{"id":26874,"date":"2021-10-08T13:00:32","date_gmt":"2021-10-08T13:00:32","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=26874"},"modified":"2024-05-27T01:59:15","modified_gmt":"2024-05-27T01:59:15","slug":"%e8%bf%9e%e7%bb%ad%e6%a6%82%e7%8e%87%e5%88%86%e5%b8%83","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/zh\/%e8%bf%9e%e7%bb%ad%e6%a6%82%e7%8e%87%e5%88%86%e5%b8%83\/","title":{"rendered":"\u8fde\u7eed\u6982\u7387\u5206\u5e03"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>\u8fde\u7eed\u6982\u7387\u5206\u5e03<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>\u6458\u8981<\/strong><br \/><em>\u5728\u8fd9\u91cc\u6211\u4eec\u5c06\u6df1\u5165\u7814\u7a76\u8fde\u7eed\u6982\u7387\u5206\u5e03\u7684\u6982\u5ff5\uff0c\u91cd\u70b9\u4ecb\u7ecd\u4e94\u79cd\u6700\u8457\u540d\u7684\u5206\u5e03\u7684\u7279\u70b9\u548c\u7528\u9014\uff1a\u6307\u6570\u5206\u5e03\u3001\u77e9\u5f62\u5747\u5300\u5206\u5e03\u3001\u6b63\u6001\u5206\u5e03\uff08\u9ad8\u65af\u5206\u5e03\uff09\u3001\u5a01\u5e03\u5c14\u5206\u5e03\u548c\u4f3d\u9a6c\u5206\u5e03\u3002\u6211\u4eec\u5c06\u63d0\u4f9b\u5b9a\u4e49\u6bcf\u79cd\u5206\u5e03\u7684\u6570\u5b66\u516c\u5f0f\uff0c\u5e76\u7814\u7a76\u8fd9\u4e9b\u5206\u5e03\u7684\u5f71\u54cd\u548c\u5b9e\u9645\u5e94\u7528\uff0c\u5982\u8bc4\u4f30\u653e\u5c04\u6027\u6837\u54c1\u4e2d\u7684\u7c92\u5b50\u53d1\u5c04\u6216\u8ba1\u7b97\u8f68\u9053\u4e0a\u7403\u7684\u4f4d\u7f6e\u3002\u6b64\u5916\uff0c\u8fd8\u8be6\u7ec6\u4ecb\u7ecd\u4e86\u901a\u8fc7\u5e94\u7528\u7279\u5b9a\u53c2\u6570\u5982\u4f55\u4fee\u6539\u548c\u8c03\u6574\u8fd9\u4e9b\u5206\u5e03\u3002<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>\u5b66\u4e60\u76ee\u6807\uff1a<\/strong><br \/>\n\u5b8c\u6210\u672c\u8bfe\u540e\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a\n<\/p>\n<ol>\n<li><strong>\u7406\u89e3<\/strong>\u4ec0\u4e48\u662f\u8fde\u7eed\u6982\u7387\u5206\u5e03\u3002<\/li>\n<li><strong>\u5e94\u7528<\/strong>\u6700\u8457\u540d\u7684\u8fde\u7eed\u6982\u7387\u5206\u5e03\uff1a\u6307\u6570\u5206\u5e03\u3001\u77e9\u5f62\u5747\u5300\u5206\u5e03\u3001\u6b63\u6001\u5206\u5e03\uff08\u9ad8\u65af\u5206\u5e03\uff09\u3001\u5a01\u5e03\u5c14\u5206\u5e03\u548c\u4f3d\u9a6c\u5206\u5e03\u3002<\/li>\n<\/ol>\n<p><center><br \/>\n<strong><u>\u5185\u5bb9\u76ee\u5f55<\/u>\uff1a<\/strong><br \/>\n<a href=\"#1\"><strong>\u4ec0\u4e48\u662f\u8fde\u7eed\u6982\u7387\u5206\u5e03\uff1f<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>\u4e94\u79cd\u6700\u8457\u540d\u7684\u8fde\u7eed\u6982\u7387\u5206\u5e03<\/strong><\/a><br \/>\n<a href=\"#3\">\u6307\u6570\u5206\u5e03<\/a><br \/>\n<a href=\"#4\">\u77e9\u5f62\u5747\u5300\u5206\u5e03<\/a><br \/>\n<a href=\"#5\">\u6b63\u6001\u5206\u5e03\uff08\u9ad8\u65af\u5206\u5e03\uff09<\/a><br \/>\n<a href=\"#6\">\u5a01\u5e03\u5c14\u5206\u5e03<\/a><br \/>\n<a href=\"#7\">\u4f3d\u9a6c\u5206\u5e03<\/a><br \/>\n<a href=\"#8\"><strong>\u7ec3\u4e60<\/strong><\/a><br \/>\n<\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/REOTUa7K8uQ\" 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;\">\u5f53\u6211\u4eec\u56de\u987e\u5173\u4e8e <a href=\"http:\/\/toposuranos.com\/material\/zh\/%e4%ba%86%e8%a7%a3%e6%a6%82%e7%8e%87%e8%ae%ba%e7%9a%84%e6%a0%b7%e6%9c%ac%e7%a9%ba%e9%97%b4\/\" target=\"_blank\" rel=\"noopener\">\u6837\u672c\u7a7a\u95f4<\/a> \u65f6\uff0c\u6211\u4eec\u770b\u5230\u5b83\u4eec\u53ef\u4ee5\u5206\u4e3a\u4e24\u79cd\u7c7b\u578b\uff1a\u79bb\u6563\u7684\u548c\u8fde\u7eed\u7684\u3002\u6211\u4eec\u8fd8\u56de\u987e\u4e86\u4ec0\u4e48\u6784\u6210\u4e86 <a href=\"http:\/\/toposuranos.com\/material\/zh\/%e7%a6%bb%e6%95%a3%e6%a6%82%e7%8e%87%e5%88%86%e5%b8%83%e5%8f%8a%e7%a4%ba%e4%be%8b\/\" target=\"_blank\" rel=\"noopener\">\u79bb\u6563\u6982\u7387\u5206\u5e03\u3002<\/a> \u73b0\u5728\u8f6e\u5230\u8fde\u7eed\u6982\u7387\u5206\u5e03\u4e86\u3002<\/p>\n<p>&nbsp;<\/p>\n<p><a name=\"1\"><\/a><\/br><\/br><\/p>\n<h2>\u4ec0\u4e48\u662f\u8fde\u7eed\u6982\u7387\u5206\u5e03\uff1f<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=86s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u6211\u4eec\u8bf4\u4e00\u4e2a\u968f\u673a\u53d8\u91cf<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u5177\u6709\u8fde\u7eed\u6982\u7387\u5206\u5e03\uff0c\u5982\u679c\u5b58\u5728\u4e00\u4e2a\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f_X : \\mathbb{R} \\longrightarrow \\mathbb{R}^+,<\/span><\/span> \u6211\u4eec\u79f0\u4e4b\u4e3a <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u7684\u5bc6\u5ea6<\/strong>\uff0c\u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\forall A \\subseteq \\mathbb{R}<\/span><\/span> \u6ee1\u8db3\u7b49\u5f0f<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X\\in A) = \\displaystyle \\int_A f_X(x)dx<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u7279\u522b\u662f\uff0c\u5982\u679c\u6211\u4eec\u53d6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A=]a,b]<\/span><\/span> \u5219\u6709<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(a\\lt X \\leq b) = \\displaystyle \\int_a^b f_X(x)dx<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=-\\infty<\/span><\/span><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x) = P( X \\leq x) = \\displaystyle \\int_{-\\infty}^x f_X(t)dt<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u6b64\u5916\uff0c\u6839\u636e\u6982\u7387\u5206\u5e03\u7684\u6027\u8d28 (c) (<a href=\"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\/\" rel=\"noopener\" target=\"_blank\">\u89c1\u8fd9\u91cc<\/a>) \u6211\u4eec\u6709<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int_{-\\infty}^{+\\infty} f_X(t)dt = 1<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u901a\u8fc7\u5e94\u7528\u5fae\u79ef\u5206\u7684\u57fa\u672c\u5b9a\u7406\uff0c\u53ef\u4ee5\u5f97\u51fa\u5bf9\u4e8e\u4e00\u4e2a\u8fde\u7eed\u5206\u5e03\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x)<\/span><\/span> \u5bf9\u6240\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u90fd\u662f\u8fde\u7eed\u7684\uff0c\u5e76\u4e14\u5b83\u7684\u5bfc\u6570\u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f_X(x)<\/span><\/span> \u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f_X(x)<\/span><\/span> \u8fde\u7eed\u7684\u6240\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u5904\u3002\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x)<\/span><\/span> \u7684\u8fde\u7eed\u6027\u548c\u6027\u8d28 (d) (<a href=\"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\/\" rel=\"noopener\" target=\"_blank\">\u89c1\u8fd9\u91cc<\/a>) \u53ef\u4ee5\u5f97\u51fa\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x=X)=0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u56e0\u6b64<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x\\leq X)= P(x\\lt X)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u662f\u4efb\u4f55\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f\\geq 0<\/span><\/span> \u5e76\u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int_{-\\infty}^{+\\infty}f(x)dx = 1,<\/span><\/span> \u7684\u51fd\u6570\uff0c\u90a3\u4e48\u5b83\u5c31\u662f\u4e00\u4e2a\u5bc6\u5ea6\u51fd\u6570\u3002<\/p>\n<p><a name=\"2\"><\/a><\/br><\/br><\/p>\n<h2>\u4e94\u79cd\u6700\u8457\u540d\u7684\u8fde\u7eed\u6982\u7387\u5206\u5e03<\/h2>\n<p><a name=\"3\"><\/a><\/br><\/br><\/p>\n<h3>\u6307\u6570\u5206\u5e03<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=714s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u4e00\u4e2a\u5177\u6709\u53c2\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha \\gt 0 <\/span><\/span> \u7684\u6307\u6570\u5206\u5e03\u51fd\u6570<\/span><\/strong><\/a> \u662f\u4e00\u4e2a\u5f62\u5982 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> \u7684\u5206\u5e03\u51fd\u6570\u3002<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(t) = \\left\\{\\begin{array}{lll}\n\n1 - e^{-t\/\\alpha} &amp; ; &amp; t\\geq 0 \\\\ \\\\\n\n0 &amp; ; &amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u56e0\u6b64\uff0c\u5b83\u7684\u5bc6\u5ea6\u51fd\u6570\u662f\u5f62\u5982<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f(t) = \\left\\{\\begin{array}{lll}\n\n\\frac{1}{\\alpha}e^{-t\/\\alpha} &amp; ; &amp; t\\geq 0 \\\\ \\\\\n\n0 &amp; ; &amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u5982\u679c\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u5177\u6709\u53c2\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> \u7684\u6307\u6570\u5206\u5e03\uff0c\u6211\u4eec\u5199\u4f5c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim Ex(\\alpha).<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u5728\u6cca\u677e\u5206\u5e03\u7684\u80cc\u666f\u4e0b\uff0c\u5982\u679c\u6211\u4eec\u6709\u4e00\u4e2a\u653e\u5c04\u6027\u6837\u54c1\u4ee5\u5e73\u5747\u53d1\u5c04\u7387 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c,<\/span><\/span> \u53d1\u5c04\u7c92\u5b50\uff0c\u90a3\u4e48\u53d1\u5c04\u7b2c\u4e00\u4e2a\u7c92\u5b50\u7684\u65f6\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T<\/span><\/span> \u5177\u6709\u53c2\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1\/c.<\/span><\/span> \u7684\u6307\u6570\u5206\u5e03\u3002\u6362\u53e5\u8bdd\u8bf4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T\\sim Ex(1\/c),<\/span><\/span> \u56e0\u6b64\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(T\\geq t)= e^{-ct}<\/span><\/span><\/p>\n<p><a name=\"4\"><\/a><\/br><\/br><\/p>\n<h3>\u77e9\u5f62\u5747\u5300\u5206\u5e03<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=930s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u4e00\u4e2a\u5728\u533a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> \u4e0a\u7684\u77e9\u5f62\u5747\u5300\u5206\u5e03<\/span><\/strong><\/a> \u662f\u7531\u5bc6\u5ea6\u51fd\u6570\u5b9a\u4e49\u7684\u5206\u5e03<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\left\\{\\begin{array}{lll}\n\n\\displaystyle\\frac{1}{b-a} &amp; ; &amp; x\\in[a,b] \\\\ \\\\\n\n0 &amp; ; &amp; \u5176\u4ed6\u60c5\u51b5\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u5982\u679c\u6211\u4eec\u5728\u533a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b],<\/span><\/span> \u7684\u4e24\u7aef\u6709\u754c\u9650\u7684\u8f68\u9053\u4e0a\u91ca\u653e\u4e00\u4e2a\u5c0f\u7403\uff0c\u5e76\u4e14\u5b83\u5f39\u6027\u78b0\u649e\u8fb9\u754c\uff0c\u90a3\u4e48\u7531\u4e8e\u6469\u64e6\u6548\u5e94\uff0c\u505c\u6b62\u4f4d\u7f6e\u7684\u968f\u673a\u53d8\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u5177\u6709\u77e9\u5f62\u5747\u5300\u5206\u5e03\uff0c\u5199\u4f5c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim Un(a,b)<\/span>.<\/span><\/p>\n<p><a name=\"5\"><\/a><\/br><\/br><\/p>\n<h3>\u6b63\u6001\u5206\u5e03\uff08\u9ad8\u65af\u5206\u5e03\uff09<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=1109s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u5728\u8fde\u7eed\u6982\u7387\u5206\u5e03\u4e2d<\/span><\/strong><\/a>\uff0c\u6b63\u6001\u5206\u5e03\u662f\u5b9e\u8df5\u4e2d\u6700\u6d41\u884c\u7684\u4e00\u79cd\u3002<\/p>\n<h4>\u6807\u51c6\u6b63\u6001\u5206\u5e03<\/h4>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=1150s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u6807\u51c6\u6b63\u6001\u5bc6\u5ea6<\/span><\/strong><\/a> \u5b9a\u4e49\u4e3a\u51fd\u6570<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\phi_{0,1}(x) = \\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u6839\u636e\u5b9a\u4e49\uff0c\u5f88\u660e\u663e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi\\gt 0.<\/span><\/span> \u56e0\u6b64\uff0c\u53ef\u4ee5\u9a8c\u8bc1\u8fd9\u662f\u4e00\u4e2a\u6982\u7387\u5bc6\u5ea6\uff0c\u53ea\u9700\u8bc1\u5b9e<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int_{-\\infty}^{+\\infty}\\phi_{0,1}(x)dx<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u6700\u540e\u4e00\u4e2a\u7b49\u5f0f\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I^2<\/span><\/span> \u7684\u503c\u6765\u8bc1\u660e\uff0c\u5f53 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I =\\int_{-\\infty}^{+\\infty}\\phi(x)dx=1.<\/span><\/span> \u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u6709\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nI^2 &amp; = \\displaystyle \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2} dx \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2}dx \\\\ \\\\\n\n&amp; = \\displaystyle \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2} dx \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-y^2\/2} dy \\\\ \\\\\n\n&amp; = \\displaystyle \\frac{1}{{2\\pi}} \\int_{-\\infty}^{+\\infty} \\int_{-\\infty}^{+\\infty} e^{-\\frac{x^2 + y^2}{2}} dxdy \\\\ \\\\\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u7ed3\u679c\u662f<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle \\int_{-\\infty}^{+\\infty} \\int_{-\\infty}^{+\\infty} e^{-\\frac{x^2 + y^2}{2}} dxdy = \\int_{0}^{2\\pi} \\int_{0}^{+\\infty} e^{-r^2\/2} rdr d\\theta = 2\\pi <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u56e0\u6b64 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I^2 = 1,<\/span><\/span> \u6240\u4ee5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I=\\int_{-\\infty}^{+\\infty}\\phi_{0,1}(x)dx = 1. <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u4ece\u6807\u51c6\u6b63\u6001\u5bc6\u5ea6\u5b9a\u4e49\u6807\u51c6\u6b63\u6001\u5206\u5e03 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(x) = \\int_{-\\infty}^x\\phi_{0,1}(t)dt.<\/span><\/span> \u5982\u679c\u4e00\u4e2a\u968f\u673a\u53d8\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u5177\u6709\u6807\u51c6\u6b63\u6001\u5206\u5e03\uff0c\u90a3\u4e48\u5199\u4f5c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim N(0,1).<\/span><\/span> \u5206\u5e03 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(x)<\/span><\/span> \u4e0d\u80fd\u663e\u5f0f\u8ba1\u7b97\uff0c\u4f46\u662f\u6709\u4e00\u4e9b\u8868\u683c\u53ef\u4ee5\u5feb\u901f\u83b7\u5f97\u8fd1\u4f3c\u503c\u3002<\/p>\n<h4>\u5177\u6709\u53c2\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma<\/span><\/span> \u7684\u6b63\u6001\u5206\u5e03<\/h4>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=1875s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u4ece\u6807\u51c6\u6b63\u6001\u5206\u5e03\u7684\u5bc6\u5ea6<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_{0,1}<\/span><\/span> \u53ef\u4ee5\u6784\u9020\u5177\u6709\u53c2\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma,<\/span><\/span> \u7684\u6b63\u6001\u5206\u5e03\u5bc6\u5ea6\uff0c\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu\\in\\mathbb{R}<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma\\gt 0 <\/span><\/span> \u5206\u522b\u662f\u5747\u503c\u548c\u6807\u51c6\u5dee\u3002\u5177\u6709\u8fd9\u4e9b\u53c2\u6570\u7684\u6b63\u6001\u5206\u5e03\u5bc6\u5ea6\u5199\u4e3a\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\phi_{\\mu,\\sigma}(x) = \\frac{1}{\\sigma}\\phi_{0,1}\\left(\\frac{x-\\mu}{\\sigma} \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u56e0\u6b64\uff0c\u5177\u6709\u53c2\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma,<\/span><\/span> \u7684\u6b63\u6001\u5206\u5e03 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{\\mu,\\sigma}(x)<\/span><\/span>\uff0c\u5f62\u5f0f\u5982\u4e0b<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\Phi_{\\mu,\\sigma}(x) = \\int_{-\\infty}^x\\frac{1}{\\sigma}\\phi_{0,1}\\left(\\frac{t-\\mu}{\\sigma} \\right)dt = \\frac{1}{\\sqrt{2\\pi\\sigma}}\\int_{-\\infty}^x e^{-\\frac{(t-\\mu)^2}{2\\sigma^2}}dt<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u5982\u679c\u968f\u673a\u53d8\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u5177\u6709\u53c2\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu, \\sigma,<\/span><\/span> \u7684\u6b63\u6001\u5206\u5e03\uff0c\u90a3\u4e48\u5199\u4f5c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim N(\\mu, \\sigma).<\/span><\/span><\/p>\n<p><a name=\"6\"><\/a><\/br><\/br><\/p>\n<h3>\u5a01\u5e03\u5c14\u5206\u5e03<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=2230s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u5a01\u5e03\u5c14\u5206\u5e03<\/span><\/strong><\/a> \u5177\u6709\u53c2\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta \\gt 0<\/span><\/span> \u7684\u5206\u5e03\u51fd\u6570\u5f62\u5f0f\u5982\u4e0b<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(t) = \\left\\{\\begin{array}{llr}\n\n\\left(1 - e^{-t\/\\alpha} \\right)^\\beta &amp; ; &amp; t\\geq 0 \\\\ \\\\\n\n0 &amp; ; &amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u5982\u679c\u968f\u673a\u53d8\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u5177\u6709\u53c2\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta<\/span><\/span> \u7684\u5a01\u5e03\u5c14\u5206\u5e03\uff0c\u5199\u4f5c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim We(\\alpha,\\beta).<\/span><\/span> \u5a01\u5e03\u5c14\u5206\u5e03\u662f\u6307\u6570\u5206\u5e03\u7684\u4e00\u822c\u5316\uff0c\u6ce8\u610f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">We(\\alpha,1) = Ex(\\alpha).<\/span><\/span><\/p>\n<p><a name=\"7\"><\/a><\/br><\/br><\/p>\n<h3>\u4f3d\u9a6c\u5206\u5e03<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=2311s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u4f3d\u9a6c\u5206\u5e03<\/span><\/strong><\/a> \u5177\u6709\u53c2\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta,\\alpha<\/span><\/span> \u7684\u5bc6\u5ea6\u51fd\u6570\u5f62\u5f0f\u5982\u4e0b<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(t) = \\left\\{\\begin{array}{llr}\n\n\\displaystyle \\frac{1}{\\alpha \\Gamma(\\beta)}\\left(\\frac{t}{\\alpha} \\right)^{\\beta-1}e^{-t\/\\alpha} &amp; ; &amp; t\\geq 0 \\\\ \\\\\n\n0 &amp; ; &amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma(s) = \\displaystyle \\int_0^{+\\infty}u^{s-1}e^{-u}du <\/span><\/span> \u88ab\u79f0\u4e3a\u201c\u4f3d\u9a6c\u51fd\u6570\u201d\u3002<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u4f3d\u9a6c\u51fd\u6570\u7684\u4e00\u4e2a\u663e\u8457\u7279\u6027\u662f\u5b83\u5141\u8bb8\u5c06\u81ea\u7136\u6570\u7684\u9636\u4e58\u63a8\u5e7f\u5230\u5b9e\u6570\uff08\u751a\u81f3\u662f\u590d\u6570\uff09\u3002\u901a\u8fc7\u5206\u90e8\u79ef\u5206\u53ef\u4ee5\u8f7b\u677e\u9a8c\u8bc1 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma(s+1) = s\\Gamma(s)<\/span><\/span>\u3002\u6b64\u5916\uff0c\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma(1)=1<\/span><\/span>\uff0c\u7ed3\u679c\u662f<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall n\\in\\mathbb{N}\\right)\\left(\\Gamma(n) = (n-1)! \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u5982\u679c\u968f\u673a\u53d8\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u5177\u6709\u53c2\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta, \\alpha<\/span><\/span> \u7684\u4f3d\u9a6c\u5206\u5e03\uff0c\u5199\u4f5c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim Ga(\\alpha,\\beta).<\/span><\/span> \u4f3d\u9a6c\u5206\u5e03\u662f\u6307\u6570\u5206\u5e03\u7684\u53e6\u4e00\u79cd\u63a8\u5e7f\uff0c\u6ce8\u610f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Ga(\\alpha,1) = Ex(\\alpha).<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u5728\u5177\u6709\u9891\u7387 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> \u7684\u6cca\u677e\u8fc7\u7a0b\u4e2d\uff08\u5982\u653e\u5c04\u6027\u8870\u53d8\uff09\uff0c\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T<\/span><\/span> \u662f\u8868\u793a\u7b2c m \u6b21\u4e8b\u4ef6\u53d1\u751f\u65f6\u523b\u7684\u968f\u673a\u53d8\u91cf\uff1b\u90a3\u4e48\uff0c\u7ed9\u5b9a\u4e00\u4e2a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t\\geq 0<\/span><\/span> \u548c\u5728\u65f6\u95f4\u95f4\u9694 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[0,t]<\/span><\/span> \u53d1\u751f\u7684\u4e8b\u4ef6\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">N<\/span><\/span>\uff0c\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t\\lt T \\leftrightarrow N\\lt m<\/span><\/span> \u5e76\u4e14\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">N\\sim Po(ct),<\/span><\/span> \u6211\u4eec\u6709\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1-F_T(t) = P(T\\gt t) = \\displaystyle \\sum_{k=0}^{m-1}Po(k; ct)=e^{-ct}\\sum_{k=0}^{m-1}\\frac{(ct)^k}{k!}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u56e0\u6b64\uff0c\u901a\u8fc7\u6c42\u5bfc\u6211\u4eec\u53ef\u4ee5\u53d1\u73b0\u5bc6\u5ea6\u51fd\u6570\u662f<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f(t) = ce^{-ct}\\frac{(ct)^{m-1}}{(m-1)!}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u56e0\u6b64\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T\\sim Ga(1\/c, m).<\/span><\/span><\/p>\n<p><a name=\"8\"><\/a><\/br><\/br><\/p>\n<h2>\u7ec3\u4e60<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>\u627e\u5230\u5e38\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> \u4f7f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f(x) = \\frac{c}{x^2+1}<\/span><\/span> \u662f\u4e00\u4e2a\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff0c\u5e76\u8ba1\u7b97\u76f8\u5e94\u7684\u6982\u7387\u5206\u5e03\u51fd\u6570\uff08\u67ef\u897f\u5206\u5e03\uff09<\/li>\n<li>\u4ece\u5206\u5e03 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Un(a.b),<\/span><\/span> \u7684\u5bc6\u5ea6\u51fd\u6570\uff0c\u786e\u5b9a\u5176\u76f8\u5e94\u7684\u5206\u5e03\u51fd\u6570\u3002<\/li>\n<li>\u8bc1\u660e\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{\\mu,\\sigma}(x)<\/span><\/span> \u662f\u4e00\u4e2a\u6982\u7387\u5206\u5e03\u51fd\u6570\u3002<\/li>\n<\/ol>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/kdxgrB1h98g\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8fde\u7eed\u6982\u7387\u5206\u5e03 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