n\n\n\n<p>\u8a3c\u660e\u306f\u304b\u306a\u308a\u96e3\u3057\u3044\u306e\u3067\u7701\u7565\u3057\u307e\u3059\uff08\u304b\u306a\u308a\u6e96\u5099\u304c\u5fc5\u8981\u3067\u3059\uff09\u3002<\/p>\n\n\n\n<p>\u3000\u7279\u6027\u95a2\u6570\u306f\u307e\u3060\u307e\u3060\u305f\u304f\u3055\u3093\u6027\u8cea\u304c\u3042\u308b\u306e\u3067\u3059\u304c\u3001\u3053\u306e\u8a18\u4e8b\u3067\u306f\u5fc5\u8981\u6700\u4f4e\u9650\u306e\u6027\u8cea\u3092\u7d39\u4ecb\u3057\u307e\u3057\u305f\u3002\u5927\u6570\u306e\u6cd5\u5247\u3084\u4e2d\u5fc3\u6975\u9650\u5b9a\u7406\u306e\u8a3c\u660e\u306b\u3082\u4f7f\u7528\u3055\u308c\u308b\u3053\u3068\u304c\u3042\u308b\u306e\u3067\u3001\u7279\u6027\u95a2\u6570\u306f\u899a\u3048\u3066\u304a\u304d\u307e\u3057\u3087\u3046\uff01<\/p>\n\n\n\n<h2><span id=\"i-7\">\u30e2\u30fc\u30e1\u30f3\u30c8\u306b\u95a2\u3059\u308b\u95a2\u9023\u8a18\u4e8b<\/span><\/h2>\n\n\n\n<p>\u30fb<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/moment\">\u30e2\u30fc\u30e1\u30f3\u30c8<\/a>\uff1a\u30e2\u30fc\u30e1\u30f3\u30c8\u306b\u3064\u3044\u3066\u5b9a\u7fa9\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u30fb<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/pgf\">\u78ba\u7387\u6bcd\u95a2\u6570<\/a>\uff1a\u96e2\u6563\u578b\u78ba\u7387\u5909\u6570\u306b\u3064\u3044\u3066\\(k\\)\u6b21\u306e\u968e\u4e57\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u95a2\u6570\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u30fb<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mgf\">\u7a4d\u7387\u6bcd\u95a2\u6570<\/a>\uff1a\\(k\\)\u6b21\u306e\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u95a2\u6570\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u30fb<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cgf\">\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570<\/a>\uff1a\u30ad\u30e5\u30df\u30e5\u30e9\u30f3\u30c8\u3092\u7528\u3044\u3066\u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u30fb<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/gf-relationship\">\u6bcd\u95a2\u6570\u9593\u306e\u95a2\u4fc2\u306e\u307e\u3068\u3081<\/a>\uff1a\u6bcd\u95a2\u6570\u9593\u306e\u95a2\u4fc2\u3092\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u7a4d\u7387\u6bcd\u95a2\u6570\u3092\u4e00\u822c\u5316\u3057\u305f\u3082\u306e\u306b\u7279\u6027\u95a2\u6570\u304c\u3042\u308a\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u306f\u671f\u5f85\u5024\u304c\u5b58\u5728\u3059\u308b\u5206\u5e03\u3067\u306a\u3044\u3068\u6c42\u3081\u308b\u3053\u3068\u304c &#8230; <\/p>\n","protected":false},"author":1,"featured_media":1202,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[76],"tags":[77,80],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/9fc17caf-9cad-4670-b857-eb43ec911ab9.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1255"}],"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=1255"}],"version-history":[{"count":15,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1255\/revisions"}],"predecessor-version":[{"id":1851,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1255\/revisions\/1851"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/1202"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=1255"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=1255"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=1255"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}