{"id":26798,"date":"2021-06-13T13:00:41","date_gmt":"2021-06-13T13:00:41","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=26798"},"modified":"2024-05-22T03:40:34","modified_gmt":"2024-05-22T03:40:34","slug":"%e9%9a%8f%e6%9c%ba%e5%8f%98%e9%87%8f%e5%92%8c%e6%a6%82%e7%8e%87%e5%88%86%e5%b8%83","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/zh\/%e9%9a%8f%e6%9c%ba%e5%8f%98%e9%87%8f%e5%92%8c%e6%a6%82%e7%8e%87%e5%88%86%e5%b8%83\/","title":{"rendered":"\u968f\u673a\u53d8\u91cf\u548c\u6982\u7387\u5206\u5e03"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>\u968f\u673a\u53d8\u91cf\u548c\u6982\u7387\u5206\u5e03<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>\u6458\u8981<\/strong><br \/><em>\u8fd9\u95e8\u8bfe\u7a0b\u6df1\u5165\u4ecb\u7ecd\u4e86\u968f\u673a\u53d8\u91cf\u548c\u6982\u7387\u5206\u5e03\u7684\u6982\u5ff5\uff0c\u8fd9\u4e9b\u662f\u6982\u7387\u8bba\u548c\u7edf\u8ba1\u5206\u6790\u7684\u57fa\u7840\u3002\u8bfe\u7a0b\u4ecb\u7ecd\u4e86\u968f\u673a\u53d8\u91cf\u7684\u5b9a\u4e49\uff0c\u5373\u4f9d\u8d56\u4e8e\u968f\u673a\u5b9e\u9a8c\u7ed3\u679c\u7684\u6570\u503c\u3002\u8ba8\u8bba\u4e86\u968f\u673a\u53d8\u91cf\u7684\u5206\u5e03\u51fd\u6570\uff0c\u5f3a\u8c03\u4e86\u5176\u91cd\u8981\u6027\u53ca\u5176\u57fa\u672c\u5c5e\u6027\u3002\u6700\u540e\uff0c\u5206\u6790\u4e86\u968f\u673a\u53d8\u91cf\u4e0e\u6982\u7387\u5206\u5e03\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u89e3\u91ca\u4e86\u4e24\u4e2a\u53d8\u91cf\u53ef\u4ee5\u5177\u6709\u76f8\u540c\u7684\u5206\u5e03\u4f46\u4e0d\u662f\u540c\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u3002<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>\u5b66\u4e60\u76ee\u6807\uff1a<\/strong><br \/>\n\u5728\u672c\u8bfe\u7a0b\u7ed3\u675f\u65f6\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a\n<\/p>\n<ol>\n<li><strong>\u7406\u89e3<\/strong>\u968f\u673a\u53d8\u91cf\u7684\u6982\u5ff5\uff1a\u5b66\u751f\u5e94\u80fd\u63cf\u8ff0\u548c\u89e3\u91ca\u4ec0\u4e48\u662f\u968f\u673a\u53d8\u91cf\u53ca\u5176\u6570\u5b66\u5b9a\u4e49\u3002<\/li>\n<li><strong>\u7406\u89e3<\/strong>\u6982\u7387\u5206\u5e03\u7684\u6982\u5ff5\uff1a\u5b66\u751f\u5e94\u80fd\u89e3\u91ca\u4ec0\u4e48\u662f\u6982\u7387\u5206\u5e03\u4ee5\u53ca\u5982\u4f55\u8868\u793a\u3002<\/li>\n<li><strong>\u63cf\u8ff0<\/strong>\u6982\u7387\u5206\u5e03\u7684\u5c5e\u6027\uff1a\u5b66\u751f\u5e94\u80fd\u8bc6\u522b\u548c\u89e3\u91ca\u6982\u7387\u5206\u5e03\u7684\u5173\u952e\u5c5e\u6027\u3002<\/li>\n<li><strong>\u5206\u6790<\/strong>\u968f\u673a\u53d8\u91cf\u4e0e\u6982\u7387\u5206\u5e03\u4e4b\u95f4\u7684\u5173\u7cfb\uff1a\u5b66\u751f\u5e94\u80fd\u8ba8\u8bba\u968f\u673a\u53d8\u91cf\u4e0e\u6982\u7387\u5206\u5e03\u4e4b\u95f4\u7684\u76f8\u4e92\u5173\u7cfb\uff0c\u4ee5\u53ca\u4e24\u4e2a\u53d8\u91cf\u5982\u4f55\u5177\u6709\u76f8\u540c\u7684\u5206\u5e03\u4f46\u4e0d\u662f\u540c\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u3002<\/li>\n<li><strong>\u8bc1\u660e<\/strong>\u5e76\u5e94\u7528\u6982\u7387\u5206\u5e03\u7684\u5c5e\u6027\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff1a\u5b66\u751f\u5e94\u80fd\u6570\u5b66\u8bc1\u660e\u6982\u7387\u5206\u5e03\u7684\u5c5e\u6027\u5e76\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\u5e94\u7528\u8fd9\u4e9b\u5c5e\u6027\u3002<\/li>\n<li><strong>\u7406\u89e3<\/strong>\u5206\u5e03\u51fd\u6570\u7684\u6982\u5ff5\uff1a\u5b66\u751f\u5e94\u80fd\u63cf\u8ff0\u4ec0\u4e48\u662f\u5206\u5e03\u51fd\u6570\u4ee5\u53ca\u5982\u4f55\u7528\u6765\u63cf\u8ff0\u968f\u673a\u53d8\u91cf\u3002<\/li>\n<\/ol>\n<p><center><br \/>\n<strong><u>\u5185\u5bb9\u76ee\u5f55<\/u>:<\/strong><br \/>\n<a href=\"#1\"><strong>\u4ec0\u4e48\u662f\u968f\u673a\u53d8\u91cf\uff1f<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>\u4ec0\u4e48\u662f\u6982\u7387\u5206\u5e03\uff1f<\/strong><\/a><br \/>\n<a href=\"#3\">\u6982\u7387\u5206\u5e03\u7684\u5c5e\u6027<\/a><br \/>\n<a href=\"#4\">\u968f\u673a\u53d8\u91cf\u4e0e\u6982\u7387\u5206\u5e03\u7684\u5173\u7cfb<\/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: #000000;\">\u6982\u7387\u8bba\u548c\u7edf\u8ba1\u5206\u6790\u7684\u5173\u952e\u6982\u5ff5\u4e4b\u4e00\u662f\u968f\u673a\u53d8\u91cf\u548c\u6982\u7387\u5206\u5e03\u3002\u867d\u7136\u6211\u4eec\u5230\u76ee\u524d\u4e3a\u6b62\u5f00\u53d1\u7684\u7406\u8bba\u5728\u67d0\u79cd\u610f\u4e49\u4e0a\u662f\u201c\u5b8c\u6574\u7684\u201d\uff0c\u4f46\u5b9e\u9645\u4e0a\u5728\u76ee\u524d\u72b6\u6001\u4e0b\u5b83\u76f8\u5f53\u539f\u59cb\uff1b\u968f\u673a\u53d8\u91cf\u548c\u6982\u7387\u5206\u5e03\u662f\u4f7f\u6211\u4eec\u80fd\u591f\u201c\u63d0\u9ad8\u5904\u7406\u6982\u7387\u548c\u8fdb\u884c\u7edf\u8ba1\u5206\u6790\u80fd\u529b\u201d\u7684\u6982\u5ff5\u3002<\/p>\n<p><a name=\"1\"><\/a><\/br><\/br><\/p>\n<h2>\u4ec0\u4e48\u662f\u968f\u673a\u53d8\u91cf\uff1f<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><strong><span style=\"color: #ff0000;\">\u4e3a\u4e86\u719f\u6089\u968f\u673a\u53d8\u91cf\u7684\u6982\u5ff5<\/span><\/strong>\uff0c\u4ece\u76f4\u89c2\u7684\u65b9\u6cd5\u5f00\u59cb\u662f\u6709\u7528\u7684\uff1a\u53ef\u4ee5\u5c06\u968f\u673a\u53d8\u91cf\u89e3\u91ca\u4e3a\u201c\u4f9d\u8d56\u4e8e\u968f\u673a\u5b9e\u9a8c\u7ed3\u679c\u7684\u6570\u503c\u201d\u3002\u7136\u800c\uff0c\u4e3a\u4e86\u66f4\u7cbe\u786e\u7684\u7406\u89e3\uff0c\u5fc5\u987b\u8fdb\u4e00\u6b65\u63a2\u7d22\u5176\u6b63\u5f0f\u5b9a\u4e49\u3002\u6765\u770b\u8fd9\u4e2a\u5b9a\u4e49\uff1a<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000080;\"><strong>\u5b9a\u4e49\uff1a<\/strong><\/span> \u96c6\u5408 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{X}<\/span><\/span> \u4e0a\u7684\u968f\u673a\u53d8\u91cf\u662f\u4e00\u4e2a\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f:\\Omega \\longmapsto \\mathcal{X}<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">\u6700\u5e38\u89c1\u7684\u60c5\u51b5\u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{X}= \\mathbb{R},<\/span><\/span> \u5e76\u4e14\uff0c\u9664\u975e\u53e6\u6709\u8bf4\u660e\uff0c\u5426\u5219\u8fd9\u662f\u6211\u4eec\u5c06\u5047\u8bbe\u7684\u60c5\u51b5\uff1b\u5373\uff0c\u6211\u4eec\u5c06\u5904\u7406\u5177\u6709\u5b9e\u6570\u503c\u7684\u968f\u673a\u53d8\u91cf\u3002\u901a\u5e38\uff0c\u968f\u673a\u53d8\u91cf\u7528\u5927\u5199\u5b57\u6bcd\u8868\u793a\uff0c\u5982 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X,Y,Z, \\cdots,<\/span><\/span>\uff0c\u800c\u5e38\u6570\u7528\u5c0f\u5199\u5b57\u6bcd\u8868\u793a\u3002\u4e3a\u7b80\u5316\u8d77\u89c1\uff0c\u6211\u4eec\u5c06\u968f\u673a\u53d8\u91cf\u7b80\u5355\u79f0\u4e3a\u201c\u53d8\u91cf\u201d\u3002<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0e0ff;\">\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000000;\"><strong>\u4f8b\u5b50\uff1a<\/strong><\/span> \u5047\u8bbe\u63b7\u4e00\u4e2a\u516d\u9762\u9ab0\u5b50\u4e24\u6b21\u3002\u7136\u540e\u6211\u4eec\u5c06\u6709\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><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: #000000;\">\u57fa\u4e8e\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u4ee5\u4e0b\u968f\u673a\u53d8\u91cf\uff1a<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=<\/span><\/span> \u201c\u51fa\u73b0\u4e00\u6b21\u7684\u6b21\u6570\u201d<\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y=<\/span><\/span> \u201c\u7ed3\u679c\u7684\u603b\u548c\u201d <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> \u201c\u7b2c\u4e8c\u6b21\u63b7\u9ab0\u5b50\u7684\u7ed3\u679c\u201d <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>\u4ec0\u4e48\u662f\u6982\u7387\u5206\u5e03\uff1f<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000080;\"><strong>\u5b9a\u4e49\uff1a<\/strong><\/span><strong><span style=\"color: #ff0000;\">\u968f\u673a\u53d8\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u7684\u5206\u5e03\u51fd\u6570\uff08\u6216\u201cDF\u201d\uff09<\/span><\/strong> \u662f\u4e00\u4e2a\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X: \\mathbb{R} \\longmapsto \\mathbb{R}<\/span><\/span>\uff0c\u7531\u5173\u7cfb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x) = P(\\{\\omega \\;|\\; X(\\omega)\\leq x\\}),<\/span><\/span> \u6216\u66f4\u7b80\u6d01\u5730\u8868\u793a\u4e3a\uff1a<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: #000000;\">\u901a\u5e38\uff0c\u5bf9\u968f\u673a\u53d8\u91cf\u611f\u5174\u8da3\u7684\u4e0d\u5728\u4e8e\u5176\u5728\u6837\u672c\u7a7a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span><\/span> \u4e2d\u7684\u660e\u786e\u8868\u8fbe\uff0c\u800c\u662f\u5176\u5206\u5e03\u51fd\u6570\u3002\u5728\u4e0a\u4e0b\u6587\u660e\u786e\u4e14\u6ca1\u6709\u6b67\u4e49\u7684\u60c5\u51b5\u4e0b\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X<\/span><\/span> \u4e2d\u7684\u4e0b\u6807 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u53ef\u4ee5\u7701\u7565\u3002\u5e38\u7528\u7b26\u53f7 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim F<\/span><\/span> \u8868\u793a\u968f\u673a\u53d8\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u5177\u6709\u5206\u5e03\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F.<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/br><\/br><\/p>\n<h3>\u6982\u7387\u5206\u5e03\u7684\u5c5e\u6027<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=gIKn9t1hnrw&amp;t=806s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> \u662f\u4e00\u4e2a\u6982\u7387\u5206\u5e03<\/span><\/strong><\/a> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b<\/span><\/span> \u662f\u4efb\u610f\u5b9e\u6570\uff0c\u90a3\u4e48\u5c06\u6ee1\u8db3\u4ee5\u4e0b\u5c5e\u6027\uff1a<\/p>\n<p style=\"text-align: justify; color: #000000;\">(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: #000000;\">(b) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b \\longrightarrow F(a) \\leq F(b),<\/span><\/span> \u5373 \u201cF \u662f\u9012\u589e\u7684\u201d\u3002<\/p>\n<p style=\"text-align: justify; color: #000000;\">(c) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to +\\infty} F(x) = 1<\/span><\/span> \u548c <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: #000000;\">(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: #000000;\">(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: #000000;\">\n<tbody>\n<tr>\n<td style=\"background-color: #ffe0e0;\"><span class=\"collapseomatic \" id=\"id69e4336748cac\"  tabindex=\"0\" title=\"\u8bc1\u660e\"    >\u8bc1\u660e<\/span><div id=\"target-id69e4336748cac\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify; color: #000000;\"><strong>(a)<\/strong> \u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/span> \u5206\u522b\u4e3a\u4e8b\u4ef6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{X\\leq a\\}<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{X\\leq b\\}<\/span><\/span>\uff0c\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b.<\/span><\/span> \u5982\u679c\u6240\u6709\u8fd9\u4e9b\u90fd\u53d1\u751f\uff0c\u90a3\u4e48\u5c06\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B<\/span><\/span>\uff0c\u56e0\u6b64\u5c06\u4f1a\u53d1\u751f\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><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: #000000;\"><strong>(b)<\/strong> \u7531\u90e8\u5206 (a) \u53ef\u5f97\uff1a\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B\\setminus A)\\geq 0,<\/span><\/span> \u56e0\u6b64\u53ef\u5f97\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><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: #000000;\">\u5373<\/p>\n<p style=\"text-align: center; color: #000000;\"><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: #000000;\"><strong>(c)<\/strong> \u8fd9\u91cc\u6211\u4eec\u5c06\u4f7f\u7528 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> \u662f\u5355\u8c03\u9012\u589e\u7684\u4e8b\u5b9e\uff08\u5728 (b) \u4e2d\u5df2\u8bc1\u660e\uff09\u4e14\u5176\u6700\u5927\u503c\u4e3a\u201c1\u201d\uff08\u56e0\u4e3a\u5206\u5e03\u662f\u6839\u636e\u6982\u7387\u5b9a\u4e49\u7684\uff09\u3002\u4ec5\u6b64\u8db3\u4ee5\u8bf4\u660e\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><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: #000000;\">\u4e00\u79cd\u8865\u5145\u7684\u65b9\u6cd5\u5141\u8bb8\u6211\u4eec\u505a\u51fa\u76f8\u540c\u7ed3\u679c\u7684\u4ee5\u4e0b\u8ba1\u7b97\u3002<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u5b9a\u4e49\u96c6\u5408 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A_n=\\{\\omega\\;|\\;X(\\omega)\\leq n\\}.<\/span><\/span> \u7531\u6b64\u53ef\u4ee5\u5bb9\u6613\u5730\u9a8c\u8bc1\uff0c\u5bf9\u4e8e\u4efb\u610f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span>\uff0c\u5c06\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A_{n}\\subseteq A_{n+1},<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\bigcup_{n\\lt +\\infty} A_n = \\Omega<\/span><\/span>\uff0c\u56e0\u6b64\u4f7f\u7528<a href=\"http:\/\/toposuranos.com\/material\/zh\/%e6%a6%82%e7%8e%87%e8%ae%a1%e7%ae%97%e7%9a%84%e6%9c%89%e7%94%a8%e5%ae%9a%e7%90%86\/\" rel=\"noopener\" target=\"_blank\">\u8fde\u7eed\u6027<\/a>\u5c5e\u6027\u53ef\u5f97\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><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: #000000;\">\u5373\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><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: #000000;\">\u76f8\u53cd\uff0c\u5bf9\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to -\\infty<\/span><\/span> \u7684\u6781\u9650\uff0c\u6709\u5982\u4e0b\u60c5\u51b5\uff1a<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u9996\u5148\u5b9a\u4e49\u96c6\u5408 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B_n=\\{\\omega\\;|\\;-n\\lt X(\\omega)\\}.<\/span><\/span> \u7531\u6b64\u53ef\u4ee5\u9a8c\u8bc1\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><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: #000000;\"><strong>(d)<\/strong> \u7406\u7531\u4e0e\u90e8\u5206 (c) \u7c7b\u4f3c\u3002\u9996\u5148\u5b9a\u4e49\u96c6\u5408<\/p>\n<p style=\"text-align: center; color: #000000;\"><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: #000000;\">\u7531\u6b64\u53ef\u5f97\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><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: #000000;\"><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: #000000;\">\u56e0\u6b64\uff0c\u4f7f\u7528<a href=\"http:\/\/toposuranos.com\/material\/zh\/%e6%a6%82%e7%8e%87%e8%ae%a1%e7%ae%97%e7%9a%84%e6%9c%89%e7%94%a8%e5%ae%9a%e7%90%86\/\" rel=\"noopener\" target=\"_blank\">\u8fde\u7eed\u6027<\/a>\u5c5e\u6027\u7684\u4e00\u4e2a\u7ed3\u679c\u53ef\u5f97\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><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: #000000;\"><strong>(e)<\/strong> \u6700\u540e\u4e00\u79cd\u60c5\u51b5\u662f\u4ece\u524d\u9762\u7684\u7ed3\u679c\u5f97\u51fa\u7684\u3002\u4e8b\u5b9e\u4e0a\uff0c\u65e2\u7136\u6211\u4eec\u5df2\u7ecf\u8bc1\u660e\u4e86\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><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: #000000;\">\u6211\u4eec\u53ef\u4ee5\u5199\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><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>\u968f\u673a\u53d8\u91cf\u4e0e\u6982\u7387\u5206\u5e03\u7684\u5173\u7cfb<\/h3>\n<p style=\"text-align: justify; color: #000000;\">\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y<\/span><\/span> \u5177\u6709\u76f8\u540c\u7684\u6982\u7387\u5206\u5e03\uff0c\u5219 <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><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u5b9a\u4e49\u5728\u76f8\u540c\u6837\u672c\u7a7a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span><\/span> \u4e0a\u7684\u4e24\u4e2a\u53d8\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y<\/span><\/span> \u53ef\u80fd\u5177\u6709\u76f8\u540c\u7684\u5206\u5e03\uff0c\u4f46\u4e0d\u4e00\u5b9a\u662f\u76f8\u540c\u7684\u968f\u673a\u53d8\u91cf\u3002\u4f8b\u5982\uff0c\u5982\u679c\u6211\u4eec\u8003\u8651\u63b7\u4e00\u679a\u516c\u5e73\u7684\u53cc\u9762\u786c\u5e01\u5b9e\u9a8c\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=1<\/span><\/span> \u4ee3\u8868\u6b63\u9762\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=0<\/span><\/span> \u4ee3\u8868\u53cd\u9762\uff0c\u53ef\u4ee5\u5b9a\u4e49\u968f\u673a\u53d8\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y=1-X<\/span><\/span>\uff0c\u90a3\u4e48\u5c06\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X=1) = P(Y=1)=0.5,<\/span><\/span> \u4e14\u4e24\u8005\u5177\u6709\u76f8\u540c\u7684\u5206\u5e03\uff0c\u4f46\u5982\u679c\u8ba1\u7b97\u4e24\u8005\u5177\u6709\u76f8\u540c\u503c\u7684\u6982\u7387\uff0c\u5c06\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X=Y)=0<\/span><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u968f\u673a\u53d8\u91cf\u548c\u6982\u7387\u5206\u5e03 \u6458\u8981\u8fd9\u95e8\u8bfe\u7a0b\u6df1\u5165\u4ecb\u7ecd\u4e86\u968f\u673a\u53d8\u91cf\u548c\u6982\u7387\u5206\u5e03\u7684\u6982\u5ff5\uff0c\u8fd9\u4e9b\u662f\u6982\u7387\u8bba\u548c\u7edf\u8ba1\u5206\u6790\u7684\u57fa\u7840\u3002\u8bfe\u7a0b\u4ecb\u7ecd\u4e86\u968f\u673a\u53d8\u91cf\u7684\u5b9a\u4e49\uff0c\u5373\u4f9d\u8d56\u4e8e\u968f\u673a\u5b9e\u9a8c\u7ed3\u679c\u7684\u6570\u503c\u3002\u8ba8\u8bba\u4e86\u968f\u673a\u53d8\u91cf\u7684\u5206\u5e03\u51fd\u6570\uff0c\u5f3a\u8c03\u4e86\u5176\u91cd\u8981\u6027\u53ca\u5176\u57fa\u672c\u5c5e\u6027\u3002\u6700\u540e\uff0c\u5206\u6790\u4e86\u968f\u673a\u53d8\u91cf\u4e0e\u6982\u7387\u5206\u5e03\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u89e3\u91ca\u4e86\u4e24\u4e2a\u53d8\u91cf\u53ef\u4ee5\u5177\u6709\u76f8\u540c\u7684\u5206\u5e03\u4f46\u4e0d\u662f\u540c\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u3002 \u5b66\u4e60\u76ee\u6807\uff1a \u5728\u672c\u8bfe\u7a0b\u7ed3\u675f\u65f6\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a \u7406\u89e3\u968f\u673a\u53d8\u91cf\u7684\u6982\u5ff5\uff1a\u5b66\u751f\u5e94\u80fd\u63cf\u8ff0\u548c\u89e3\u91ca\u4ec0\u4e48\u662f\u968f\u673a\u53d8\u91cf\u53ca\u5176\u6570\u5b66\u5b9a\u4e49\u3002 \u7406\u89e3\u6982\u7387\u5206\u5e03\u7684\u6982\u5ff5\uff1a\u5b66\u751f\u5e94\u80fd\u89e3\u91ca\u4ec0\u4e48\u662f\u6982\u7387\u5206\u5e03\u4ee5\u53ca\u5982\u4f55\u8868\u793a\u3002 \u63cf\u8ff0\u6982\u7387\u5206\u5e03\u7684\u5c5e\u6027\uff1a\u5b66\u751f\u5e94\u80fd\u8bc6\u522b\u548c\u89e3\u91ca\u6982\u7387\u5206\u5e03\u7684\u5173\u952e\u5c5e\u6027\u3002 \u5206\u6790\u968f\u673a\u53d8\u91cf\u4e0e\u6982\u7387\u5206\u5e03\u4e4b\u95f4\u7684\u5173\u7cfb\uff1a\u5b66\u751f\u5e94\u80fd\u8ba8\u8bba\u968f\u673a\u53d8\u91cf\u4e0e\u6982\u7387\u5206\u5e03\u4e4b\u95f4\u7684\u76f8\u4e92\u5173\u7cfb\uff0c\u4ee5\u53ca\u4e24\u4e2a\u53d8\u91cf\u5982\u4f55\u5177\u6709\u76f8\u540c\u7684\u5206\u5e03\u4f46\u4e0d\u662f\u540c\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u3002 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