{"id":34192,"date":"2021-06-13T13:00:53","date_gmt":"2021-06-13T13:00:53","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34192"},"modified":"2025-08-16T11:20:50","modified_gmt":"2025-08-16T11:20:50","slug":"%e7%a2%ba%e7%8e%87%e5%a4%89%e6%95%b0%e3%81%a8%e7%a2%ba%e7%8e%87%e5%88%86%e5%b8%83","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/ja\/%e7%a2%ba%e7%8e%87%e5%a4%89%e6%95%b0%e3%81%a8%e7%a2%ba%e7%8e%87%e5%88%86%e5%b8%83\/","title":{"rendered":"\u78ba\u7387\u5909\u6570\u3068\u78ba\u7387\u5206\u5e03"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>\u78ba\u7387\u5909\u6570\u3068\u78ba\u7387\u5206\u5e03<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>\u8981\u7d04<\/strong><br \/><em>\u672c\u8b1b\u7fa9\u3067\u306f\u3001\u78ba\u7387\u8ad6\u304a\u3088\u3073\u7d71\u8a08\u89e3\u6790\u306e\u57fa\u790e\u3092\u6210\u3059\u78ba\u7387\u5909\u6570\u3068\u78ba\u7387\u5206\u5e03\u306e\u6982\u5ff5\u3092\u6df1\u304f\u6398\u308a\u4e0b\u3052\u308b\u3002\u78ba\u7387\u5909\u6570\u3092\u300c\u78ba\u7387\u5b9f\u9a13\u306e\u7d50\u679c\u306b\u4f9d\u5b58\u3059\u308b\u6570\u300d\u3068\u3057\u3066\u5b9a\u7fa9\u3057\u3001\u305d\u306e\u5206\u5e03\u95a2\u6570\u3092\u5c0e\u5165\u3057\u3001\u305d\u306e\u91cd\u8981\u6027\u304a\u3088\u3073\u672c\u8cea\u7684\u6027\u8cea\u3092\u5f37\u8abf\u3059\u308b\u3002\u6700\u5f8c\u306b\u3001\u78ba\u7387\u5909\u6570\u3068\u78ba\u7387\u5206\u5e03\u306e\u95a2\u4fc2\u3092\u5206\u6790\u3057\u3001\u4e8c\u3064\u306e\u78ba\u7387\u5909\u6570\u304c\u540c\u3058\u5206\u5e03\u3092\u6301\u3061\u306a\u304c\u3089\u3082\u540c\u4e00\u306e\u78ba\u7387\u5909\u6570\u3067\u306f\u306a\u3044\u5834\u5408\u304c\u3042\u308b\u3053\u3068\u3092\u8aac\u660e\u3059\u308b\u3002<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>\u5b66\u7fd2\u76ee\u6a19:<\/strong><br \/>\n\u672c\u8b1b\u7fa9\u3092\u7d42\u3048\u308b\u6642\u70b9\u3067\u3001\u5b66\u751f\u306f\u6b21\u306e\u3053\u3068\u304c\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308b:\n<\/p>\n<ol>\n<li><strong>\u7406\u89e3\u3059\u308b<\/strong> \u78ba\u7387\u5909\u6570\u306e\u6982\u5ff5: \u5b66\u751f\u306f\u78ba\u7387\u5909\u6570\u3068\u306f\u4f55\u3067\u3042\u308a\u3001\u3069\u306e\u3088\u3046\u306b\u6570\u5b66\u7684\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u304b\u3092\u8a18\u8ff0\u3057\u3001\u8aac\u660e\u3067\u304d\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u3002<\/li>\n<li><strong>\u7406\u89e3\u3059\u308b<\/strong> \u78ba\u7387\u5206\u5e03\u306e\u6982\u5ff5: \u5b66\u751f\u306f\u78ba\u7387\u5206\u5e03\u3068\u306f\u4f55\u3067\u3042\u308a\u3001\u3069\u306e\u3088\u3046\u306b\u8868\u73fe\u3055\u308c\u308b\u304b\u3092\u8aac\u660e\u3067\u304d\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u3002<\/li>\n<li><strong>\u8a18\u8ff0\u3059\u308b<\/strong> \u78ba\u7387\u5206\u5e03\u306e\u6027\u8cea: \u5b66\u751f\u306f\u78ba\u7387\u5206\u5e03\u306e\u4e3b\u8981\u306a\u6027\u8cea\u3092\u8a8d\u8b58\u3057\u3001\u8aac\u660e\u3067\u304d\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u3002<\/li>\n<li><strong>\u5206\u6790\u3059\u308b<\/strong> \u78ba\u7387\u5909\u6570\u3068\u78ba\u7387\u5206\u5e03\u306e\u95a2\u4fc2: \u5b66\u751f\u306f\u78ba\u7387\u5909\u6570\u3068\u78ba\u7387\u5206\u5e03\u304c\u3069\u306e\u3088\u3046\u306b\u76f8\u4e92\u306b\u95a2\u9023\u3057\u3066\u3044\u308b\u304b\u3001\u307e\u305f\u4e8c\u3064\u306e\u78ba\u7387\u5909\u6570\u304c\u540c\u3058\u5206\u5e03\u3092\u6301\u3061\u306a\u304c\u3089\u3082\u540c\u4e00\u306e\u78ba\u7387\u5909\u6570\u3067\u306f\u306a\u3044\u53ef\u80fd\u6027\u306b\u3064\u3044\u3066\u8b70\u8ad6\u3067\u304d\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u3002<\/li>\n<li><strong>\u8a3c\u660e\u3057\u5fdc\u7528\u3059\u308b<\/strong> \u78ba\u7387\u5206\u5e03\u306e\u6027\u8cea\u3092\u5b9f\u969b\u306e\u72b6\u6cc1\u306b\u304a\u3044\u3066: \u5b66\u751f\u306f\u78ba\u7387\u5206\u5e03\u306e\u6027\u8cea\u3092\u6570\u5b66\u7684\u306b\u8a3c\u660e\u3057\u3001\u3053\u308c\u3089\u306e\u6027\u8cea\u3092\u73fe\u5b9f\u306e\u72b6\u6cc1\u306b\u5fdc\u7528\u3067\u304d\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u3002<\/li>\n<li><strong>\u7406\u89e3\u3059\u308b<\/strong> \u5206\u5e03\u95a2\u6570\u306e\u6982\u5ff5: \u5b66\u751f\u306f\u5206\u5e03\u95a2\u6570\u3068\u306f\u4f55\u3067\u3042\u308a\u3001\u78ba\u7387\u5909\u6570\u3092\u8a18\u8ff0\u3059\u308b\u305f\u3081\u306b\u3069\u306e\u3088\u3046\u306b\u7528\u3044\u3089\u308c\u308b\u304b\u3092\u8aac\u660e\u3067\u304d\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u3002<\/li>\n<\/ol>\n<p><center><br \/>\n<strong><u>\u76ee\u6b21<\/u>:<\/strong><br \/>\n<a href=\"#1\"><strong>\u78ba\u7387\u5909\u6570\u3068\u306f\u4f55\u304b?<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>\u78ba\u7387\u5206\u5e03\u3068\u306f\u4f55\u304b?<\/strong><\/a><br \/>\n<a href=\"#3\">\u78ba\u7387\u5206\u5e03\u306e\u6027\u8cea<\/a><br \/>\n<a href=\"#4\">\u78ba\u7387\u5909\u6570\u3068\u78ba\u7387\u5206\u5e03\u306e\u95a2\u4fc2<\/a><br \/>\n<\/center><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/gIKn9t1hnrw\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p style=\"text-align: justify; color:\">\u78ba\u7387\u8ad6\u304a\u3088\u3073\u7d71\u8a08\u89e3\u6790\u306b\u304a\u3051\u308b\u91cd\u8981\u306a\u6982\u5ff5\u306e\u4e00\u3064\u304c\u3001\u78ba\u7387\u5909\u6570\u3068\u78ba\u7387\u5206\u5e03\u3067\u3042\u308b\u3002\u3053\u308c\u307e\u3067\u306b\u5c55\u958b\u3057\u3066\u304d\u305f\u7406\u8ad6\u306f\u3042\u308b\u610f\u5473\u3067\u300c\u5b8c\u6210\u300d\u3057\u3066\u3044\u308b\u3082\u306e\u306e\u3001\u73fe\u72b6\u3067\u306f\u304b\u306a\u308a\u521d\u6b69\u7684\u306a\u6bb5\u968e\u306b\u3068\u3069\u307e\u3063\u3066\u3044\u308b\u3002\u78ba\u7387\u5909\u6570\u3068\u78ba\u7387\u5206\u5e03\u306f\u3001\u8a00\u308f\u3070\u300c\u78ba\u7387\u3092\u6271\u3044\u3001\u7d71\u8a08\u89e3\u6790\u3092\u884c\u3046\u305f\u3081\u306e\u80fd\u529b\u306b\u6f64\u6ed1\u6cb9\u3092\u4e0e\u3048\u308b\u300d\u6982\u5ff5\u3067\u3042\u308b\u3002<\/p>\n<p><a name=\"1\"><\/a><\/br><\/br><\/p>\n<h2>\u78ba\u7387\u5909\u6570\u3068\u306f\u4f55\u304b?<\/h2>\n<p style=\"text-align: justify; color:\"><strong><span style=\"color: #ff0000;\">\u78ba\u7387\u5909\u6570\u306e\u6982\u5ff5\u306b\u6163\u308c\u308b\u305f\u3081\u306b<\/span><\/strong>\u3001\u76f4\u89b3\u7684\u306a\u30a2\u30d7\u30ed\u30fc\u30c1\u304b\u3089\u59cb\u3081\u308b\u306e\u304c\u6709\u7528\u3067\u3042\u308b\u3002\u78ba\u7387\u5909\u6570\u306f\u300c\u78ba\u7387\u5b9f\u9a13\u306e\u7d50\u679c\u306b\u4f9d\u5b58\u3059\u308b\u6570\u300d\u3068\u89e3\u91c8\u3067\u304d\u308b\u3002\u3057\u304b\u3057\u3001\u3088\u308a\u6b63\u78ba\u306b\u7406\u89e3\u3059\u308b\u305f\u3081\u306b\u306f\u3001\u305d\u306e\u5f62\u5f0f\u7684\u5b9a\u7fa9\u3092\u63a2\u308b\u3053\u3068\u304c\u4e0d\u53ef\u6b20\u3067\u3042\u308b\u3002\u6b21\u306b\u305d\u306e\u5b9a\u7fa9\u3092\u793a\u3059:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify; color:\"><span style=\"color: #000080;\"><strong>\u5b9a\u7fa9:<\/strong><\/span> \u96c6\u5408 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{X}<\/span><\/span> \u4e0a\u306e\u78ba\u7387\u5909\u6570\u3068\u306f\u3001\u95a2\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f:\\Omega \\longmapsto \\mathcal{X}<\/span><\/span> \u3067\u3042\u308b\u3002<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color:\">\u6700\u3082\u4e00\u822c\u7684\u306a\u5834\u5408\u306f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{X}= \\mathbb{R},<\/span><\/span> \u3067\u3042\u308a\u3001\u7279\u306b\u65ad\u3089\u306a\u3044\u9650\u308a\u4eca\u5f8c\u306f\u3053\u308c\u3092\u4eee\u5b9a\u3059\u308b\u3002\u3059\u306a\u308f\u3061\u3001\u5b9f\u6570\u5024\u3092\u53d6\u308b\u78ba\u7387\u5909\u6570\u3092\u6271\u3046\u3002\u901a\u5e38\u3001\u78ba\u7387\u5909\u6570\u306f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X,Y,Z, \\cdots,<\/span><\/span> \u306e\u3088\u3046\u306a\u5927\u6587\u5b57\u3067\u8868\u3055\u308c\u3001\u5b9a\u6570\u306f\u5c0f\u6587\u5b57\u3067\u8868\u3055\u308c\u308b\u3002\u7c21\u5358\u306e\u305f\u3081\u3001\u78ba\u7387\u5909\u6570\u3092\u5358\u306b\u300c\u5909\u6570\u300d\u3068\u547c\u3076\u3053\u3068\u306b\u3059\u308b\u3002<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0e0ff;\">\n<p style=\"text-align: justify; color:\"><span style=\"color:\"><strong>\u4f8b:<\/strong><\/span> 6\u9762\u4f53\u306e\u30b5\u30a4\u30b3\u30ed\u30922\u56de\u6295\u3052\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3002\u3053\u306e\u3068\u304d\u6b21\u306e\u3088\u3046\u306b\u306a\u308b:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2d6} = \\{(\\omega_1, \\omega_2)\\;|\\; \\omega_1,\\omega_2 \\in \\{1,2,3,4,5,6\\}\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">\u3053\u3053\u304b\u3089\u6b21\u306e\u78ba\u7387\u5909\u6570\u3092\u5b9a\u7fa9\u3067\u304d\u308b:<\/p>\n<ul style=\"text-align: justify; color:\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=<\/span><\/span> \u300c1\u304c\u51fa\u305f\u56de\u6570\u300d<\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y=<\/span><\/span> \u300c\u5f97\u3089\u308c\u305f\u7d50\u679c\u306e\u548c\u300d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">=\\omega_1 + \\omega_2<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Z=<\/span><\/span> \u300c2\u56de\u76ee\u306e\u6295\u64f2\u306e\u7d50\u679c\u300d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">= \\omega_2<\/span><\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"2\"><\/a><\/br><\/br><\/p>\n<h2>\u78ba\u7387\u5206\u5e03\u3068\u306f\u4f55\u304b?<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify; color:\"><span style=\"color: #000080;\"><strong>\u5b9a\u7fa9: <\/strong><\/span><strong><span style=\"color: #ff0000;\">\u5206\u5e03\u95a2\u6570\uff08\u307e\u305f\u306f\u300cFD\u300d\uff09<\/span><\/strong> \u78ba\u7387\u5909\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u306e\u5206\u5e03\u95a2\u6570\u3068\u306f\u3001\u6b21\u306e\u95a2\u4fc2\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u95a2\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X: \\mathbb{R} \\longmapsto \\mathbb{R}<\/span><\/span> \u3067\u3042\u308b: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x) = P(\\{\\omega \\;|\\; X(\\omega)\\leq x\\}),<\/span><\/span> \u3042\u308b\u3044\u306f\u3088\u308a\u7c21\u6f54\u306b\u8868\u305b\u3070 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X\\leq x).<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color:\">\u4e00\u822c\u7684\u306b\u3001\u78ba\u7387\u5909\u6570\u306b\u95a2\u3057\u3066\u91cd\u8981\u306a\u306e\u306f\u3001\u6a19\u672c\u7a7a\u9593 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span><\/span> \u306b\u304a\u3051\u308b\u660e\u793a\u7684\u306a\u8868\u73fe\u305d\u306e\u3082\u306e\u3067\u306f\u306a\u304f\u3001\u305d\u306e\u5206\u5e03\u95a2\u6570\u3067\u3042\u308b\u3002\u6dfb\u5b57 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u306f\u3001\u6587\u8108\u304c\u660e\u78ba\u3067\u66d6\u6627\u3055\u304c\u306a\u3044\u5834\u5408\u306b\u306f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X<\/span><\/span> \u304b\u3089\u7701\u7565\u3055\u308c\u308b\u3053\u3068\u304c\u3042\u308b\u3002\u78ba\u7387\u5909\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u304c\u5206\u5e03\u95a2\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> \u3092\u6301\u3064\u3053\u3068\u3092\u793a\u3059\u305f\u3081\u306b\u3001\u3057\u3070\u3057\u3070 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim F<\/span><\/span> \u3068\u3044\u3046\u8868\u8a18\u304c\u7528\u3044\u3089\u308c\u308b\u3002<\/p>\n<p><a name=\"3\"><\/a><\/br><\/br><\/p>\n<h3>\u78ba\u7387\u5206\u5e03\u306e\u6027\u8cea<\/h3>\n<p style=\"text-align: justify; color:\"><a href=\"https:\/\/www.youtube.com\/watch?v=gIKn9t1hnrw&amp;t=806s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u3082\u3057 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> \u304c\u78ba\u7387\u5206\u5e03\u3067\u3042\u308b\u306a\u3089\u3070<\/span><\/strong><\/a>\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b<\/span><\/span> \u304c\u4efb\u610f\u306e\u5b9f\u6570\u3067\u3042\u308b\u3068\u304d\u3001\u6b21\u306e\u6027\u8cea\u304c\u6210\u308a\u7acb\u3064:<\/p>\n<p style=\"text-align: justify; color:\">(a) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b \\longrightarrow [P(a\\lt X \\leq b) = F(b) - F(a)]<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">(b) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b \\longrightarrow F(a) \\leq F(b),<\/span><\/span> \u3059\u306a\u308f\u3061\u300cF \u306f\u5358\u8abf\u5897\u52a0\u3067\u3042\u308b\u300d\u3002<\/p>\n<p style=\"text-align: justify; color:\">(c) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to +\\infty} F(x) = 1<\/span><\/span> \u304a\u3088\u3073 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to -\\infty} F(x) = 0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">(d) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X=x)=\\lim_{t\\to x^+}F(t) - \\lim_{t\\to x^-}F(t)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">(e) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle F(x)=\\lim_{t\\to x^+}F(t)<\/span><\/span><\/p>\n<table style=\"text-align: justify; color:\">\n<tbody>\n<tr>\n<td style=\"background-color: #ffe0e0;\"><span class=\"collapseomatic \" id=\"id69e42d17d738b\"  tabindex=\"0\" title=\"\u8a3c\u660e\"    >\u8a3c\u660e<\/span><div id=\"target-id69e42d17d738b\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify; color:\"><strong>(a)<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b<\/span><\/span> \u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\u3001\u30a4\u30d9\u30f3\u30c8 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A=\\{X\\leq a\\}<\/span><\/span> \u304a\u3088\u3073 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B=\\{X\\leq b\\}<\/span><\/span> \u3092\u8003\u3048\u308b\u3068\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B<\/span><\/span> \u304c\u6210\u308a\u7acb\u3064\u3002\u3057\u305f\u304c\u3063\u3066\u6b21\u306e\u3088\u3046\u306b\u306a\u308b:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{blue}{P(a\\lt X\\leq b)} = P(B\\setminus A) = P(B) - P(B\\cap A) = P(B)-P(A) =\\color{blue}{F(b) - F(a)}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\"><strong>(b)<\/strong> \u90e8\u5206 (a) \u3088\u308a\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B\\setminus A)\\geq 0<\/span><\/span> \u3067\u3042\u308b\u305f\u3081\u3001\u6b21\u304c\u6210\u308a\u7acb\u3064:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(b) - F(a) \\geq 0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">\u3053\u308c\u306f\u6b21\u306e\u3053\u3068\u3068\u540c\u5024\u3067\u3042\u308b:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(a) \\leq F(b)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\"><strong>(c)<\/strong> \u3053\u3053\u3067\u306f\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> \u304c\u5358\u8abf\u5897\u52a0\u3067\u3042\u308a\uff08(b) \u3067\u8a3c\u660e\u6e08\u307f\uff09\u3001\u6700\u5927\u5024\u304c\u300c1\u300d\u3067\u3042\u308b\uff08\u5206\u5e03\u306f\u78ba\u7387\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u305f\u3081\uff09\u3053\u3068\u3092\u7528\u3044\u308b\u3002\u3053\u306e\u3053\u3068\u304b\u3089\u6b21\u304c\u5f93\u3046:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty} F(x) = 1<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">\u3053\u308c\u306b\u88dc\u8db3\u7684\u306a\u30a2\u30d7\u30ed\u30fc\u30c1\u3092\u52a0\u3048\u308b\u3068\u3001\u540c\u3058\u7d50\u679c\u3092\u5f97\u308b\u6b21\u306e\u8a08\u7b97\u304c\u53ef\u80fd\u3068\u306a\u308b\u3002<\/p>\n<p style=\"text-align: justify; color:\">\u96c6\u5408\u3092\u5b9a\u7fa9\u3059\u308b: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A_n=\\{\\omega\\;|\\;X(\\omega)\\leq n\\}.<\/span><\/span> \u3053\u3053\u304b\u3089\u5bb9\u6613\u306b\u78ba\u8a8d\u3067\u304d\u308b\u306e\u306f\u3001\u4efb\u610f\u306e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> \u306b\u5bfe\u3057\u3066 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A_{n}\\subseteq A_{n+1}<\/span><\/span> \u304c\u6210\u308a\u7acb\u3061\u3001\u307e\u305f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\bigcup_{n\\lt +\\infty} A_n = \\Omega<\/span><\/span> \u3067\u3042\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u3001<a href=\"http:\/\/toposuranos.com\/material\/es\/teoremas-utiles-para-el-calculo-de-probabilidades\/\" rel=\"noopener\" target=\"_blank\">\u9023\u7d9a\u6027<\/a>\u306e\u6027\u8cea\u3092\u7528\u3044\u308b\u3068\u6b21\u304c\u5f97\u3089\u308c\u308b:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 1=P(\\Omega) = P\\left( \\bigcup_{n\\lt +\\infty} A_n \\right) = \\lim_{n\\to +\\infty} P(A_n) = \\lim_{n\\to +\\infty} P(\\{\\omega\\;|\\;X(\\omega)\\leq n\\}) = \\lim_{n\\to +\\infty} P(X\\leq n)=\\lim_{n\\to +\\infty}F(n)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">\u3059\u306a\u308f\u3061:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\lim_{x\\to +\\infty} F(x) = 1}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">\u4e00\u65b9\u3067\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to -\\infty<\/span><\/span> \u306e\u5834\u5408\u306b\u306f\u6b21\u306e\u3053\u3068\u304c\u6210\u308a\u7acb\u3064:<\/p>\n<p style=\"text-align: justify; color:\">\u307e\u305a\u96c6\u5408\u3092\u5b9a\u7fa9\u3059\u308b: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B_n=\\{\\omega\\;|\\;-n\\lt X(\\omega)\\}.<\/span><\/span> \u3053\u3053\u304b\u3089\u6b21\u304c\u78ba\u8a8d\u3067\u304d\u308b:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{n \\to -\\infty}F(n) = \\lim_{n\\to -\\infty} P(X\\leq n) = \\lim_{n\\to \\infty} P(X\\leq -n)= 1 - \\lim_{n\\to \\infty} P(-n \\lt X) = 1 - \\lim_{n\\to \\infty}P(B_n)) = 1 - P(\\Omega) = 1-1=0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\"><strong>(d)<\/strong> \u90e8\u5206 (c) \u3068\u540c\u69d8\u306b\u8b70\u8ad6\u3059\u308b\u3002\u307e\u305a\u6b21\u306e\u96c6\u5408\u3092\u5b9a\u7fa9\u3059\u308b:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle C_n = \\left\\{x - \\frac{1}{n} \\leq X \\leq x + \\frac{1}{n}\\right\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">\u3053\u3053\u304b\u3089\u6b21\u304c\u5f97\u3089\u308c\u308b:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> C_{n+1}\\subseteq C_n<\/span><\/span><\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcap_{n\\gt 0} C_n = \\{X=x\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">\u3057\u305f\u304c\u3063\u3066\u3001<a href=\"http:\/\/toposuranos.com\/material\/es\/teoremas-utiles-para-el-calculo-de-probabilidades\/\" rel=\"noopener\" target=\"_blank\">\u9023\u7d9a\u6027\u306e\u6027\u8cea\u306b\u57fa\u3065\u304f\u7d50\u679c<\/a>\u3092\u7528\u3044\u308b\u3068\u6b21\u304c\u5f97\u3089\u308c\u308b:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X=x)=P\\left(\\bigcap_{n\\gt 0} C_n \\right) = \\lim_{n\\to \\infty} P(C_n) = \\lim_{x+1\/n \\to x^+}F\\left(x+1\/n\\right) - \\lim_{x-1\/n \\to x^-}F\\left(x-1\/n\\right)= \\lim_{t \\to x^+}F\\left(t\\right) - \\lim_{t \\to x^-}F\\left(t\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\"><strong>(e)<\/strong> \u3053\u306e\u6700\u5f8c\u306e\u30b1\u30fc\u30b9\u306f\u524d\u306e\u7d50\u679c\u304b\u3089\u5f97\u3089\u308c\u308b\u3002\u5b9f\u969b\u3001\u3059\u3067\u306b\u6b21\u3092\u8a3c\u660e\u3057\u305f\u306e\u3067:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X=x)= \\lim_{t \\to x^+}F\\left(t\\right) - \\lim_{t \\to x^-}F\\left(t\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">\u3053\u308c\u3092\u66f8\u304d\u63db\u3048\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{t \\to x^+}F\\left(t\\right) = P(X=x) + \\lim_{t \\to x^-}F\\left(t\\right) = P(X=x) + \\lim_{t\\to x^-}P(X\\leq t)= P(X\\leq x) = F(x)<\/span><\/span><\/p>\n<\/div><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"4\"><\/a><\/br><\/br><\/p>\n<h3>\u78ba\u7387\u5909\u6570\u3068\u78ba\u7387\u5206\u5e03\u306e\u95a2\u4fc2<\/h3>\n<p style=\"text-align: justify; color:\">2\u3064\u306e\u78ba\u7387\u5909\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u3068 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y<\/span><\/span> \u304c\u540c\u3058\u78ba\u7387\u5206\u5e03\u3092\u6301\u3064\u3068\u306f\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall A\\subseteq \\mathbb{R})(P(X\\in A) = P(Y\\in A))<\/span><\/span> \u304c\u6210\u308a\u7acb\u3064\u5834\u5408\u3092\u3044\u3046\u3002<\/p>\n<p style=\"text-align: justify; color:\">\u540c\u3058\u6a19\u672c\u7a7a\u9593 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span><\/span> \u4e0a\u306b\u5b9a\u7fa9\u3055\u308c\u305f2\u3064\u306e\u78ba\u7387\u5909\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u3068 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y<\/span><\/span> \u306f\u3001\u540c\u3058\u5206\u5e03\u3092\u6301\u3064\u3053\u3068\u304c\u3042\u3063\u3066\u3082\u3001\u305d\u308c\u3060\u3051\u3067\u540c\u4e00\u306e\u78ba\u7387\u5909\u6570\u3067\u3042\u308b\u3068\u306f\u9650\u3089\u306a\u3044\u3002\u4f8b\u3048\u3070\u3001\u516c\u5e73\u306a\u30b3\u30a4\u30f3\u6295\u3052\u306e\u5b9f\u9a13\u3092\u8003\u3048\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=1<\/span><\/span> \u304c\u8868\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=0<\/span><\/span> \u304c\u88cf\u306b\u5bfe\u5fdc\u3059\u308b\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\u78ba\u7387\u5909\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y=1-X<\/span><\/span> \u3092\u5b9a\u7fa9\u3059\u308c\u3070\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X=1) = P(Y=1)=0.5<\/span><\/span> \u3068\u306a\u308a\u3001\u4e21\u8005\u306f\u540c\u3058\u5206\u5e03\u3092\u6301\u3064\u3002\u3057\u304b\u3057\u4e21\u8005\u304c\u540c\u3058\u5024\u3092\u3068\u308b\u78ba\u7387\u3092\u8a08\u7b97\u3059\u308b\u3068\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X=Y)=0<\/span><\/span> \u3068\u306a\u308b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u78ba\u7387\u5909\u6570\u3068\u78ba\u7387\u5206\u5e03 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\u5b66\u7fd2\u76ee\u6a19: \u672c\u8b1b\u7fa9\u3092\u7d42\u3048\u308b\u6642\u70b9\u3067\u3001\u5b66\u751f\u306f\u6b21\u306e\u3053\u3068\u304c\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308b: \u7406\u89e3\u3059\u308b \u78ba\u7387\u5909\u6570\u306e\u6982\u5ff5: \u5b66\u751f\u306f\u78ba\u7387\u5909\u6570\u3068\u306f\u4f55\u3067\u3042\u308a\u3001\u3069\u306e\u3088\u3046\u306b\u6570\u5b66\u7684\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u304b\u3092\u8a18\u8ff0\u3057\u3001\u8aac\u660e\u3067\u304d\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u3002 \u7406\u89e3\u3059\u308b \u78ba\u7387\u5206\u5e03\u306e\u6982\u5ff5: \u5b66\u751f\u306f\u78ba\u7387\u5206\u5e03\u3068\u306f\u4f55\u3067\u3042\u308a\u3001\u3069\u306e\u3088\u3046\u306b\u8868\u73fe\u3055\u308c\u308b\u304b\u3092\u8aac\u660e\u3067\u304d\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u3002 \u8a18\u8ff0\u3059\u308b \u78ba\u7387\u5206\u5e03\u306e\u6027\u8cea: 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