{"id":1005,"date":"2020-06-09T11:00:00","date_gmt":"2020-06-09T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1005"},"modified":"2020-06-06T19:27:07","modified_gmt":"2020-06-06T10:27:07","slug":"cumulative-distribution-function","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cumulative-distribution-function","title":{"rendered":"\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u3010\u610f\u5473\u3068\u6027\u8cea\u3092\u7d39\u4ecb\uff01\u3011"},"content":{"rendered":"\n<p class=\"has-text-align-right\"><span style=\"text-decoration: underline;\">\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u9ad8\u6821\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2606\u2606\u2606\u2606<\/span><\/p>\n\n\n\n\n\n\n\n<p>\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306f\u3042\u308b\u5024\u4ee5\u4e0b\u3092\u3068\u308b\u78ba\u7387\u306e\u3053\u3068\u3067\u3059\u3002\u30b5\u30a4\u30b3\u30ed\u306e\u4f8b\u3067\u3044\u304f\u3068\u30014\u4ee5\u4e0b\u304c\u51fa\u308b\u78ba\u7387\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u95a2\u6570\u304c\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306b\u3042\u305f\u308a\u307e\u3059\u3002\u3053\u306e\u8a18\u4e8b\u3067\u306f\u3001\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306e\u5b9a\u7fa9\u304a\u3088\u3073\u6027\u8cea\u3092\u7d39\u4ecb\u3057\u3066\u3044\u304d\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> \u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306e\u5b9a\u7fa9<\/a><ul><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_2\">1.1<\/span> \u96e2\u6563\u578b\u306e\u4f8b\uff08\u30b5\u30a4\u30b3\u30ed\u306e\u5834\u5408\uff09<\/a><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_2\">1.2<\/span> \u9023\u7d9a\u578b\u306e\u4f8b<\/a><\/li><\/ul><\/li><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_1\">2<\/span> \u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306e\u6027\u8cea<\/a><\/li><li><a href=\"#i-5\"><span class=\"toc_number toc_depth_1\">3<\/span> \u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306e\u307e\u3068\u3081<\/a><\/li><\/ul><\/div>\n<h3><span id=\"i\">\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306e\u5b9a\u7fa9<\/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>\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570(cumulative distribution function)<\/strong><\/div>\n<div style=\"background: #ffffea; padding: 10px; border: 2px solid #33cc33;\">\u78ba\u7387\u5909\u6570\\(X\\)\u306b\u5bfe\u5fdc\u3059\u308b\u78ba\u7387\u95a2\u6570\u30fb\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\\(f(x)\\)\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u78ba\u7387\u5909\u6570\u304c\\(x\\)\u4ee5\u4e0b\u306b\u306a\u308b\u78ba\u7387\u3092\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u3068\u3044\u3044\u3001\u6b21\u306e\u5f0f\u3067\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\n<div style=\"overflow-x: auto;\">\\begin{align}\n\uff08\u96e2\u6563\u578b\uff09&#038; F(x) = \\mathrm{P}(X\\leq x) = \\sum_{i=1}^{\\infty}f(x_{i})\\mathbf{1}_{\\{x_{i}\\leq x\\}} \\\\\n\uff08\u9023\u7d9a\u578b\uff09&#038; F(x) = \\mathrm{P}(X\\leq x) = \\int_{-\\infty}^{x}f(u)du\n\\end{align}<\/div>\n\u3053\u3053\u3067\u3001\\(\\mathbf{1}_{\\{x_{i}\\leq x\\}}\\)\u306f\\(x_{i}\\leq x\\)\u3092\u6e80\u305f\u3059\u3068\u304d\\(1\\)\u3001\u305d\u308c\u4ee5\u5916\u306f\\(0\\)\u3092\u3068\u308b\u5b9a\u7fa9\u95a2\u6570\u3067\u3059\u3002\n<\/div>\n<p>&nbsp;<\/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>\u5f0f\u3067\u898b\u308b\u3068\u96e3\u3057\u305d\u3046\u3067\u3059\u304c\u3001\u5b9f\u969b\u306b\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u3092\u6c42\u3081\u3066\u307f\u308b\u3068\u3001\u975e\u5e38\u306b\u7c21\u5358\u3067\u3059\u3002<\/p><div class=\"liquid-speech-balloon-arrow\" style=\"border-color:transparent #d6ebf8 transparent transparent\"><\/div><\/div><\/div>\n\n\n\n<h4><span id=\"i-2\">\u96e2\u6563\u578b\u306e\u4f8b\uff08\u30b5\u30a4\u30b3\u30ed\u306e\u5834\u5408\uff09<\/span><\/h4>\n\n\n\n<p>\u30b5\u30a4\u30b3\u30ed\u306e\u4f8b\u3067\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u3092\u6c42\u3081\u3066\u307f\u307e\u3057\u3087\u3046\uff01<br>\u307e\u305a\u3001\u78ba\u7387\u5909\u6570\u3068\u78ba\u7387\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u3066\u307f\u308b\u3068\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-table aligncenter\"><table><tbody><tr><td>\u78ba\u7387\u5909\u6570\\(X\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(1\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(2\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(3\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(4\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(5\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(6\\)<\/td><\/tr><tr><td>\u78ba\u7387\\(\\mathrm{P}(X)\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac{1}{6}\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac{1}{6}\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac{1}{6}\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac{1}{6}\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac{1}{6}\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac{1}{6}\\)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>\u30b5\u30a4\u30b3\u30ed\u306e\u76ee\u304c3\u4ee5\u4e0b\u306b\u306a\u308b\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff08\u3064\u307e\u308a\\(x\\leq 3\\)\u3068\u306a\u308b\u78ba\u7387\uff09\u306f$$\\mathrm{P}(X\\leq 3)=\\mathrm{P}(X=1)+\\mathrm{P}(X=2)+\\mathrm{P}(X=3)=\\frac{1}{2}$$\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h4><span id=\"i-3\">\u9023\u7d9a\u578b\u306e\u4f8b<\/span><\/h4>\n\n\n\n<p>\u6b21\u306e\u3088\u3046\u306a\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u8003\u3048\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n<p>\\[<br>f(x)=\\left\\{<div style=\"overflow-x: auto;\">\\begin{align}<br>&amp;2x\\ \\ (0\\leq x\\leq 1) \\\\<br>&amp;0\\ \\ \\ (x\u304c\u305d\u308c\u4ee5\u5916)<br>\\end{align}<\/div>\\right.<br>\\]<\/p>\n\n\n\n<p>\u3053\u308c\u306f\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u3002\u307e\u305a\u3069\u3093\u306a\\(x\\)\u306b\u5bfe\u3057\u3066\u3082\\(f(x)\\geq0\\)\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u305d\u3057\u3066\u5168\u7a4d\u5206\u3059\u308b\u3068<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\int_{-\\infty}^{\\infty}f(x)dx &amp; =\\int_{-\\infty}^{0}0dx+\\int_{0}^{1}2xdx+\\int_{1}^{\\infty}0dx \\\\<br>&amp;= \\left[ x^{2} \\right]_{0}^{1} \\\\<br>&amp;= 1<br>\\end{align}<\/div><br>\u3068\u306a\u308b\u306e\u3067\u3001\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306b\u306a\u3063\u3066\u3044\u308b\u3044\u308b\u3053\u3068\u306f\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u203b\u3000\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306b\u95a2\u3057\u3066\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/probability-function\">\uff1c\u78ba\u7387\u95a2\u6570\u30fb\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306e\u8a18\u4e8b\uff1e<\/a>\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<p>\u78ba\u7387\u5909\u6570\\(X\\)\u304c\\(u\\)\u4ee5\u4e0b\u306b\u306a\u308b\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\\(F(u)\\)\u3092\u8a08\u7b97\u3057\u3066\u307f\u308b\u3068<\/p>\n\n\n\n<p>\\[<br>F(u) = P(X\\leq u) =\\int_{-\\infty}^{u}f(x)dx <br>=<br>\\left\\{<div style=\"overflow-x: auto;\">\\begin{align}<br>&amp;0\\ \\ \\ (u\\leq 0)\\\\<br>&amp;u^{2}\\ \\ \\ (0&lt;u\\leq1)\\\\<br>&amp;1\\ \\ \\ (1&lt;u)<br>\\end{align}<\/div>\\right.<br>\\]<\/p>\n\n\n\n<p>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h3><span id=\"i-4\">\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306e\u6027\u8cea<\/span><\/h3>\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>\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306e\u6027\u8cea\u3092\u3044\u304f\u3064\u304b\u7d39\u4ecb\u3057\u3066\u3044\u304d\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>\uff11\uff0e\u78ba\u7387\u5909\u6570\u3067\\(X_{1}\\leq X_{2}\\)\u304c\u6210\u308a\u7acb\u3064\u3068\u304d\u3001\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\\(F(x)\\)\u306f\u5fc5\u305a$$F(X_{1})\\leq F(X_{2})$$\u3068\u306a\u308a\u307e\u3059\u3002\uff08\u3064\u307e\u308a<strong>\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306f\u5e83\u7fa9\u5358\u8abf\u5897\u52a0\u306b\u306a\u308a\u307e\u3059\u3002<\/strong>\uff09<\/p>\n\n\n\n<p>\u203b\u3000\u5e83\u7fa9\u5358\u8abf\u5897\u52a0\u306b\u3064\u3044\u3066\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/math\/2020\/05\/24\/05242255\/\" target=\"_blank\" rel=\"noreferrer noopener\">\uff1c\u5358\u8abf\u5897\u52a0\u30fb\u5358\u8abf\u6e1b\u5c11\u306e\u8a18\u4e8b\uff1e<\/a>\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<p>\uff12\uff0e\u78ba\u7387\u5909\u6570\\(X\\)\u306b\u3064\u3044\u3066\u3001\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\\(F(x)\\)\u3092\u7528\u3044\u3066\u6b21\u306e\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002$$\\mathrm{P}(a\\leq X\\leq b)=F(b)-F(a)$$<\/p>\n\n\n\n<h3><span id=\"i-5\">\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306e\u307e\u3068\u3081<\/span><\/h3>\n\n\n\n<p>\u3000\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306f\u3042\u308b\u5024\u4ee5\u4e0b\u3092\u3068\u308b\u78ba\u7387\u306e\u3053\u3068\u3067\u3042\u308a\u3001\u78ba\u7387\u95a2\u6570\u30fb\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u7528\u3044\u3066\u5b9a\u7fa9\u3067\u304d\u307e\u3059\u3002\u540c\u6642\u78ba\u7387\u3084\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u3067\u3082\u7528\u3044\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5909\u6570\u306b\u5bfe\u3057\u3066\u3001\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306f\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306e\u7a4d\u5206\u3067\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u304c\u3001\u9006\u306b\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u3092\u5fae\u5206\u3059\u308b\u3068\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u203b\u3000\u3064\u307e\u308a\u3001\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\\(f(x)\\)\u306f\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\\(F(x)\\)\u3092\u7528\u3044\u3066\u6b21\u306e\u3088\u3046\u306b\u8868\u305b\u307e\u3059\u3002$$f(x)=\\frac{d}{dx}F(x)$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u9ad8\u6821\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2606\u2606\u2606\u2606 \u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\u306f\u3042\u308b\u5024\u4ee5\u4e0b\u3092\u3068\u308b\u78ba\u7387\u306e\u3053\u3068\u3067\u3059\u3002\u30b5\u30a4\u30b3\u30ed\u306e\u4f8b\u3067\u3044\u304f\u3068\u30014\u4ee5\u4e0b\u304c\u51fa\u308b\u78ba\u7387\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u95a2 &#8230; <\/p>\n","protected":false},"author":1,"featured_media":829,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[54],"tags":[55,70],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/05\/135-money-images.jpg","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1005"}],"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=1005"}],"version-history":[{"count":29,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1005\/revisions"}],"predecessor-version":[{"id":1051,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1005\/revisions\/1051"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/829"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=1005"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=1005"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=1005"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}