{"id":1476,"date":"2024-07-25T21:35:27","date_gmt":"2024-07-25T13:35:27","guid":{"rendered":"https:\/\/www.gnn.club\/?p=1476"},"modified":"2024-07-29T19:35:22","modified_gmt":"2024-07-29T11:35:22","slug":"%e6%a6%82%e7%8e%87%e4%b8%8e%e7%bb%9f%e8%ae%a1","status":"publish","type":"post","link":"http:\/\/www.gnn.club\/?p=1476","title":{"rendered":"\u6982\u7387\u4e0e\u7edf\u8ba1"},"content":{"rendered":"<h1><img decoding=\"async\" src=\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729192657875.png\" style=\"height:50px;display:inline\">  Deep Learning Math<\/h1>\n<hr \/>\n<h2>\u6982\u7387\u4e0e\u7edf\u8ba1\uff08Probability and Statistics\uff09<\/h2>\n<p>\u6982\u7387\u8bba\u548c\u7edf\u8ba1\u5b66\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\u81f3\u5173\u91cd\u8981\u3002\u6982\u7387\u8bba\u4e3a\u6a21\u578b\u7684\u4e0d\u786e\u5b9a\u6027\u548c\u9884\u6d4b\u63d0\u4f9b\u7406\u8bba\u57fa\u7840\u3002\u6570\u636e\u4f4d\u7f6e\u548c\u6570\u636e\u6563\u5e03\u7684\u6982\u5ff5\u5e2e\u52a9\u6211\u4eec\u7406\u89e3\u548c\u63cf\u8ff0\u6570\u636e\u7684\u4e2d\u5fc3\u8d8b\u52bf\u548c\u53d8\u5f02\u6027\u3002\u56fe\u5f62\u8868\u793a\u6280\u672f\uff0c\u5982\u76f4\u65b9\u56fe\u548c\u6563\u70b9\u56fe\uff0c\u7528\u4e8e\u6570\u636e\u7684\u53ef\u89c6\u5316\u548c\u63a2\u7d22\u3002\u79bb\u6563\u578b\u6982\u7387\u5206\u5e03\u548c\u8fde\u7eed\u578b\u6982\u7387\u5206\u5e03\u5219\u7528\u4e8e\u63cf\u8ff0\u4e0d\u540c\u7c7b\u578b\u7684\u6570\u636e\u548c\u5176\u5206\u5e03\u6a21\u5f0f\u3002<\/p>\n<p>\u5728\u7edf\u8ba1\u5b66\u4e2d\uff0c\u70b9\u4f30\u8ba1\u548c\u533a\u95f4\u4f30\u8ba1\u7528\u4e8e\u63a8\u65ad\u6570\u636e\u7684\u53c2\u6570\uff0c\u4ece\u800c\u5e2e\u52a9\u6211\u4eec\u7406\u89e3\u6a21\u578b\u7684\u9884\u6d4b\u51c6\u786e\u6027\u548c\u7f6e\u4fe1\u533a\u95f4\u3002\u5047\u8bbe\u6027\u68c0\u9a8c\u7528\u4e8e\u8bc4\u4f30\u6a21\u578b\u5047\u8bbe\u7684\u6709\u6548\u6027\u548c\u663e\u8457\u6027\uff0c\u786e\u4fdd\u7ed3\u679c\u7684\u53ef\u9760\u6027\u3002\u76f8\u5173\u6027\u5206\u6790\u5219\u7528\u4e8e\u53d1\u73b0\u53d8\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u548c\u4f9d\u8d56\u6027\uff0c\u4ece\u800c\u5e2e\u52a9\u6211\u4eec\u4f18\u5316\u6a21\u578b\u548c\u63d0\u5347\u6027\u80fd\u3002\u901a\u8fc7\u6982\u7387\u8bba\u548c\u7edf\u8ba1\u5b66\u5de5\u5177\uff0c\u6df1\u5ea6\u5b66\u4e60\u80fd\u591f\u66f4\u597d\u5730\u5904\u7406\u6570\u636e\u7684\u4e0d\u786e\u5b9a\u6027\u3001\u8fdb\u884c\u6a21\u578b\u8bc4\u4f30\u548c\u4f18\u5316\uff0c\u63d0\u5347\u9884\u6d4b\u7684\u51c6\u786e\u6027\u548c\u53ef\u9760\u6027\u3002<\/p>\n<hr \/>\n<h3><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/bubbles\/50\/000000\/checklist.png\" style=\"height:50px;display:inline\"> Agenda<\/h3>\n<hr \/>\n<ul>\n<li>\n<p>\u6982\u7387\u8bba(Probability Theory)<\/p>\n<ul>\n<li>\u6570\u636e\u4f4d\u7f6e<\/li>\n<li>\u6570\u636e\u6563\u5e03<\/li>\n<li>\u56fe\u5f62\u8868\u793a<\/li>\n<li>\u79bb\u6563\u578b\u6982\u7387\u5206\u5e03<\/li>\n<li>\u8fde\u7eed\u578b\u6982\u7387\u5206\u5e03<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u7edf\u8ba1\u5b66(Statistics)<\/p>\n<ul>\n<li>\u70b9\u4f30\u8ba1<\/li>\n<li>\u8499\u7279\u5361\u6d1b\u91c7\u6837<\/li>\n<li>\u533a\u95f4\u4f30\u8ba1<\/li>\n<li>\u5047\u8bbe\u6027\u68c0\u9a8c<\/li>\n<li>\u76f8\u5173\u6027\u5206\u6790<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h4><img decoding=\"async\" src=\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729192916175.png\" style=\"height:30px;display:inline\"> Additional Packages for Google Colab<\/h4>\n<hr \/>\n<p>If you are using <a href=\"https:\/\/colab.research.google.com\/\">Google Colab<\/a>, you have to install additional packages. To do this, simply run the following cell.<\/p>\n<h2><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/?size=100&id=BE6lAShe24JV&format=png&color=5C7CFA\" style=\"height:50px;display:inline\"> \u6982\u7387\u8bba<\/h2>\n<hr \/>\n<hr \/>\n<h3><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/nolan\/64\/000000\/categorize.png\" style=\"height:50px;display:inline\"> \u6570\u636e\u4f4d\u7f6e<\/h3>\n<p><strong>\u5e73\u5747\u6570<\/strong>\uff08Average\uff09\u662f\u8868\u793a\u4e00\u7ec4\u6570\u636e\u96c6\u4e2d\u8d8b\u52bf\u7684\u91cf\u6570\uff0c\u662f\u6307\u5728\u4e00\u7ec4\u6570\u636e\u4e2d\u6240\u6709\u6570\u636e\u4e4b\u548c\u518d\u9664\u4ee5\u8fd9\u7ec4\u6570\u636e\u7684\u4e2a\u6570\uff0c\u662f\u53cd\u6620\u6570\u636e\u96c6\u4e2d\u8d8b\u52bf\u7684\u4e00\u9879\u6307\u6807\u3002<br \/>\n\u7279\u70b9\uff1a<\/p>\n<ul>\n<li>\uff081\uff09\u6613\u53d7\u6781\u7aef\u503c\u5f71\u54cd\u3002<\/li>\n<li>\uff082\uff09\u6570\u5b66\u6027\u8d28\u4f18\u826f\u3002<\/li>\n<li>\uff083\uff09\u6570\u636e\u5bf9\u79f0\u5206\u5e03\u6216\u63a5\u8fd1\u5bf9\u79f0\u5206\u5e03\u65f6\u5e94\u7528\u3002<\/li>\n<\/ul>\n<p><strong>\u4f17\u6570<\/strong>\uff08Mode\uff09\u662f\u6307\u5728\u7edf\u8ba1\u5206\u5e03\u4e0a\u5177\u6709\u660e\u663e\u96c6\u4e2d\u8d8b\u52bf\u70b9\u7684\u6570\u503c\uff0c\u4ee3\u8868\u6570\u636e\u7684\u4e00\u822c\u6c34\u5e73\u3002 \u4e5f\u662f\u4e00\u7ec4\u6570\u636e\u4e2d\u51fa\u73b0\u6b21\u6570\u6700\u591a\u7684\u6570\u503c\uff0c\u6709\u65f6\u4f17\u6570\u5728\u4e00\u7ec4\u6570\u4e2d\u6709\u597d\u51e0\u4e2a\uff0c\u7528M\u8868\u793a\u3002<br \/>\n\u7279\u70b9\uff1a<\/p>\n<ul>\n<li>\uff081\uff09\u7ec4\u6570\u636e\u4e2d\u51fa\u73b0\u6b21\u6570\u6700\u591a\u7684\u53d8\u91cf\u503c\u3002<\/li>\n<li>\uff082\uff09\u9002\u5408\u4e8e\u6570\u636e\u91cf\u8f83\u591a\u65f6\u4f7f\u7528\u3002<\/li>\n<li>\uff083\uff09\u4e0d\u53d7\u6781\u7aef\u503c\u7684\u5f71\u54cd\u3002<\/li>\n<li>\uff084\uff09\u4e00\u7ec4\u6570\u636e\u53ef\u80fd\u6ca1\u6709\u4f17\u6570\u4e5f\u53ef\u80fd\u6709\u51e0\u4e2a\u4f17\u6570\u3002<\/li>\n<\/ul>\n<p><strong>\u4e2d\u4f4d\u6570<\/strong>\uff08Median\uff09\u53c8\u79f0\u4e2d\u503c\uff0c\u7edf\u8ba1\u5b66\u4e2d\u7684\u4e13\u6709\u540d\u8bcd\uff0c\u662f\u6309\u987a\u5e8f\u6392\u5217\u7684\u4e00\u7ec4\u6570\u636e\u4e2d\u5c45\u4e8e\u4e2d\u95f4\u4f4d\u7f6e\u7684\u6570\uff0c\u4ee3\u8868\u4e00\u4e2a\u6837\u672c\u3001\u79cd\u7fa4\u6216\u6982\u7387\u5206\u5e03\u4e2d\u7684\u4e00\u4e2a\u6570\u503c\uff0c\u5176\u53ef\u5c06\u6570\u503c\u96c6\u5408\u5212\u5206\u4e3a\u76f8\u7b49\u7684\u4e0a\u4e0b\u4e24\u90e8\u5206\u3002\u5bf9\u4e8e\u6709\u9650\u7684\u6570\u96c6\uff0c\u53ef\u4ee5\u901a\u8fc7\u628a\u6240\u6709\u89c2\u5bdf\u503c\u9ad8\u4f4e\u6392\u5e8f\u540e\u627e\u51fa\u6b63\u4e2d\u95f4\u7684\u4e00\u4e2a\u4f5c\u4e3a\u4e2d\u4f4d\u6570\u3002\u5982\u679c\u89c2\u5bdf\u503c\u6709\u5076\u6570\u4e2a\uff0c\u901a\u5e38\u53d6\u6700\u4e2d\u95f4\u7684\u4e24\u4e2a\u6570\u503c\u7684\u5e73\u5747\u6570\u4f5c\u4e3a\u4e2d\u4f4d\u6570\u3002<br \/>\n\u7279\u70b9\uff1a<\/p>\n<ul>\n<li>\uff081\uff09\u4e0d\u53d7\u6781\u7aef\u503c\u7684\u5f71\u54cd\u3002\u5728\u6709\u6781\u7aef\u6570\u503c\u51fa\u73b0\u65f6\uff0c\u4e2d\u4f4d\u6570\u4f5c\u4e3a\u5206\u6790\u73b0\u8c61\u4e2d\u96c6\u4e2d\u8d8b\u52bf\u7684\u6570\u503c\uff0c\u6bd4\u5e73\u5747\u6570\u66f4\u5177\u6709\u4ee3\u8868\u6027\u3002<\/li>\n<li>\uff082\uff09\u4e3b\u8981\u7528\u4e8e\u987a\u5e8f\u6570\u636e\uff0c\u4e5f\u53ef\u4ee5\u7528\u4e8e\u6570\u503c\u578b\u6570\u636e\uff0c\u4f46\u4e0d\u80fd\u7528\u4e8e\u5206\u7c7b\u6570\u636e\u3002<\/li>\n<li>\uff083\uff09\u5404\u53d8\u91cf\u503c\u4e0e\u4e2d\u4f4d\u6570\u7684\u79bb\u5dee\u7edd\u5bf9\u503c\u4e4b\u548c\u6700\u5c0f\u3002<\/li>\n<\/ul>\n<p><strong>\u56db\u5206\u4f4d\u6570<\/strong>\uff08Quartile\uff09\u4e5f\u79f0\u56db\u5206\u4f4d\u70b9\uff0c\u662f\u6307\u5728\u7edf\u8ba1\u5b66\u4e2d\u628a\u6240\u6709\u6570\u503c\u7531\u5c0f\u5230\u5927\u6392\u5217\u5e76\u5206\u6210\u56db\u7b49\u4efd\uff0c\u5904\u4e8e\u4e09\u4e2a\u5206\u5272\u70b9\u4f4d\u7f6e\u7684\u6570\u503c\u5c31\u662f\u56db\u5206\u4f4d\u6570\u3002\u56db\u5206\u4f4d\u6570\u591a\u5e94\u7528\u4e8e\u7edf\u8ba1\u5b66\u4e2d\u7684\u7bb1\u7ebf\u56fe\u7ed8\u5236\u3002\u5b83\u662f\u4e00\u7ec4\u6570\u636e\u6392\u5e8f\u540e\u5904\u4e8e25%\u548c75%\u4f4d\u7f6e\u4e0a\u7684\u503c\u3002\u56db\u5206\u4f4d\u6570\u662f\u901a\u8fc73\u4e2a\u70b9\u5c06\u5168\u90e8\u6570\u636e\u7b49\u5206\u4e3a4\u90e8\u5206\uff0c\u5176\u4e2d\u6bcf\u90e8\u5206\u5305\u542b25%\u7684\u6570\u636e\u3002\u5f88\u663e\u7136\uff0c\u4e2d\u95f4\u7684\u56db\u5206\u4f4d\u6570\u5c31\u662f\u4e2d\u4f4d\u6570\uff0c\u56e0\u6b64\u901a\u5e38\u6240\u8bf4\u7684\u56db\u5206\u4f4d\u6570\u662f\u6307\u5904\u572825%\u4f4d\u7f6e\u4e0a\u7684\u6570\u503c\uff08\u79f0\u4e3a\u4e0b\u56db\u5206\u4f4d\u6570\uff09\u548c\u5904\u572875%\u4f4d\u7f6e\u4e0a\u7684\u6570\u503c\uff08\u79f0\u4e3a\u4e0a\u56db\u5206\u4f4d\u6570\uff09\u3002\u4e0e\u4e2d\u4f4d\u6570\u7684\u8ba1\u7b97\u65b9\u6cd5\u7c7b\u4f3c\uff0c\u6839\u636e\u672a\u5206\u7ec4\u6570\u636e\u8ba1\u7b97\u56db\u5206\u4f4d\u6570\u65f6\uff0c\u9996\u5148\u5bf9\u6570\u636e\u8fdb\u884c\u6392\u5e8f\uff0c\u7136\u540e\u786e\u5b9a\u56db\u5206\u4f4d\u6570\u6240\u5728\u7684\u4f4d\u7f6e\uff0c\u8be5\u4f4d\u7f6e\u4e0a\u7684\u6570\u503c\u5c31\u662f\u56db\u5206\u4f4d\u6570\u3002<\/p>\n<p>\u5047\u8bbe\u6211\u4eec\u6709\u4ee5\u4e0b\u4e00\u7ec4\u6570\u636e:<br \/>\n$$<br \/>\n7,15,36,39,40,41,42,43,44,49,50<br \/>\n$$<\/p>\n<p>\u5bf9\u4e8e\u5947\u6570\u4e2a\u6570\u636e\uff0c\u4e2d\u4f4d\u6570\u662f\u7b2c $\\frac{n+1}{2}$ \u4e2a\u6570\u636e\u3002<br \/>\n\u8fd9\u91cc\u6570\u636e\u4e2a\u6570 $n=11$ \uff0c\u6240\u4ee5\u4e2d\u4f4d\u6570\u4f4d\u7f6e\u662f\u7b2c $\\frac{11+1}{2}=6$ \u4e2a\u6570\u636e\u3002\u6240\u4ee5\u4e2d\u4f4d\u6570 $Q_2$ \u662f\u7b2c 6 \u4e2a\u6570\u636e\uff0c\u4e5f\u5c31\u662f 41 \u3002<\/p>\n<p>\u4e0b\u56db\u5206\u4f4d\u6570\u662f\u4f4d\u4e8e\u7b2c $\\frac{n+1}{4}$ \u4e2a\u6570\u636e\u3002<br \/>\n\u8fd9\u91cc\u6570\u636e\u4e2a\u6570 $n=11$ \uff0c\u6240\u4ee5\u4e0b\u56db\u5206\u4f4d\u6570\u4f4d\u7f6e\u662f\u7b2c $\\frac{11+1}{4}=3$ \u4e2a\u6570\u636e\u3002<br \/>\n\u6240\u4ee5\u4e0b\u56db\u5206\u4f4d\u6570 $Q_1$ \u662f\u7b2c 3 \u4e2a\u6570\u636e\uff0c\u4e5f\u5c31\u662f 36 \u3002<\/p>\n<p>\u4e0a\u56db\u5206\u4f4d\u6570\u662f\u4f4d\u4e8e\u7b2c $\\frac{3(n+1)}{4}$ \u4e2a\u6570\u636e\u3002<br \/>\n\u8fd9\u91cc\u6570\u636e\u4e2a\u6570 $n=11$ \uff0c\u6240\u4ee5\u4e0a\u56db\u5206\u4f4d\u6570\u4f4d\u7f6e\u662f\u7b2c $\\frac{3(11+1)}{4}=9$ \u4e2a\u6570\u636e\u3002\u6240\u4ee5\u4e0a\u56db\u5206\u4f4d\u6570 $Q_3$ \u662f\u7b2c 9 \u4e2a\u6570\u636e\uff0c\u4e5f\u5c31\u662f 44 \u3002<\/p>\n<h2><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/plasticine\/100\/000000\/mind-map.png\" style=\"height:50px;display:inline\"> \u6570\u636e\u6563\u5e03<\/h2>\n<hr \/>\n<p><strong>\u6570\u5b66\u671f\u671b<\/strong><\/p>\n<p>\u6570\u5b66\u671f\u671b\uff08Mathematical expectations\uff09\u662f\u5bf9<strong>\u957f\u671f\u4ef7\u503c\u7684\u6570\u5b57\u5316\u8861\u91cf<\/strong>\u3002<\/p>\n<p>\u6570\u5b66\u671f\u671b\u503c\u662f\u7406\u60f3\u72b6\u6001\u4e0b\u5f97\u5230\u7684\u5b9e\u9a8c\u7ed3\u679c\u7684\u5e73\u5747\u503c\uff0c\u662f\u8bd5\u9a8c\u4e2d\u6bcf\u6b21\u53ef\u80fd\u7684\u7ed3\u679c\u6982\u7387\u4e58\u4ee5\u5176\u7ed3\u679c\u7684\u603b\u548c\uff0c\u662f\u6700\u57fa\u672c\u7684\u6570\u5b66\u7279\u5f81\u4e4b\u4e00\uff0c\u5b83\u53cd\u6620\u968f\u673a\u53d8\u91cf\u5e73\u5747\u53d6\u503c\u7684\u5927\u5c0f\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u671f\u671b\u503c\u50cf\u662f\u968f\u673a\u8bd5\u9a8c\u5728\u540c\u6837\u7684\u673a\u4f1a\u4e0b\u91cd\u590d\u591a\u6b21\uff0c\u6240\u6709\u90a3\u4e9b\u53ef\u80fd\u72b6\u6001\u7684\u5e73\u5747\u7ed3\u679c\u3002<\/p>\n<ul>\n<li>\n<p>\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u6570\u5b66\u671f\u671b\u4e25\u683c\u7684\u5b9a\u4e49\u4e3a: \u8bbe\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf $X$ \u7684\u5206\u5e03\u5217\u4e3a$P\\lbrace X=x_i \\rbrace = p_i, \\quad i=1,2, \\cdots$\u3002\u82e5\u7ea7\u6570 $\\sum_{i=1}^{+\\infty} x_i p_i$  \u7edd\u5bf9\u6536\u655b, \u5219\u79f0\u7ea7\u6570 $\\sum_{i=1}^{+\\infty} x_i p_i$ \u7684\u548c\u4e3a\u968f\u673a\u53d8\u91cf $X$ \u7684\u6570\u5b66\u671f\u671b(\u4e5f\u79f0\u671f\u671b\u6216\u5747\u503c), \u8bb0\u4e3a $E(X)$ \u3002\u5373$E(X)=x_1 p_1+x_2 p_2+\\cdots+x_i p_i+\\cdots=\\sum_{i=1}^{+\\infty} x_i p_i$  \u3002<\/p>\n<\/li>\n<li>\n<p>\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u6570\u5b66\u671f\u671b\u4e25\u683c\u7684\u5b9a\u4e49\u4e3a: \u8bbe\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf $X$ \u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3a $f(x)$,\u79ef\u5206 $\\int_{-\\infty}^{+\\infty} x f(x) d x$  \u7edd\u5bf9\u6536\u655b, \u5219\u5b9a\u4e49 $x$ \u7684\u6570\u5b66\u671f\u671b $E(X)$ \u4e3a $E(X)=\\int_{-\\infty}^{+\\infty} x f(x) d x$ \u3002<\/p>\n<\/li>\n<\/ul>\n<p>\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u7684\u6570\u5b66\u671f\u671b\u662f\u4e00\u4e2a\u5e38\u6570\uff0c\u5b83\u8868\u793a\u968f\u673a\u53d8\u91cf\u53d6\u503c\u7684\u4e00\u4e2a\u5e73\u5747\uff1b\u8fd9\u91cc\u7528\u7684\u4e0d\u662f\u7b97\u672f\u5e73\u5747\uff0c\u800c\u662f<strong>\u4ee5\u6982\u7387\u4e3a\u6743\u91cd\u7684\u52a0\u6743\u5e73\u5747<\/strong>\u3002\u6570\u5b66\u671f\u671b\u53cd\u6620\u4e86\u968f\u673a\u53d8\u91cf\u7684\u4e00\u5927\u7279\u5f81\uff0c<strong>\u5373\u968f\u673a\u53d8\u91cf\u7684\u53d6\u503c\u5c06\u96c6\u4e2d\u5728\u5176\u671f\u671b\u503c\u9644\u8fd1\uff0c\u8fd9\u7c7b\u4f3c\u4e8e\u7269\u7406\u4e2d\u8d28\u70b9\u7ec4\u6210\u7684\u8d28\u5fc3<\/strong>\u3002<\/p>\n<p>\u6700\u540e\uff0c\u5f3a\u8c03\u4e00\u4e0b\u5e73\u5747\u6570\u548c\u6570\u5b66\u671f\u671b\u7684\u8054\u7cfb\uff1a\u5e73\u5747\u6570\u662f\u4e00\u4e2a\u7edf\u8ba1\u5b66\u6982\u5ff5\uff0c\u671f\u671b\u662f\u4e00\u4e2a\u6982\u7387\u8bba\u6982\u5ff5\u3002<strong>\u5e73\u5747\u6570\u662f\u5b9e\u9a8c\u540e\u6839\u636e\u5b9e\u9645\u7ed3\u679c\u7edf\u8ba1\u5f97\u5230\u7684\u6837\u672c\u7684\u5e73\u5747\u503c\uff0c\u671f\u671b\u662f\u5b9e\u9a8c\u524d\u6839\u636e\u6982\u7387\u5206\u5e03\u201c\u9884\u6d4b\u201d\u7684\u6837\u672c\u7684\u5e73\u5747\u503c\u3002<\/strong>\u4e4b\u6240\u4ee5\u8bf4\u201c\u9884\u6d4b\u201d\u662f\u56e0\u4e3a\u5728\u5b9e\u9a8c\u524d\u80fd\u5f97\u5230\u7684\u671f\u671b\u4e0e\u5b9e\u9645\u5b9e\u9a8c\u5f97\u5230\u7684\u6837\u672c\u7684\u5e73\u5747\u6570\u603b\u4f1a\u4e0d\u53ef\u907f\u514d\u5730\u5b58\u5728\u504f\u5dee\uff0c\u6bd5\u7adf\u968f\u673a\u5b9e\u9a8c\u7684\u7ed3\u679c\u6c38\u8fdc\u5145\u6ee1\u7740\u4e0d\u786e\u5b9a\u6027\u3002<strong>\u671f\u671b\u5c31\u662f\u5e73\u5747\u6570\u968f\u6837\u672c\u8d8b\u4e8e\u65e0\u7a77\u7684\u6781\u9650\u3002<\/strong><\/p>\n<p><strong>\u65b9\u5dee<\/strong><\/p>\n<p>\u65b9\u5dee\uff08Variance\uff09\u7528\u6765\u63cf\u8ff0<strong>\u968f\u673a\u53d8\u91cf\u4e0e\u6570\u5b66\u671f\u671b\u7684\u504f\u79bb\u7a0b\u5ea6<\/strong>\u3002<\/p>\n<p>\u5982\u679c\u628a\u5355\u4e2a\u6570\u636e\u70b9\u79f0\u4e3a\u201c $X_i$ \u201d, \u90a3\u4e48 \u201c $X_1$ \u201d \u662f\u7b2c\u4e00\u4e2a\u503c, \u201c $X_2$ \u201d \u662f\u7b2c\u4e8c\u4e2a\u503c, \u4ee5\u6b64\u7c7b\u63a8, \u4e00\u5171\u6709 $n$ \u4e2a\u503c\u3002\u5747\u503c\u79f0\u4e3a \u201c $\\mathrm{M}$ \u201d\u3002<\/p>\n<ul>\n<li>\n<p>\u521d\u770b\u4e0a\u53bb $\\sum\\left(X_i-\\mathrm{M}\\right)$ \u5c31\u53ef\u4ee5\u4f5c\u4e3a\u63cf\u8ff0\u6570\u636e\u70b9\u6563\u5e03\u60c5\u51b5\u7684\u6307\u6807, \u4e5f\u5c31\u662f\u628a\u6bcf\u4e2a $X_i$\u4e0e$M$\u7684\u504f\u5dee\u6c42\u548c\u3002\u6362\u53e5\u8bdd\u8bb2\uff0c\u662f\u5355\u4e2a\u6570\u636e\u70b9\u51cf\u53bb\u6570\u636e\u70b9\u7684\u5e73\u5747\u7684\u603b\u548c\u3002<\/p>\n<ul>\n<li>\u6b64\u65b9\u6cd5\u770b\u4e0a\u53bb\u5f88\u6709\u903b\u8f91\u6027\uff0c\u4f46\u5374\u6709\u4e00\u4e2a\u81f4\u547d\u7684\u7f3a\u70b9\uff1a\u9ad8\u51fa\u5747\u503c\u7684\u503c\u548c\u4f4e\u4e8e\u5747\u503c\u53ef\u4ee5\u76f8\u4e92\u62b5\u6d88\uff0c\u56e0\u6b64\u4e0a\u8ff0\u5b9a\u4e49\u7684\u7ed3\u679c\u8d8b\u8fd1\u4e8e0\u3002<\/li>\n<li>\u8fd9\u4e2a\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u53d6\u5dee\u503c\u7684\u7edd\u5bf9\u503c\u6765\u89e3\u51b3\uff08\u4e5f\u5c31\u662f\u8bf4\uff0c\u5ffd\u7565\u8d1f\u503c\u7684\u7b26\u53f7\uff09\uff0c\u4f46\u662f\u7531\u4e8e\u5404\u79cd\u539f\u56e0\uff0c\u7edf\u8ba1\u5b66\u5bb6\u4e0d\u559c\u6b22\u7edd\u5bf9\u503c\u3002\u53e6\u5916\u4e00\u4e2a\u5254\u9664\u8d1f\u53f7\u7684\u65b9\u6cd5\u662f\u53d6\u5e73\u65b9\uff0c\u56e0\u4e3a\u4efb\u4f55\u6570\u7684\u5e73\u65b9\u80af\u5b9a\u662f\u6b63\u7684\uff0c\u56e0\u6b64\u4fbf\u5f97\u5230\u4e86\u65b9\u5dee\u7684\u5206\u5b50$\\sum\\left(X_i-\\mathrm{M}\\right)^2$<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u518d\u8003\u8651\u4e00\u4e2a\u95ee\u9898: \u6bd4\u5982\u6709 25 \u4e2a\u503c\u7684\u6837\u672c, \u6839\u636e\u65b9\u5dee\u8ba1\u7b97\u51fa\u6807\u51c6\u5dee\u662f 10 \u3002\u5982\u679c\u628a\u8fd9 25 \u4e2a\u503c\u590d\u5236\u4e00\u4e0b\u53d8\u6210 50 \u4e2a\u6837\u672c\u5462, \u76f4\u89c9\u4e0a 50 \u4e2a\u6837\u672c\u7684\u6570\u636e\u70b9\u5206\u5e03\u60c5\u51b5\u5e94\u8be5\u4e0d\u53d8\u7684, \u4f46\u662f\u516c\u5f0f\u4e2d\u7684\u7d2f\u52a0\u4f1a\u4ea7\u751f\u66f4\u5927\u7684\u65b9\u5dee\u503c\u3002<\/p>\n<ul>\n<li>\u6240\u4ee5\u9700\u8981\u901a\u8fc7\u9664\u4ee5\u6570\u636e\u70b9\u6570\u91cf $n$ \u6765\u5f25\u8865\u8fd9\u4e2a\u6f0f\u6d1e\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>\u56e0\u6b64, \u65b9\u5dee\u7684\u5b9a\u4e49\u5982\u4e0b:<br \/>\n$$<br \/>\nD(X)=\\frac{\\sum_{i=1}^N\\left(x_i-\\bar{x}_i\\right)^2}{n}<br \/>\n$$<\/p>\n<p><strong>\u6807\u51c6\u5dee<\/strong><\/p>\n<p>\u6807\u51c6\u5dee\uff08standard deviation\uff09\u662f\u901a\u8fc7\u65b9\u5dee\u9664\u4ee5\u6837\u672c\u91cf\u518d\u5f00\u6839\u53f7\u5f97\u5230\u7684\uff0c\u5177\u4f53\u516c\u5f0f\u5982\u4e0b\uff1a<\/p>\n<p>$$<br \/>\n\\sigma=\\sqrt{\\frac{\\sum_{i=1}^N\\left(x_i-\\bar{x}_i\\right)^2}{n}}<br \/>\n$$<\/p>\n<p>\u4e0e\u65b9\u5dee\u7684\u4f5c\u7528\u7c7b\u4f3c, \u6807\u51c6\u5dee\u4e5f\u80fd\u53cd\u6620\u4e00\u4e2a\u6570\u636e\u96c6\u7684\u79bb\u6563\u7a0b\u5ea6, \u5b83\u662f\u5404\u70b9\u4e0e\u5747\u503c\u7684\u5e73\u5747\u8ddd\u79bb\u3002\u5e73\u5747\u6570\u76f8\u540c\u7684\u6570\u636e, \u6807\u51c6\u5dee\u672a\u5fc5\u76f8\u540c\u3002<\/p>\n<p><strong>\u6781\u5dee<\/strong><\/p>\n<p>\u6781\u5dee\u53c8\u79f0\u8303\u56f4\u8bef\u5dee\u6216\u5168\u8ddd(Range)\uff0c\u4ee5R\u8868\u793a\uff0c\u8ba1\u7b97\u65b9\u6cd5\u662f\u5176\u6700\u5927\u503c\u4e0e\u6700\u5c0f\u503c\u4e4b\u95f4\u7684\u5dee\u8ddd\uff0c\u5373\u6700\u5927\u503c\u51cf\u6700\u5c0f\u503c\u540e\u6240\u5f97\u6570\u636e\u3002<\/p>\n<p><strong>\u56db\u5206\u6570\u8303\u56f4<\/strong><\/p>\n<p>\u56db\u5206\u4f4d\u6570\uff0c\u5373\u628a\u6240\u6709\u6570\u503c\u7531\u5c0f\u5230\u5927\u6392\u5217\u5e76\u5206\u6210\u56db\u7b49\u4efd\uff0c\u5904\u4e8e\u4e09\u4e2a\u5206\u5272\u70b9\u4f4d\u7f6e\u7684\u6570\u503c\u5c31\u662f\u56db\u5206\u4f4d\u6570\u3002\u7b2c\u4e09\u56db\u5206\u4f4d\u6570\u4e0e\u7b2c\u4e00\u56db\u5206\u4f4d\u6570\u7684\u5dee\u503c\u79f0\u4e3a\u56db\u5206\u4f4d\u6570\u95f4\u8ddd\uff08Interquartile Range\uff0cIQR\uff09\uff0c\u7b80\u79f0\u56db\u5206\u4f4d\u8ddd\u3002<br \/>\n\u56db\u5206\u4f4d\u8ddd\u662f\u63cf\u8ff0\u7edf\u8ba1\u5b66\u4e2d\u7684\u4e00\u79cd\u65b9\u6cd5\uff0c\u4f46\u7531\u4e8e\u56db\u5206\u4f4d\u8ddd\u4e0d\u53d7\u6781\u5927\u503c\u6216\u6781\u5c0f\u503c\u7684\u5f71\u54cd\uff0c\u5e38\u7528\u4e8e\u63cf\u8ff0\u975e\u6b63\u6001\u5206\u5e03\u8d44\u6599\u7684\u79bb\u6563\u7a0b\u5ea6\uff0c\u5176\u6570\u503c\u8d8a\u5927\uff0c\u6570\u636e\u79bb\u6563\u7a0b\u5ea6\u8d8a\u5927\uff0c\u53cd\u4e4b\u79bb\u6563\u7a0b\u5ea6\u8d8a\u5c0f\u3002<\/p>\n<h2><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/?size=100&id=TQjbBaTB3Sqp&format=png&color=000000\" style=\"height:50px;display:inline\"> \u56fe\u5f62\u8868\u793a<\/h2>\n<hr \/>\n<p>\u5e38\u89c1\u7684\u6570\u636e\u56fe\u5f62\u5316\u8868\u793a\u65b9\u5f0f\u6709\u5f88\u591a\uff0c\u6bcf\u79cd\u65b9\u5f0f\u90fd\u9002\u7528\u4e8e\u5c55\u793a\u4e0d\u540c\u7c7b\u578b\u7684\u6570\u636e\u548c\u63ed\u793a\u4e0d\u540c\u7684\u5173\u7cfb\u3002\u4ee5\u4e0b\u662f\u51e0\u79cd\u5e38\u89c1\u7684\u6570\u636e\u56fe\u5f62\u5316\u8868\u793a\u65b9\u5f0f\uff1a<\/p>\n<ul>\n<li>\n<p>\u67f1\u72b6\u56fe\uff08Bar Chart\uff09\uff1a\u7528\u4e8e\u6bd4\u8f83\u4e0d\u540c\u7c7b\u522b\u7684\u6570\u636e\u503c\u3002<\/p>\n<\/li>\n<li>\n<p>\u6298\u7ebf\u56fe\uff08Line Chart\uff09\uff1a\u7528\u4e8e\u5c55\u793a\u6570\u636e\u968f\u65f6\u95f4\u7684\u53d8\u5316\u8d8b\u52bf\u3002<\/p>\n<\/li>\n<li>\n<p>\u6563\u70b9\u56fe\uff08Scatter Plot\uff09\uff1a\u7528\u4e8e\u5c55\u793a\u4e24\u4e2a\u53d8\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u3002<\/p>\n<\/li>\n<li>\n<p>\u997c\u56fe\uff08Pie Chart\uff09\uff1a\u7528\u4e8e\u5c55\u793a\u6570\u636e\u5728\u4e00\u4e2a\u6574\u4f53\u4e2d\u7684\u5360\u6bd4\u3002<\/p>\n<\/li>\n<li>\n<p>\u7bb1\u7ebf\u56fe\uff08Box Plot\uff09\uff1a\u7528\u4e8e\u5c55\u793a\u6570\u636e\u7684\u5206\u5e03\u53ca\u5176\u5f02\u5e38\u503c\uff0c\u5c24\u5176\u9002\u5408\u5c55\u793a\u56db\u5206\u4f4d\u6570\u3002<\/p>\n<\/li>\n<li>\n<p>\u76f4\u65b9\u56fe\uff08Histogram\uff09\uff1a\u7528\u4e8e\u5c55\u793a\u6570\u636e\u7684\u9891\u7387\u5206\u5e03\u3002<\/p>\n<\/li>\n<li>\n<p>\u70ed\u56fe\uff08Heatmap\uff09\uff1a\u7528\u4e8e\u5c55\u793a\u6570\u503c\u53d8\u91cf\u7684\u5bc6\u5ea6\u5206\u5e03\u3002<\/p>\n<\/li>\n<li>\n<p>\u96f7\u8fbe\u56fe\uff08Radar Chart\uff09\uff1a\u7528\u4e8e\u5c55\u793a\u591a\u53d8\u91cf\u6570\u636e\u7684\u6bd4\u8f83\u3002<\/p>\n<\/li>\n<\/ul>\n<p>\u6bcf\u79cd\u56fe\u5f62\u5316\u8868\u793a\u65b9\u5f0f\u90fd\u6709\u5176\u72ec\u7279\u7684\u7528\u9014\u548c\u4f18\u52bf\uff0c\u9009\u62e9\u9002\u5408\u7684\u56fe\u8868\u7c7b\u578b\u53ef\u4ee5\u66f4\u6e05\u6670\u5730\u4f20\u8fbe\u6570\u636e\u7684\u542b\u4e49\u3002<\/p>\n<pre><code class=\"language-python\">import numpy as np\nimport pandas as pd\nfrom scipy import stats\nfrom sklearn.datasets import load_iris\n\n# Load the Iris dataset\niris = load_iris()\ndf = pd.DataFrame(data=iris.data, columns=iris.feature_names)\n\n# Calculate statistics\nstatistics = {}\n\nfor column in df.columns:\n    data = df[column]\n    statistics[column] = {\n        &#039;mean&#039;: np.mean(data),\n        &#039;mode&#039;: stats.mode(data, keepdims=True)[0][0],\n        &#039;median&#039;: np.median(data),\n        &#039;quartiles&#039;: np.percentile(data, [25, 50, 75]),\n        &#039;variance&#039;: np.var(data),\n        &#039;standard_deviation&#039;: np.std(data),\n        &#039;range&#039;: np.ptp(data),\n        &#039;interquartile_range&#039;: stats.iqr(data)\n    }\n\n# Convert statistics to DataFrame for better visualization\nstats_df = pd.DataFrame(statistics).T\nstats_df\n<\/code><\/pre>\n<div>\n<style scoped>\n    .dataframe tbody tr th:only-of-type {\n        vertical-align: middle;\n    }<\/p>\n<p>    .dataframe tbody tr th {\n        vertical-align: top;\n    }<\/p>\n<p>    .dataframe thead th {\n        text-align: right;\n    }\n<\/style>\n<table border=\"1\" class=\"dataframe\">\n<thead>\n<tr style=\"text-align: right;\">\n<th><\/th>\n<th>mean<\/th>\n<th>mode<\/th>\n<th>median<\/th>\n<th>quartiles<\/th>\n<th>variance<\/th>\n<th>standard_deviation<\/th>\n<th>range<\/th>\n<th>interquartile_range<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<th>sepal length (cm)<\/th>\n<td>5.843333<\/td>\n<td>5.0<\/td>\n<td>5.8<\/td>\n<td>[5.1, 5.8, 6.4]<\/td>\n<td>0.681122<\/td>\n<td>0.825301<\/td>\n<td>3.6<\/td>\n<td>1.3<\/td>\n<\/tr>\n<tr>\n<th>sepal width (cm)<\/th>\n<td>3.057333<\/td>\n<td>3.0<\/td>\n<td>3.0<\/td>\n<td>[2.8, 3.0, 3.3]<\/td>\n<td>0.188713<\/td>\n<td>0.434411<\/td>\n<td>2.4<\/td>\n<td>0.5<\/td>\n<\/tr>\n<tr>\n<th>petal length (cm)<\/th>\n<td>3.758<\/td>\n<td>1.4<\/td>\n<td>4.35<\/td>\n<td>[1.6, 4.35, 5.1]<\/td>\n<td>3.095503<\/td>\n<td>1.759404<\/td>\n<td>5.9<\/td>\n<td>3.5<\/td>\n<\/tr>\n<tr>\n<th>petal width (cm)<\/th>\n<td>1.199333<\/td>\n<td>0.2<\/td>\n<td>1.3<\/td>\n<td>[0.3, 1.3, 1.8]<\/td>\n<td>0.577133<\/td>\n<td>0.759693<\/td>\n<td>2.4<\/td>\n<td>1.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<pre><code class=\"language-python\">import matplotlib.pyplot as plt\nimport seaborn as sns\nimport numpy as np\nimport pandas as pd\nfrom sklearn import datasets\n\n# Load the iris dataset\niris = datasets.load_iris()\ndf = pd.DataFrame(iris.data, columns=iris.feature_names)\ndf[&#039;species&#039;] = iris.target\n\n# Plotting all charts as subplots in a single figure\nfig, axes = plt.subplots(4, 2, figsize=(10, 15), subplot_kw=dict(polar=False))\n\n# Bar Chart\nsns.barplot(x=&#039;species&#039;, y=&#039;sepal length (cm)&#039;, data=df, ax=axes[0, 0])\naxes[0, 0].set_title(&#039;Mean Sepal Length by Species&#039;)\naxes[0, 0].set_xlabel(&#039;Species&#039;)\naxes[0, 0].set_ylabel(&#039;Sepal Length (cm)&#039;)\n\n# Line Chart\naxes[0, 1].plot(df[&#039;sepal length (cm)&#039;])\naxes[0, 1].set_title(&#039;Trend of Sepal Length Across Samples&#039;)\naxes[0, 1].set_xlabel(&#039;Sample Index&#039;)\naxes[0, 1].set_ylabel(&#039;Sepal Length (cm)&#039;)\n\n# Scatter Plot\nsns.scatterplot(x=&#039;sepal length (cm)&#039;, y=&#039;petal length (cm)&#039;, hue=&#039;species&#039;, data=df, ax=axes[1, 0])\naxes[1, 0].set_title(&#039;Relationship Between Sepal Length and Petal Length&#039;)\naxes[1, 0].set_xlabel(&#039;Sepal Length (cm)&#039;)\naxes[1, 0].set_ylabel(&#039;Petal Length (cm)&#039;)\n\n# Pie Chart\nspecies_counts = df[&#039;species&#039;].value_counts()\naxes[1, 1].pie(species_counts, labels=iris.target_names, autopct=&#039;%1.1f%%&#039;, startangle=140)\naxes[1, 1].set_title(&#039;Proportion of Each Species&#039;)\n\n# Box Plot\nsns.boxplot(x=&#039;species&#039;, y=&#039;sepal length (cm)&#039;, data=df, ax=axes[2, 0])\naxes[2, 0].set_title(&#039;Distribution of Sepal Length by Species&#039;)\naxes[2, 0].set_xlabel(&#039;Species&#039;)\naxes[2, 0].set_ylabel(&#039;Sepal Length (cm)&#039;)\n\n# Histogram\naxes[2, 1].hist(df[&#039;sepal length (cm)&#039;], bins=20, edgecolor=&#039;k&#039;)\naxes[2, 1].set_title(&#039;Frequency Distribution of Sepal Length&#039;)\naxes[2, 1].set_xlabel(&#039;Sepal Length (cm)&#039;)\naxes[2, 1].set_ylabel(&#039;Frequency&#039;)\n\n# Heatmap\nsns.heatmap(df.iloc[:, :-1].corr(), annot=True, cmap=&#039;coolwarm&#039;, vmin=-1, vmax=1, ax=axes[3, 0])\naxes[3, 0].set_title(&#039;Heatmap of Feature Correlations&#039;)\n\n# Radar Chart\n# Create a subplot with polar coordinates for the radar chart\nax_radar = fig.add_subplot(4, 2, 8, polar=True)\n\n# Calculate mean values for each species\nmean_values = df.groupby(&#039;species&#039;).mean()\ncategories = list(mean_values.columns)\nvalues = mean_values.values\n\n# Number of variables\nnum_vars = len(categories)\n# Compute angle for each axis\nangles = np.linspace(0, 2 * np.pi, num_vars, endpoint=False).tolist()\nangles += angles[:1]\n\n# Plot data on the radar chart\nfor i, value in enumerate(values):\n    data = np.append(value, value[0])\n    ax_radar.plot(angles, data, linewidth=2, linestyle=&#039;solid&#039;, label=iris.target_names[i])\n    ax_radar.fill(angles, data, alpha=0.25)\n\n# Set the axis labels\nax_radar.set_yticklabels([])\nax_radar.set_xticks(angles[:-1])\nax_radar.set_xticklabels(categories)\n# Add title and legend\nax_radar.set_title(&#039;Radar Chart of Mean Feature Values by Species&#039;)\nax_radar.legend(loc=&#039;upper right&#039;, bbox_to_anchor=(1.3, 0))\n\nplt.tight_layout()\nplt.show()\n<\/code><\/pre>\n<p align=\"center\">\n  <img decoding=\"async\" src=\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_14_0-1.png\" style=\"height:700px\">\n<\/p>\n<h3><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/?size=100&id=aTCbZl5kcFhM&format=png&color=5C7CFA\" style=\"height:50px;display:inline\"> \u79bb\u6563\u578b\u6982\u7387\u5206\u5e03<\/h3>\n<hr \/>\n<p>\u79bb\u6563\u6570\u636e\u5373\u6570\u636e\u7684\u53d6\u503c\u662f\u4e0d\u8fde\u7eed\u7684\u3002\u4f8b\u5982\u63b7\u786c\u5e01\u5c31\u662f\u4e00\u4e2a\u5178\u578b\u7684\u79bb\u6563\u6570\u636e\uff0c\u56e0\u4e3a\u629b\u786c\u5e01\u53ea\u67092\u79cd\u6570\u503c\uff08\u4e5f\u5c31\u662f2\u79cd\u7ed3\u679c\uff0c\u8981\u4e48\u662f\u6b63\u9762\uff0c\u8981\u4e48\u662f\u53cd\u9762\uff09\u3002\u6982\u7387\u5206\u5e03\u6e05\u695a\u800c\u5b8c\u6574\u5730\u8868\u793a\u4e86\u968f\u673a\u53d8\u91cf $X$ \u6240\u53d6\u503c\u7684\u6982\u7387\u5206\u5e03\u60c5\u51b5\u3002\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u7684\u6982\u7387\u5206\u5e03\u53ef\u7528\u8868\u683c\u5f62\u5f0f\u6765\u8868\u793a, \u79f0\u4e4b\u4e3a\u5206\u5e03\u5217, \u89c1\u4e0b\u8868\u3002<br \/>\n$$<br \/>\n\\begin{array}{|c|c|c|c|c|}<br \/>\n\\hline X &amp; x_1 &amp; x_2 &amp; \\cdots &amp; x_k \\\\<br \/>\n\\hline P &amp; p_1 &amp; p_2 &amp; \\cdots &amp; p_k \\\\<br \/>\n\\hline<br \/>\n\\end{array}<br \/>\n$$<\/p>\n<p>\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u7684\u6982\u7387\u5206\u5e03\u5217\u5177\u6709\u4e0b\u5217\u6027\u8d28\uff1a<br \/>\n$$<br \/>\n\\begin{aligned}<br \/>\n&amp; \\sum_{k=1}^{+\\infty} p_k=1 \\\\<br \/>\n&amp; p_k \\geq 0, k=1,2, \\cdots<br \/>\n\\end{aligned}<br \/>\n$$<\/p>\n<p>\u90a3\u4e48\u4e3a\u4ec0\u4e48\u8981\u53bb\u7edf\u8ba1\u6982\u7387\u5206\u5e03\u5462\uff1f\u5f53\u7edf\u8ba1\u5b66\u5bb6\u4eec\u5f00\u59cb\u7814\u7a76\u6982\u7387\u5206\u5e03\u65f6\uff0c\u4ed6\u4eec\u770b\u5230\uff0c\u6709\u51e0\u79cd\u5f62\u72b6\u53cd\u590d\u51fa\u73b0\uff0c\u4e8e\u662f\u5c31\u7814\u7a76\u5b83\u4eec\u7684\u89c4\u5f8b\uff0c\u6839\u636e\u8fd9\u4e9b\u89c4\u5f8b\u6765\u89e3\u51b3\u7279\u5b9a\u6761\u4ef6\u4e0b\u7684\u95ee\u9898\u3002\u5927\u5bb6\u60f3\u60f3\u5f53\u5e74\u9ad8\u8003\u7684\u65f6\u5019\uff0c\u4e3a\u4e86\u5907\u6218\u8bed\u6587\u4f5c\u6587\uff0c\u53ef\u4ee5\u51c6\u5907\u4e00\u4e2a\u81ea\u5df1\u7684\u201c\u4e07\u80fd\u6a21\u677f\u201d\uff0c\u4efb\u4f55\u4f5c\u6587\u9898\u76ee\u90fd\u53ef\u4ee5\u5957\u7528\u8be5\u6a21\u677f\uff0c\u5feb\u901f\u89e3\u51b3\u4f5c\u6587\u8fd9\u4e2a\u96be\u9898\u3002\u540c\u6837\u7684\uff0c\u8bb0\u4f4f\u6982\u7387\u91cc\u8fd9\u4e9b\u7279\u6b8a\u5206\u5e03\u7684\u597d\u5904\u5c31\u662f\uff1a\u4e0b\u6b21\u9047\u5230\u7c7b\u4f3c\u7684\u95ee\u9898\uff0c\u5c31\u53ef\u4ee5\u76f4\u63a5\u5957\u7528\u201c\u6a21\u677f\u201d\uff08\u8fd9\u4e9b\u7279\u6b8a\u5206\u5e03\u7684\u89c4\u5f8b\uff09\u6765\u89e3\u51b3\u95ee\u9898\u4e86\u3002\u800c\u8fd9\u5c31\u662f\u7814\u7a76\u6982\u7387\u5206\u5e03\u7684\u610f\u4e49\u6240\u5728\u3002<\/p>\n<p><strong>\u4e24\u70b9\u5206\u5e03<\/strong><\/p>\n<p>\u5982\u679c\u968f\u673a\u53d8\u91cf $X$ \u7684\u5206\u5e03\u5217\u5982\u4e0b:<br \/>\n$$<br \/>\n\\begin{aligned}<br \/>\n&amp; P{X=1}=p(0&lt;p&lt;1) \\\\<br \/>\n&amp; P{X=0}=q=1-p<br \/>\n\\end{aligned}<br \/>\n$$<\/p>\n<p>\u5219\u79f0 $X$ \u670d\u4ece\u4e24\u70b9\u5206\u5e03\u3002\u4e24\u70b9\u5206\u5e03\u4e5f\u53eb\u4f2f\u52aa\u5229\u5206\u5e03\uff08Bernoulli\uff09\u6216 0-1 \u5206\u5e03\u3002<br \/>\n\u4e24\u70b9\u5206\u5e03\u867d\u7b80\u5355\u4f46\u5f88\u6709\u7528\u3002\u5f53\u968f\u673a\u8bd5\u9a8c\u53ea\u6709 2 \u4e2a\u53ef\u80fd\u7ed3\u679c, \u4e14\u90fd\u6709\u6b63\u6982\u7387\u65f6, \u5c31\u786e\u5b9a\u4e00\u4e2a\u670d\u4ece\u4e24\u70b9\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\u3002\u4f8b\u5982\u68c0\u67e5\u4ea7\u54c1\u8d28\u91cf\u662f\u5426\u5408\u683c; \u68c0\u67e5\u67d0\u8f66\u95f4\u7684\u7535\u529b\u6d88\u8017\u662f\u5426\u8d85\u8fc7\u8d1f\u8377; \u67d0\u5c04\u624b\u5bf9\u76ee\u6807\u7684\u4e00\u6b21\u5c04\u51fb\u662f\u5426\u4e2d\u9776\u7b49\u8bd5\u9a8c\u90fd\u53ef\u4ee5\u7528\u670d\u4ece\u4e8c\u70b9\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\u6765\u63cf\u8ff0\u3002<\/p>\n<p><strong>\u4e8c\u9879\u5206\u5e03<\/strong><\/p>\n<p>\u5982\u679c\u968f\u673a\u53d8\u91cf $X$ \u7684\u6982\u7387\u5206\u5e03\u4e3a:<br \/>\n$$<br \/>\nP(X)=k=C_n^k p^k q^{n-k}, \\mathrm{k}=0,1,2, \\cdots, n, 0&lt;p&lt;1, q=1-p<br \/>\n$$<\/p>\n<p>\u5219\u79f0 $X$ \u670d\u4ece\u53c2\u6570\u4e3a $n, p$ \u7684\u4e8c\u9879\u5206\u5e03\u3002\u5176\u4e2d, \u4e8c\u9879\u5b9a\u7406\u7684\u7cfb\u6570\u8ba1\u7b97\u65b9\u6cd5\u5982\u4e0b:<br \/>\n$$<br \/>\nC_n^k=\\frac{n!}{k!(n-k)!}<br \/>\n$$<\/p>\n<p>\u4e8c\u9879\u5206\u5e03\u8bb0\u4e3a $X \\sim B(n, p)$ \u6216 $X \\sim b(n, p)$ \u3002<\/p>\n<ul>\n<li>\n<p>\u670d\u4ece\u4e8c\u9879\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\u7684\u76f4\u89c2\u80cc\u666f\u53ef\u89e3\u91ca\u4e3a: <\/p>\n<ul>\n<li>\u91cd\u590d\u670d\u4ece\u4e8c\u9879\u5206\u5e03\u7684\u5b9e\u9a8c $n$ \u6b21, \u67d0\u4e8b\u4ef6 $A$ \u53d1\u751f\u7684\u6b21\u6570 $X$ \u662f\u670d\u4ece\u4e8c\u9879\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\u3002<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u4e8c\u9879\u5206\u5e03\u6709\u4ec0\u4e48\u7528\u5462\uff1f<\/p>\n<ul>\n<li>\u5047\u8bbe\u9047\u5230\u4e00\u4e2a\u4e8b\u60c5\uff0c\u5982\u679c\u8be5\u4e8b\u60c5\u53d1\u751f\u6b21\u6570\u56fa\u5b9a\uff0c\u800c\u60f3\u8981\u7edf\u8ba1\u7684\u662f\u6210\u529f\u7684\u6b21\u6570\uff0c\u90a3\u4e48\u5c31\u53ef\u4ee5\u7528\u4e8c\u9879\u5206\u5e03\u7684\u516c\u5f0f\u5feb\u901f\u8ba1\u7b97\u51fa\u6982\u7387\u6765\u3002<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u5982\u4f55\u5224\u65ad\u662f\u4e0d\u662f\u4e8c\u9879\u5206\u5e03\uff1f<\/p>\n<ul>\n<li>\u987e\u540d\u601d\u4e49\uff0c\u4e8c\u9879\u4ee3\u8868\u4e8b\u4ef6\u67092\u79cd\u53ef\u80fd\u7684\u7ed3\u679c\uff0c\u628a\u4e00\u79cd\u79f0\u4e3a\u6210\u529f\uff1b\u53e6\u5916\u4e00\u79cd\u79f0\u4e3a\u5931\u8d25\u3002<\/li>\n<li>\u751f\u6d3b\u4e2d\u6709\u5f88\u591a\u8fd9\u68372\u79cd\u7ed3\u679c\u7684\u4e8c\u9879\u60c5\u51b5\uff0c\u4f8b\u5982\u8868\u767d\u7ed3\u679c\u662f\u4e8c\u9879\u7684\uff0c\u4e00\u79cd\u6210\u529f\uff1b\u53e6\u4e00\u79cd\u662f\u5931\u8d25\u3002<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u4e8c\u9879\u5206\u5e03\u7b26\u5408\u4e0b\u97624\u4e2a\u7279\u70b9\uff1a<\/p>\n<ul>\n<li>\u505a\u67d0\u4ef6\u4e8b\u7684\u6b21\u6570\uff08\u4e5f\u53eb\u8bd5\u9a8c\u6b21\u6570\uff09\u662f\u56fa\u5b9a\u7684\uff0c\u7528n\u8868\u793a\u3002<\/li>\n<li>\u6bcf\u4e00\u6b21\u4e8b\u4ef6\u90fd\u6709\u4e24\u4e2a\u53ef\u80fd\u7684\u7ed3\u679c\uff08\u6210\u529f\uff0c\u6216\u8005\u5931\u8d25\uff09\u3002<\/li>\n<li>\u6bcf\u4e00\u6b21\u6210\u529f\u7684\u6982\u7387\u90fd\u662f\u76f8\u7b49\u7684\uff0c\u6210\u529f\u7684\u6982\u7387p\u7528\u8868\u793a<\/li>\n<li>\u611f\u5174\u8da3\u7684\u662f\u6210\u529fx\u6b21\u7684\u6982\u7387\u662f\u591a\u5c11\u3002<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u4e8c\u9879\u5206\u5e03\u7684\u671f\u671b<\/p>\n<ul>\n<li>$E(x)=n p$ <\/li>\n<li>\u8868\u793a\u67d0\u4e8b\u60c5\u53d1\u751f $n$ \u6b21, \u9884\u671f\u6210\u529f\u591a\u5c11\u6b21\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>\u90a3\u4e48\u77e5\u9053\u8fd9\u4e2a\u671f\u671b\u6709\u4ec0\u4e48\u7528\u5462? <\/p>\n<p>\u505a\u4efb\u4f55\u4e8b\u60c5\u4e4b\u524d, \u77e5\u9053\u9884\u671f\u7ed3\u679c\u80af\u5b9a\u4f1a\u5bf9\u540e\u9762\u7684\u51b3\u7b56\u6709\u5e2e\u52a9\u3002\u6bd4\u5982\u62cb\u786c\u5e01 5 \u6b21, \u6bcf\u6b21\u6982\u7387\u662f $1 \/ 2$, \u90a3\u4e48\u671f\u671b $E(x)=5 \\times \\frac{1}{2}=2.5$ \u6b21, \u4e5f\u5c31\u662f\u6709\u5927\u7ea6 3 \u6b21\u53ef\u4ee5\u629b\u51fa\u6b63\u9762\u3002<\/p>\n<p>\u518d\u6bd4\u5982\u6295\u8d44\u4e86 5 \u652f\u80a1\u7968\uff0c\u5047\u8bbe\u6bcf\u652f\u80a1\u7968\u8d5a\u5230\u94b1\u7684\u6982\u7387\u662f $80 \\%$, \u90a3\u4e48\u671f\u671b $E(x)=5 \\times 80 \\%=4$, \u4e5f\u5c31\u662f\u9884\u671f\u4f1a\u6709 4 \u53ea\u80a1\u7968\u6295\u8d44\u6210\u529f\u8d5a\u5230\u94b1\u3002<\/p>\n<p><strong>\u51e0\u4f55\u5206\u5e03<\/strong><\/p>\n<p>\u51e0\u4f55\u5206\u5e03\u5b9e\u9645\u4e0a\u4e0e\u4e8c\u9879\u5206\u5e03\u975e\u5e38\u7684\u50cf\uff0c\u5148\u6765\u770b\u51e0\u4f55\u5206\u5e03\u76844\u4e2a\u7279\u70b9\uff1a<\/p>\n<ol>\n<li>\u505a\u67d0\u4ef6\u4e8b\u7684\u6b21\u6570\uff08\u4e5f\u53eb\u8bd5\u9a8c\u6b21\u6570\uff09\u662f\u56fa\u5b9a\u7684\uff0c\u7528n\u8868\u793a\u3002<\/li>\n<li>\u6bcf\u4e00\u6b21\u4e8b\u4ef6\u90fd\u6709\u4e24\u4e2a\u53ef\u80fd\u7684\u7ed3\u679c\uff08\u6210\u529f\uff0c\u6216\u8005\u5931\u8d25\uff09\u3002<\/li>\n<li>\u6bcf\u4e00\u6b21\u6210\u529f\u7684\u6982\u7387\u90fd\u662f\u76f8\u7b49\u7684\uff0c\u6210\u529f\u7684\u6982\u7387p\u7528\u8868\u793a<\/li>\n<li>\u611f\u5174\u8da3\u7684\u662f\u8fdb\u884c\u6b21\u5c1d\u8bd5\u8fd9\u4e2a\u4e8b\u60c5\uff0c\u53d6\u5f97\u7b2c1\u6b21\u6210\u529f\u7684\u6982\u7387\u662f\u591a\u5927\u3002<br \/>\n\u6b63\u5982\u8bfb\u8005\u6240\u770b\u5230\u7684\uff0c\u51e0\u4f55\u5206\u5e03\u548c\u4e8c\u9879\u5206\u5e03\u7684\u533a\u522b\u53ea\u6709\u7b2c4\u70b9\uff0c\u4e5f\u5c31\u662f\u89e3\u51b3\u95ee\u9898\u76ee\u7684\u4e0d\u540c\u3002\u51e0\u4f55\u5206\u5e03\u7684\u6570\u5b66\u516c\u5f0f\u5982\u4e0b\uff1a<br \/>\n$$<br \/>\np(x)=(1-p)^{x-1} p<br \/>\n$$<\/li>\n<\/ol>\n<p>\u5176\u4e2d $p$ \u4e3a\u6210\u529f\u6982\u7387, \u5373\u4e3a\u4e86\u5728\u7b2c $x$ \u6b21\u5c1d\u8bd5\u53d6\u5f97\u7b2c 1 \u6b21\u6210\u529f, \u9996\u5148\u8981\u5931\u8d25 $(x-1)$ \u6b21\u3002<\/p>\n<p>\u7ee7\u7eed\u521a\u624d\u7684\u4f8b\u5b50\uff0c\u5047\u5982\u5728\u8868\u767d\u4e4b\u524d, \u8ba1\u7b97\u51fa\u5373\u4f7f\u5c1d\u8bd5\u8868\u767d 3 \u6b21, \u5728\u6700\u540e 1 \u6b21\u6210\u529f\u7684\u6982\u7387\u8fd8\u662f\u5c0f\u4e8e $50 \\%$,\u8fd8\u6ca1\u6709\u629b\u786c\u5e01\u7684\u6982\u7387\u9ad8\u3002\u90a3\u5c31\u8981\u8003\u8651\u6362\u4e2a\u8ffd\u6c42\u5bf9\u8c61\u3002\u6216\u8005\u9996\u5148\u63d0\u5347\u4e0b\u81ea\u5df1, \u63d0\u9ad8\u81ea\u5df1\u6bcf\u4e00\u6b21\u8868\u767d\u7684\u6982\u7387\u3002<\/p>\n<p>\u6700\u540e, \u51e0\u4f55\u5206\u5e03\u7684\u671f\u671b\u662f $E(x)=1 \/ p$ \u3002\u5047\u5982\u6bcf\u6b21\u8868\u767d\u7684\u6210\u529f\u6982\u7387\u662f $60 \\%$, \u540c\u65f6\u4e5f\u7b26\u5408\u51e0\u4f55\u5206\u5e03\u7684\u7279\u70b9, \u6240\u4ee5\u671f\u671b $E(x)=1 \/ p=1 \/ 0.6=1.67$ \u3002\u8fd9\u610f\u5473\u7740\u8868\u767d 1.67 \u6b21\uff08\u7ea6\u7b49\u4e8e 2 \u6b21\uff09\u4f1a\u6210\u529f\u3002<\/p>\n<p><strong>\u6cca\u677e\u5206\u5e03<\/strong><\/p>\n<p>\u5982\u679c\u968f\u673a\u53d8\u91cf$X$\u7684\u6982\u7387\u5206\u5e03\u4e3a\uff1a<br \/>\n$$<br \/>\nP{X}=k=\\frac{\\lambda^k e^{-\\lambda}}{k!}, k=0,1,2, \\cdots<br \/>\n$$<\/p>\n<p>\u5176\u4e2d\u5e38\u6570 $\\lambda&gt;0$, \u5219\u79f0 $X$ \u670d\u4ece\u53c2\u6570\u4e3a $\\lambda$ \u7684\u6cca\u677e\u5206\u5e03, \u8bb0\u4e3a $X \\sim P(\\lambda)$ \u3002 $k$ \u4ee3\u8868\u4e8b\u60c5\u53d1\u751f\u7684\u6b21\u6570, $\\lambda$ \u4ee3\u8868\u7ed9\u5b9a\u65f6\u95f4\u8303\u56f4\u5185\u4e8b\u60c5\u53d1\u751f\u7684\u5e73\u5747\u6b21\u6570\u3002<\/p>\n<ul>\n<li>\n<p>\u90a3\u4e48\u6cca\u677e\u5206\u5e03\u6709\u4ec0\u4e48\u7528\uff1f<\/p>\n<ul>\n<li>\u5982\u679c\u60f3\u77e5\u9053\u67d0\u4e2a\u65f6\u95f4\u8303\u56f4\u5185\uff0c\u53d1\u751f\u67d0\u4ef6\u4e8b\u60c5$x$\u6b21\u7684\u6982\u7387\u662f\u591a\u5927\u3002\u8fd9\u65f6\u5019\u5c31\u53ef\u4ee5\u7528\u6cca\u677e\u5206\u5e03\u8f7b\u677e\u641e\u5b9a\u3002<\/li>\n<li>\u6bd4\u5982\u4e00\u5929\u5185\u4e2d\u5956\u7684\u6b21\u6570\uff0c\u4e00\u4e2a\u6708\u5185\u67d0\u673a\u5668\u635f\u574f\u7684\u6b21\u6570\u7b49\u3002<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u77e5\u9053\u8fd9\u4e9b\u4e8b\u60c5\u7684\u6982\u7387\u6709\u4ec0\u4e48\u7528\u5462\uff1f<br \/>\n*\u5f53\u7136\u662f\u6839\u636e\u6982\u7387\u7684\u5927\u5c0f\u6765\u505a\u51fa\u51b3\u7b56\u4e86\u3002\u6bd4\u5982\u7ec4\u7ec7\u4e00\u6b21\u62bd\u5956\u6d3b\u52a8\uff0c\u6700\u540e\u7b97\u51fa\u6765\u4e00\u5929\u5185\u4e2d\u595610\u6b21\u7684\u6982\u7387\u90fd\u8d85\u8fc7\u4e8690%\uff0c\u7136\u540e\u8fdb\u884c\u671f\u671b\u548c\u6d3b\u52a8\u6210\u672c\u7684\u6bd4\u8f83\uff0c\u53d1\u73b0\u8981\u8d54\u4e0d\u5c11\u94b1\uff0c\u90a3\u8fd9\u4e2a\u6d3b\u52a8\u5c31\u522b\u7ec4\u7ec7\u4e86\u3002<\/p>\n<\/li>\n<\/ul>\n<p>\u6cca\u677e\u5206\u5e03\u7b26\u5408\u4ee5\u4e0b3\u4e2a\u7279\u70b9\uff1a<\/p>\n<ol>\n<li>\u4e8b\u4ef6\u662f\u72ec\u7acb\u4e8b\u4ef6\u3002<\/li>\n<li>\u5728\u4efb\u610f\u76f8\u540c\u7684\u65f6\u95f4\u8303\u56f4\u5185\uff0c\u4e8b\u4ef6\u53d1\u7684\u6982\u7387\u76f8\u540c\u3002<\/li>\n<li>\u60f3\u77e5\u9053\u67d0\u4e2a\u65f6\u95f4\u8303\u56f4\u5185\uff0c\u53d1\u751f\u67d0\u4ef6\u4e8b\u60c5\u6b21\u7684\u6982\u7387\u662f\u591a\u5927\u3002<\/li>\n<\/ol>\n<p>\u4f8b\u5982\u7ec4\u7ec7\u4e86\u4e00\u4e2a\u4fc3\u9500\u62bd\u5956\u6d3b\u52a8, \u53ea\u77e5\u9053 1 \u5929\u5185\u4e2d\u5956\u7684\u5e73\u5747\u4e2a\u6570\u4e3a 4 \u4e2a, \u60f3\u77e5\u9053 1 \u5929\u5185\u6070\u5de7\u4e2d\u5956\u6b21\u6570\u4e3a 8 \u7684\u6982\u7387\u662f\u591a\u5c11?<br \/>\n$$<br \/>\nP(X)=8=\\frac{4^8 e^{-4}}{8!}=0.0298<br \/>\n$$<\/p>\n<p>\u6cca\u677e\u6982\u7387\u8fd8\u6709\u4e00\u4e2a\u91cd\u8981\u6027\u8d28\uff0c\u5b83\u7684\u6570\u5b66\u671f\u671b\u548c\u65b9\u5dee\u76f8\u7b49\uff0c\u90fd\u7b49\u4e8e$\\lambda$\u3002<\/p>\n<p><strong>\u79bb\u6563\u578b\u6570\u636e\u5206\u5e03\u5c0f\u7ed3<\/strong><\/p>\n<ul>\n<li>\u4e8c\u70b9\u5206\u5e03: \u8868\u793a\u4e00\u6b21\u8bd5\u9a8c\u53ea\u6709\u4e24\u79cd\u7ed3\u679c\u5373\u968f\u673a\u53d8\u91cf $X$ \u53ea\u6709\u4e24\u4e2a\u53ef\u80fd\u7684\u53d6\u503c\u3002<\/li>\n<li>\u4e8c\u9879\u5206\u5e03: \u611f\u5174\u8da3\u7684\u662f\u6210\u529f $x$ \u6b21\u7684\u6982\u7387\u662f\u591a\u5c11\u3002<\/li>\n<li>\u51e0\u4f55\u5206\u5e03: \u611f\u5174\u8da3\u7684\u662f\u8fdb\u884c $x$ \u6b21\u5c1d\u8bd5\u8fd9\u4e2a\u4e8b\u60c5, \u53d6\u5f97\u7b2c 1 \u6b21\u6210\u529f\u7684\u6982\u7387\u662f\u591a\u5927\u3002<\/li>\n<li>\u6cca\u677e\u5206\u5e03: \u60f3\u77e5\u9053\u67d0\u4e2a\u8303\u56f4\u5185, \u53d1\u751f\u67d0\u4ef6\u4e8b\u60c5 $x$ \u6b21\u7684\u6982\u7387\u662f\u591a\u5927\u3002<\/li>\n<\/ul>\n<h3><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/?size=100&id=aTCbZl5kcFhM&format=png&color=5C7CFA\" style=\"height:50px;display:inline\"> \u8fde\u7eed\u578b\u6982\u7387\u5206\u5e03<\/h3>\n<hr \/>\n<p><strong>\u6982\u7387\u5bc6\u5ea6\u51fd\u6570<\/strong><\/p>\n<p>\u5bf9\u4e8e\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u7531\u4e8e\u5176\u53d6\u503c\u4e0d\u80fd\u4e00\u4e00\u5217\u4e3e\u51fa\u6765\uff0c\u56e0\u800c\u4e0d\u80fd\u7528\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u7684\u5206\u5e03\u5217\u6765\u63cf\u8ff0\u5176\u53d6\u503c\u7684\u6982\u7387\u5206\u5e03\u60c5\u51b5\u3002\u4f46\u4eba\u4eec\u5728\u5927\u91cf\u7684\u793e\u4f1a\u5b9e\u8df5\u4e2d\u53d1\u73b0\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u843d\u5728\u4efb\u4e00\u533a\u95f4 $[a, b]$ \u4e0a\u7684\u6982\u7387, \u53ef\u7528\u67d0\u4e00\u51fd\u6570 $f(x)$ \u5728 $[a, b]$ \u4e0a\u7684\u5b9a\u79ef\u5206\u6765\u8ba1\u7b97\u3002\u4e8e\u662f\u6709\u4e0b\u5217\u5b9a\u4e49:\u5bf9\u4e8e\u968f\u673a\u53d8\u91cf $X$, \u5982\u679c\u5b58\u5728\u975e\u8d1f\u53ef\u79ef\u51fd\u6570 $f(x)(-\\infty&lt;x&lt;+\\infty)$, \u4f7f\u5bf9\u4efb\u610f $a, b(a&lt;b)$ \u90fd\u6709 $P{a \\leq X \\leq b}=\\int_a^b f(x) d x$ \u3002\u5219\u79f0 $X$ \u4e3a\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf, \u5e76\u79f0 $f(x)$ \u4e3a\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf $X$ \u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff08Probability Density Function, PDF\uff09\uff0c\u7b80\u79f0\u6982\u7387\u5bc6\u5ea6\u6216\u5bc6\u5ea6\u51fd\u6570\u3002<\/p>\n<p><strong>\u7d2f\u79ef\u5206\u5e03\u51fd\u6570<\/strong><\/p>\n<p>\u4e0d\u7ba1 $X$ \u662f\u4ec0\u4e48\u7c7b\u578b\uff08\u8fde\u7eed\/\u79bb\u6563\/\u5176\u4ed6\uff09\u7684\u968f\u673a\u53d8\u91cf, \u90fd\u53ef\u4ee5\u5b9a\u4e49\u5b83\u7684\u7d2f\u79ef\u5206\u5e03\u51fd\u6570 $F_x(x)$ (cumulative distribution function, CDF), \u6709\u65f6\u7b80\u79f0\u4e3a\u5206\u5e03\u51fd\u6570\u3002\u5bf9\u4e8e\u8fde\u7eed\u6027\u968f\u673a\u53d8\u91cf, CDF \u5c31\u662f PDF \u7684\u79ef\u5206, PDF \u5c31\u662f CDF \u7684\u5bfc\u6570:<br \/>\n$$<br \/>\nF_X(x)=\\operatorname{Pr}(X \\leq x)=\\int_{- \\text {inf }}^x f_X(t) \\mathrm{d} t<br \/>\n$$<\/p>\n<p><strong>\u5747\u5300\uff08uniform\uff09\u5206\u5e03<\/strong><\/p>\n<p>\u8bbe\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf $X$ \u5728\u6709\u9650\u533a\u95f4 $[a, b]$ \u4e0a\u53d6\u503c, \u4e14\u5b83\u7684\u6982\u7387\u5bc6\u5ea6\u4e3a:<br \/>\n$$<br \/>\nf(x)=\\begin{cases}<br \/>\n\\frac{1}{b-a} &amp; a \\leq x \\leq b \\\\<br \/>\n0 &amp; \\text { \u5176\u5b83 }<br \/>\n\\end{cases}<br \/>\n$$<\/p>\n<p>\u5219\u79f0 $X$ \u670d\u4ece\u533a\u95f4 $[a, b]$ \u4e0a\u7684\u5747\u5300\u5206\u5e03, \u53ef\u8bb0\u6210 $X \\sim U[a, b]$, \u5982\u56fe 1-32 \u6240\u793a\u3002\u5176\u4e2d\u7b2c\u4e00\u79cd\u5206\u5e03\u4f7f\u7528\u5b9e\u7ebf\u8868\u793a, \u8303\u56f4\u4e3a $[0,0.5]$, \u6982\u7387\u5bc6\u5ea6\u4e3a 2 ; \u7b2c\u4e8c\u79cd\u5206\u5e03\u4f7f\u7528\u865a\u7ebf\u8868\u793a, \u8303\u56f4\u4e3a $[0.5,1.5]$,\u6982\u7387\u5bc6\u5ea6\u4e3a 1 \u3002<\/p>\n<pre><code class=\"language-python\">import numpy as np\nimport matplotlib.pyplot as plt\n\n# Define the ranges for the two distributions\nx1 = np.linspace(0, 0.5, 100)\nx2 = np.linspace(0.5, 1.5, 100)\n\n# Define the probability densities\nf1 = np.full_like(x1, 2)  # Probability density for first distribution\nf2 = np.full_like(x2, 1)  # Probability density for second distribution\n\n# Plot the first distribution\nplt.plot(x1, f1, label=&#039;U[0, 0.5]&#039;, color=&#039;blue&#039;)\n\n# Plot the second distribution\nplt.plot(x2, f2, label=&#039;U[0.5, 1.5]&#039;, color=&#039;red&#039;, linestyle=&#039;dashed&#039;)\n\n# Add labels and legend\nplt.xlabel(&#039;x&#039;)\nplt.ylabel(&#039;Probability Density&#039;)\nplt.title(&#039;Uniform Distributions&#039;)\nplt.legend()\nplt.grid(True)\nplt.show()\n<\/code><\/pre>\n<p align=\"center\">\n  <img decoding=\"async\" src=\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_24_0-1.png\" style=\"height:400px\">\n<\/p>\n<p>\u4f8b: \u8bbe\u516c\u5171\u6c7d\u8f66\u6bcf\u9694 5 \u5206\u949f\u4e00\u73ed, \u4e58\u5ba2\u5230\u7ad9\u662f\u968f\u673a\u7684, \u5219\u7b49\u8f66\u65f6\u95f4 $X$ \u670d\u4ece [0,5]\u4e0a\u7684\u5747\u5300\u5206\u5e03, \u6c42 $X$ \u7684\u5bc6\u5ea6\u51fd\u6570\u5e76\u6c42\u67d0\u4e58\u5ba2\u968f\u673a\u5730\u53bb\u4e58\u8f66\u800c\u5019\u8f66\u65f6\u95f4\u4e0d\u8d85\u8fc7 3 \u5206\u949f\u7684\u6982\u7387?<\/p>\n<p>\u89e3: $X$ \u670d\u4ece $[0,5]$ \u4e0a\u7684\u5747\u5300\u5206\u5e03, \u6545\u5176\u5bc6\u5ea6\u51fd\u6570\u4e3a:<\/p>\n<p>$$<br \/>\nf(x)=\\begin{cases}<br \/>\n\\frac{1}{5} &amp; 0 \\leq x \\leq 5 \\\\<br \/>\n0 &amp; \\text { \u5176\u5b83 }<br \/>\n\\end{cases}<br \/>\n$$<\/p>\n<p>\u5019\u8f66\u65f6\u95f4\u4e0d\u8d85\u8fc7 3 \u5206\u949f\u7684\u6982\u7387\u4e3a:<br \/>\n$$<br \/>\nP{0 \\leq X \\leq 3}=\\int_0^3 \\frac{1}{5} d x=\\frac{3}{5}<br \/>\n$$<\/p>\n<p><strong>\u6307\u6570\uff08exponential\uff09\u5206\u5e03<\/strong><\/p>\n<p>\u6307\u6570\u5206\u5e03\u53ef\u4ee5\u7528\u6765\u8868\u793a\u72ec\u7acb\u968f\u673a\u4e8b\u4ef6\u53d1\u751f\u7684\u65f6\u95f4\u95f4\u9694\uff0c\u6bd4\u5982\u65c5\u5ba2\u8fdb\u673a\u573a\u7684\u65f6\u95f4\u95f4\u9694\u7b49\u3002 \u8bb8\u591a\u7535\u5b50\u4ea7\u54c1\u7684\u5bff\u547d\u5206\u5e03\u4e00\u822c\u670d\u4ece\u6307\u6570\u5206\u5e03\u3002\u6709\u7684\u7cfb\u7edf\u7684\u5bff\u547d\u5206\u5e03\u4e5f\u53ef\u7528\u6307\u6570\u5206\u5e03\u6765\u8fd1\u4f3c\u3002\u5b83\u5728\u53ef\u9760\u6027\u7814\u7a76\u4e2d\u662f\u6700\u5e38\u7528\u7684\u4e00\u79cd\u5206\u5e03\u5f62\u5f0f\u3002<\/p>\n<p>\u8bbe\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf$X$\u7684\u6982\u7387\u5bc6\u5ea6\u4e3a\uff1a<\/p>\n<p>$$<br \/>\nf(x)=\\begin{cases}<br \/>\n\\lambda e^{-\\lambda x} &amp; x \\geq 0 \\\\<br \/>\n0 &amp; x&lt;0<br \/>\n\\end{cases}<br \/>\n$$<\/p>\n<p>\u5176\u4e2d\u5e38\u6570 $\\lambda&gt;0$, \u5219\u79f0 $X$ \u670d\u4ece\u53c2\u6570\u4e3a $\\lambda$ \u7684\u6307\u6570\u5206\u5e03, \u53ef\u8bb0\u6210 $X \\sim E(\\lambda)$<\/p>\n<p>\u5982\u4e0b\u56fe\u6240\u793a\u3002\u5176\u4e2d\u7b2c\u4e00\u79cd\u5206\u5e03\u4f7f\u7528\u5b9e\u7ebf\u8868\u793a $(\\lambda=2)$; \u7b2c\u4e8c\u79cd\u5206\u5e03\u4f7f\u7528\u865a\u7ebf\u8868\u793a $(\\lambda=1)$ \u3002<\/p>\n<pre><code class=\"language-python\">import numpy as np\nimport matplotlib.pyplot as plt\n\n# Define the exponential distribution functions\ndef exponential_pdf(x, lambda_):\n    return lambda_ * np.exp(-lambda_ * x)\n\n# Define the range for x\nx = np.linspace(0, 5, 1000)\n\n# Define the lambda values\nlambda_1 = 2\nlambda_2 = 1\n\n# Calculate the probability densities\ny1 = exponential_pdf(x, lambda_1)\ny2 = exponential_pdf(x, lambda_2)\n\n# Plot the first distribution\nplt.plot(x, y1, label=r&#039;$\\lambda=2$&#039;, color=&#039;blue&#039;)\n\n# Plot the second distribution\nplt.plot(x, y2, label=r&#039;$\\lambda=1$&#039;, color=&#039;red&#039;, linestyle=&#039;dashed&#039;)\n\n# Add labels and legend\nplt.xlabel(&#039;x&#039;)\nplt.ylabel(&#039;Probability Density&#039;)\nplt.title(&#039;Exponential Distributions&#039;)\nplt.legend()\nplt.grid(True)\nplt.show()\n<\/code><\/pre>\n<p align=\"center\">\n  <img decoding=\"async\" src=\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_27_0-1.png\" style=\"height:400px\">\n<\/p>\n<p>\u4f8b: \u8bbe\u67d0\u4eba\u9020\u536b\u661f\u7684\u5bff\u547d $X$ (\u5355\u4f4d: \u5e74)\u670d\u4ece\u53c2\u6570\u4e3a $2 \/ 3$ \u7684\u6307\u6570\u5206\u5e03\u3002\u82e5 3 \u9897\u8fd9\u6837\u7684\u536b\u661f\u540c\u65f6\u5347\u7a7a\u6295\u5165\u4f7f\u7528, \u6c42 2 \u5e74\u540e 3 \u9897\u536b\u661f\u90fd\u6b63\u5e38\u8fd0\u884c\u7684\u6982\u7387?<\/p>\n<p>\u89e3: $X$ \u7684\u5bc6\u5ea6\u51fd\u6570\u4e3a:<br \/>\n$$<br \/>\nf(x)=\\begin{cases}<br \/>\n\\frac{2}{3} \\mathrm{e}^{-\\frac{2}{3} x} &amp; x \\geq 0 \\\\<br \/>\n0 &amp; x&lt;0<br \/>\n\\end{cases}<br \/>\n$$<\/p>\n<p>\u6545 1 \u9897\u536b\u661f 2 \u5e74\u540e\u8fd8\u6b63\u5e38\u8fd0\u884c\u7684\u6982\u7387\u4e3a:<br \/>\n$$<br \/>\nP{X \\geq 2}=\\int_2^{+\\infty} \\frac{2}{3} \\mathrm{e}^{-\\frac{2}{3} x} \\mathrm{~d} x=\\mathrm{e}^{-\\frac{4}{3}}<br \/>\n$$<\/p>\n<p>\u56e0\u6b64, 2 \u5e74\u540e 3 \u9897\u536b\u661f\u90fd\u6b63\u5e38\u7684\u6982\u7387\u4e3a:<br \/>\n$$<br \/>\nP{Y=3}=\\left(\\mathrm{e}^{-\\frac{4}{3}}\\right)^3=\\mathrm{e}^{-4} \\approx 0.0183<br \/>\n$$<\/p>\n<p><strong>\u6b63\u6001\u5206\u5e03<\/strong><\/p>\n<p>\u6b63\u6001\u5206\u5e03\u53c8\u540d\u9ad8\u65af\u5206\u5e03\uff0c\u662f\u4e00\u4e2a\u5728\u6570\u5b66\u3001\u7269\u7406\u53ca\u5de5\u7a0b\u7b49\u9886\u57df\u90fd\u975e\u5e38\u91cd\u8981\u7684\u6982\u7387\u5206\u5e03\uff0c\u5728\u7edf\u8ba1\u5b66\u7684\u8bb8\u591a\u65b9\u9762\u6709\u7740\u91cd\u5927\u7684\u5f71\u54cd\u529b\u3002\u82e5\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf $X$ \u7684\u5bc6\u5ea6\u51fd\u6570\u4e3a\uff1a<\/p>\n<p>$$<br \/>\nf(x)=\\frac{1}{\\sqrt{2 \\pi} \\sigma} \\exp \\left(-\\frac{(x-\\mu)^2}{2 \\sigma^2}\\right),-\\infty&lt;x&lt;+\\infty<br \/>\n$$<br \/>\n\u5176\u4e2d, $\\mu$ \u4e3a\u5747\u503c\u3001 $\\sigma$ \u4e3a\u6807\u51c6\u5dee, $\\mu, \\sigma(\\sigma&gt;0)$ \u90fd\u4e3a\u5e38\u6570, \u5219\u79f0 $X$ \u670d\u4ece\u53c2\u6570\u4e3a $\\mu, \\sigma$ \u7684\u6b63\u6001\u5206\u5e03,\u7b80\u8bb0\u4e3a $X \\sim N\\left(\\mu, \\sigma^2\\right)$ \u3002\u56e0\u5176\u66f2\u7ebf\u5448\u949f\u5f62, \u56e0\u6b64\u53c8\u7ecf\u5e38\u79f0\u4e4b\u4e3a\u949f\u5f62\u66f2\u7ebf\u3002\u901a\u5e38\u6240\u8bf4\u7684\u6807\u51c6\u6b63\u6001\u5206\u5e03\u662f $\\mu=0, \\sigma=1$ \u7684\u6b63\u6001\u5206\u5e03\u3002<\/p>\n<p>\u6b63\u6001\u5206\u5e03\u7684\u53c2\u6570\u4e2d, $\\mu$ \u51b3\u5b9a\u4e86\u5176\u4f4d\u7f6e, \u6807\u51c6\u5dee $\\sigma^2$ \u51b3\u5b9a\u4e86\u5206\u5e03\u7684\u5e45\u5ea6\u3002<\/p>\n<p>\u5177\u4f53\u6765\u8bf4, \u82e5\u56fa\u5b9a $\\sigma$ \u800c\u6539\u53d8 $\\mu$ \u7684\u503c, \u5219\u6b63\u6001\u5206\u5e03\u5bc6\u5ea6\u66f2\u7ebf\u6cbf\u7740 $x$ \u8f74\u5e73\u884c\u79fb\u52a8, \u800c\u4e0d\u6539\u53d8\u5176\u5f62\u72b6, \u53ef\u89c1\u66f2\u7ebf\u7684\u4f4d\u7f6e\u5b8c\u5168\u7531\u53c2\u6570 $\\mu$ \u786e\u5b9a\u3002<\/p>\n<p>\u82e5\u56fa\u5b9a $\\mu$ \u800c\u6539\u53d8 $\\sigma$ \u7684\u503c, \u5219\u5f53 $\\sigma$ \u8d8a\u5c0f\u65f6\u56fe\u5f62\u53d8\u5f97\u8d8a\u9661\u5ced; \u53cd\u4e4b,\u5f53 $\\sigma$ \u8d8a\u5927\u65f6\u56fe\u5f62\u53d8\u5f97\u8d8a\u5e73\u7f13,<\/p>\n<p>\u5982\u4e0b\u56fe\u6240\u793a\u3002\u5176\u4e2d\u7b2c\u4e00\u79cd\u5206\u5e03\u4f7f\u7528\u5b9e\u7ebf\u8868\u793a $(\\mu=0, \\sigma=0.5$ );\u7b2c\u4e8c\u79cd\u5206\u5e03\u4f7f\u7528\u865a\u7ebf\u8868\u793a $(\\mu=1, \\sigma=1)$ \u3002<\/p>\n<pre><code class=\"language-python\">import numpy as np\nimport matplotlib.pyplot as plt\n\n# Define the normal distribution function\ndef normal_pdf(x, mu, sigma):\n    return (1 \/ (sigma * np.sqrt(2 * np.pi))) * np.exp(-0.5 * ((x - mu) \/ sigma)**2)\n\n# Define the range for x\nx = np.linspace(-3, 5, 1000)\n\n# Parameters for the normal distributions\nmu1, sigma1 = 0, 0.5\nmu2, sigma2 = 1, 1\n\n# Calculate the probability densities\ny1 = normal_pdf(x, mu1, sigma1)\ny2 = normal_pdf(x, mu2, sigma2)\n\n# Plot the first distribution\nplt.plot(x, y1, label=r&#039;$\\mu=0, \\sigma=0.5$&#039;, color=&#039;blue&#039;)\n\n# Plot the second distribution\nplt.plot(x, y2, label=r&#039;$\\mu=1, \\sigma=1$&#039;, color=&#039;red&#039;, linestyle=&#039;dashed&#039;)\n\n# Add labels and legend\nplt.xlabel(&#039;x&#039;)\nplt.ylabel(&#039;Probability Density&#039;)\nplt.title(&#039;Normal Distributions&#039;)\nplt.legend()\nplt.grid(True)\nplt.show()\n<\/code><\/pre>\n<p align=\"center\">\n  <img decoding=\"async\" src=\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_30_0.png\" style=\"height:400px\">\n<\/p>\n<p>\u6b63\u6001\u5206\u5e03\u4e2d\u4e00\u4e9b\u503c\u5f97\u6ce8\u610f\u7684\u91cf\uff1a<\/p>\n<ul>\n<li>\u5bc6\u5ea6\u51fd\u6570\u5173\u4e8e\u5e73\u5747\u503c\u5bf9\u79f0\u3002<\/li>\n<li>\u5e73\u5747\u503c\u4e0e\u5b83\u7684\u4f17\u6570\u4ee5\u53ca\u4e2d\u4f4d\u6570\u540c\u4e00\u6570\u503c\u3002<\/li>\n<li>68.268949%\u7684\u9762\u79ef\u5728\u5e73\u5747\u6570\u5de6\u53f3\u7684\u4e00\u4e2a\u6807\u51c6\u5dee\u8303\u56f4\u5185\u3002<\/li>\n<li>95.449974%\u7684\u9762\u79ef\u5728\u5e73\u5747\u6570\u5de6\u53f3\u4e24\u4e2a\u6807\u51c6\u5dee\u7684\u8303\u56f4\u5185\u3002<\/li>\n<li>99.730020%\u7684\u9762\u79ef\u5728\u5e73\u5747\u6570\u5de6\u53f3\u4e09\u4e2a\u6807\u51c6\u5dee\u7684\u8303\u56f4\u5185\u3002<\/li>\n<li>99.993666%\u7684\u9762\u79ef\u5728\u5e73\u5747\u6570\u5de6\u53f3\u56db\u4e2a\u6807\u51c6\u5dee\u7684\u8303\u56f4\u5185\u3002<\/li>\n<\/ul>\n<p>\u5728\u5b9e\u9645\u5e94\u7528\u4e0a\uff0c\u5e38\u8003\u8651\u4e00\u7ec4\u6570\u636e\u5177\u6709\u8fd1\u4f3c\u4e8e\u6b63\u6001\u5206\u5e03\u7684\u6982\u7387\u5206\u5e03\u3002<\/p>\n<p>\u82e5\u5176\u5047\u8bbe\u6b63\u786e\uff0c\u5219\u7ea668.3%\u6570\u503c\u5206\u5e03\u5728\u8ddd\u79bb\u5e73\u5747\u503c\u67091\u4e2a\u6807\u51c6\u5dee\u4e4b\u5185\u7684\u8303\u56f4\uff0c\u7ea695.4%\u6570\u503c\u5206\u5e03\u5728\u8ddd\u79bb\u5e73\u5747\u503c\u67092\u4e2a\u6807\u51c6\u5dee\u4e4b\u5185\u7684\u8303\u56f4\uff0c\u4ee5\u53ca\u7ea699.7%\u6570\u503c\u5206\u5e03\u5728\u8ddd\u79bb\u5e73\u5747\u503c\u67093\u4e2a\u6807\u51c6\u5dee\u4e4b\u5185\u7684\u8303\u56f4\u3002<\/p>\n<p>\u79f0\u4e3a\u201c68-95-99.7\u6cd5\u5219\u201d\u6216\u201c\u7ecf\u9a8c\u6cd5\u5219\u201d\u3002<\/p>\n<p><strong>Z\u503c\u4e0e\u6807\u51c6\u5316<\/strong><\/p>\n<p>\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97\u968f\u673a\u53d8\u91cf\u7684Z\u503c\uff08z-score\uff09\uff0c\u5f97\u77e5\u5176\u8ddd\u79bb\u5747\u503c\u6709\u591a\u5c11\u4e2a\u6807\u51c6\u5dee\u3002Z\u503c\u7684\u8ba1\u7b97\u516c\u5f0f\u4e3a\uff1a<\/p>\n<p>$$<br \/>\n\\mathrm{Z}=\\frac{x-\\mu}{\\sigma}<br \/>\n$$<\/p>\n<p>\u5176\u4e2d $x$ \u662f\u968f\u673a\u53d8\u91cf\u7684\u503c, $\\mu$ \u662f\u603b\u4f53\u5747\u503c, $\\sigma$ \u662f\u603b\u4f53\u6807\u51c6\u5dee\u3002\u5f53 $\\mu=0, \\sigma=1$ \u65f6, \u6b63\u6001\u5206\u5e03\u5c31\u6210\u4e3a\u6807\u51c6\u6b63\u6001\u5206\u5e03, \u8bb0\u4f5c $N(0,1)$ \u3002 $\\mathrm{Z}$ \u503c\u5c06\u4e24\u7ec4\u6216\u591a\u7ec4\u6570\u636e\u8f6c\u5316\u4e3a\u65e0\u5355\u4f4d\u7684 $\\mathrm{Z}$ score \u5206\u503c, \u4f7f\u5f97\u6570\u636e\u6807\u51c6\u7edf\u4e00\u5316, \u63d0\u9ad8\u4e86\u6570\u636e\u53ef\u6bd4\u6027, \u540c\u65f6\u4e5f\u524a\u5f31\u4e86\u6570\u636e\u89e3\u91ca\u6027\u3002 $\\mathrm{z}$ \u503c\u7684\u91cf\u4ee3\u8868\u7740\u5b9e\u6d4b\u503c\u548c\u603b\u4f53\u5e73\u5747\u503c\u4e4b\u95f4\u7684\u8ddd\u79bb, \u662f\u4ee5\u6807\u51c6\u5dee\u4e3a\u5355\u4f4d\u8ba1\u7b97\u3002\u5927\u4e8e\u5747\u503c\u7684\u5b9e\u6d4b\u503c\u4f1a\u5f97\u5230\u4e00\u4e2a\u6b63\u6570\u7684 $\\mathrm{z}$ \u503c,\u5c0f\u4e8e\u5747\u503c\u7684\u5b9e\u6d4b\u503c\u4f1a\u5f97\u5230\u4e00\u4e2a\u8d1f\u6570\u7684 $\\mathrm{z}$ \u503c\u3002<\/p>\n<p>\u6570\u636e\u5206\u6790\u4e0e\u6316\u6398\u4e2d\uff0c\u5f88\u591a\u65b9\u6cd5\u9700\u8981\u6837\u672c\u7b26\u5408\u4e00\u5b9a\u7684\u6807\u51c6\uff0c\u5982\u679c\u9700\u8981\u5206\u6790\u7684\u8bf8\u591a\u81ea\u53d8\u91cf\u4e0d\u662f\u540c\u4e00\u4e2a\u91cf\u7ea7\uff0c\u5c31\u4f1a\u7ed9\u5206\u6790\u5de5\u4f5c\u9020\u6210\u56f0\u96be\uff0c\u751a\u81f3\u5f71\u54cd\u540e\u671f\u5efa\u6a21\u7684\u7cbe\u51c6\u5ea6\u3002\u4e3e\u4e2a\u4f8b\u5b50\uff1a<\/p>\n<ul>\n<li>\u5047\u8bbe\uff1aA\u73ed\u7ea7\u7684\u5e73\u5747\u5206\u662f80\uff0c\u6807\u51c6\u5dee\u662f10\uff0cA\u8003\u4e8690\u5206\uff1bB\u73ed\u7684\u5e73\u5747\u5206\u662f400\uff0c\u6807\u51c6\u5dee\u662f100\uff0cB\u8003\u4e86600\u5206\u3002A\u548cB\u8c01\u7684\u6210\u7ee9\u597d\uff1f<\/li>\n<li>\u8fd9\u53ef\u4ee5\u8ba1\u7b97\u5f97\u51fa\uff0cA\u7684Z-score\u662f\uff0890-80\uff09\/10=1\uff0cB\u7684Z-score\uff08600-400\uff09\/100=2\u662f\u3002\u56e0\u6b64B\u7684\u6210\u7ee9\u66f4\u4e3a\u4f18\u5f02\u3002<\/li>\n<\/ul>\n<p>\u901a\u8fc7Z-score\u53ef\u4ee5\u6709\u6548\u7684\u628a\u6570\u636e\u8f6c\u6362\u4e3a\u7edf\u4e00\u7684\u6807\u51c6\uff0c\u5e76\u8fdb\u884c\u6bd4\u8f83\u3002\u4f46\u662f\u9700\u8981\u6ce8\u610f\uff0cZ-score\u672c\u8eab\u6ca1\u6709\u5b9e\u9645\u610f\u4e49\uff0c\u5b83\u7684\u73b0\u5b9e\u610f\u4e49\u9700\u8981\u5728\u6bd4\u8f83\u4e2d\u5f97\u4ee5\u5b9e\u73b0\uff0c\u8fd9\u4e5f\u662fZ-score\u7684\u7f3a\u70b9\u4e4b\u4e00\u3002<\/p>\n<p>Z-score\u7684\u53e6\u4e00\u4e2a\u7f3a\u70b9\u662f\u4f30\u7b97Z-score\u9700\u8981\u603b\u4f53\u7684\u5e73\u5747\u503c\u4e0e\u65b9\u5dee\uff0c\u4f46\u662f\u8fd9\u4e00\u503c\u5728\u771f\u5b9e\u7684\u5206\u6790\u4e0e\u6316\u6398\u4e2d\u5f88\u96be\u5f97\u5230\uff0c\u5927\u591a\u6570\u60c5\u51b5\u4e0b\u662f\u7528\u6837\u672c\u7684\u5747\u503c\u4e0e\u6807\u51c6\u5dee\u66ff\u4ee3\u3002<\/p>\n<p><strong>\u6b63\u6001\u5206\u5e03\u4e3a\u4ec0\u4e48\u5982\u6b64\u91cd\u8981\uff1f<\/strong><\/p>\n<p>\u6b63\u6001\u5206\u5e03\u662f\u6700\u5e38\u89c1\u4e5f\u662f\u6700\u91cd\u8981\u7684\u4e00\u79cd\u5206\u5e03\uff0c\u81ea\u7136\u754c\u53ca\u793e\u4f1a\u751f\u6d3b\u3001\u751f\u4ea7\u5b9e\u9645\u4e2d\u5f88\u591a\u968f\u673a\u53d8\u91cf\u90fd\u670d\u4ece\u6216\u8fd1\u4f3c\u670d\u4ece\u6b63\u6001\u5206\u5e03\uff0c\u4f8b\u5982\u4ea7\u54c1\u7684\u5404\u79cd\u8d28\u91cf\u6307\u6807\u3001\u6d4b\u91cf\u8bef\u5dee\u3001\u67d0\u5730\u533a\u7684\u5e74\u964d\u96e8\u91cf\u548c\u6210\u5e74\u4eba\u7684\u8eab\u9ad8\u7b49\u3002<\/p>\n<ul>\n<li>\n<p>\u4e00\u4e2a\u7ecf\u5178\u7684\u4f8b\u5b50\u662f\u9ad8\u5c14\u987f\u9876\u677f\uff1a<\/p>\n<ul>\n<li>\n<p>\u9ad8\u5c14\u987f\u9876\u677f\u7531\u4e00\u4e2a\u5782\u76f4\u653e\u7f6e\u7684\u677f\u5b50\u6784\u6210\uff0c\u677f\u5b50\u4e0a\u6709\u8bb8\u591a\u6c34\u5e73\u6392\u5217\u7684\u5c0f\u9489\u5b50\uff0c\u9489\u5b50\u4e4b\u95f4\u6709\u4e00\u5b9a\u95f4\u9694\u3002\u677f\u5b50\u7684\u9876\u90e8\u6709\u4e00\u4e2a\u6f0f\u6597\uff0c\u7528\u4e8e\u8ba9\u5c0f\u7403\uff08\u901a\u5e38\u662f\u94a2\u73e0\uff09\u901a\u8fc7\u3002\u5c0f\u7403\u4ece\u6f0f\u6597\u53e3\u4f9d\u6b21\u843d\u4e0b\uff0c\u78b0\u5230\u9489\u5b50\u540e\u968f\u673a\u5730\u5411\u5de6\u6216\u5411\u53f3\u5f39\u8df3\uff0c\u7ee7\u7eed\u5411\u4e0b\u6389\u843d\uff0c\u76f4\u5230\u5230\u5e95\u90e8\u7684\u591a\u4e2a\u683c\u5b50\u4e2d\u3002<\/p>\n<\/li>\n<li>\n<p>\u5f53\u5927\u91cf\u5c0f\u7403\u4ece\u6f0f\u6597\u4e2d\u843d\u4e0b\u65f6\uff0c\u5b83\u4eec\u7ecf\u8fc7\u9489\u5b50\u7684\u968f\u673a\u5f39\u8df3\u540e\uff0c\u6700\u7ec8\u843d\u5728\u6bcf\u4e2a\u683c\u5b50\u4e2d\u7684\u5c0f\u7403\u6570\u91cf\u5f62\u6210\u4e86\u4e00\u4e2a\u949f\u5f62\u66f2\u7ebf\uff0c\u5373\u6b63\u6001\u5206\u5e03\u3002<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p align=\"center\">\n  <img decoding=\"async\" src=\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729193312543.png\" style=\"height:300px\">\n<\/p>\n<ul>\n<li>\u9ad8\u5c14\u987f\u9876\u677f\u7684\u5173\u952e\u70b9\uff1a\n<ul>\n<li>\u5f53\u5c0f\u7403\u6570\u91cf\u8db3\u591f\u5927\u65f6\uff0c\u4e2a\u4f53\u884c\u4e3a\u7684\u968f\u673a\u6027\u88ab\u7fa4\u4f53\u884c\u4e3a\u7684\u7a33\u5b9a\u6027\u6240\u8986\u76d6\uff0c\u5f62\u6210\u4e86\u6709\u89c4\u5f8b\u7684\u5206\u5e03\u3002\u8fd9\u88ab\u79f0\u4e3a<strong>\u5927\u6570\u5b9a\u5f8b<\/strong><\/li>\n<li>\u65e0\u8bba\u5c0f\u7403\u7684\u521d\u59cb\u5206\u5e03\u5982\u4f55\uff0c\u5f53\u5176\u7ecf\u8fc7\u591a\u6b21\u72ec\u7acb\u7684\u968f\u673a\u8fc7\u7a0b\u540e\uff0c\u6700\u7ec8\u7684\u5206\u5e03\u8d8b\u5411\u4e8e\u6b63\u6001\u5206\u5e03\u3002\u8fd9\u662f\u6b63\u6001\u5206\u5e03\u5728\u7edf\u8ba1\u5b66\u548c\u81ea\u7136\u73b0\u8c61\u4e2d\u5982\u6b64\u666e\u904d\u7684\u91cd\u8981\u539f\u56e0\u3002\u8fd9\u88ab\u79f0\u4e3a<strong>\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u5927\u6570\u5b9a\u5f8b\u4e0e\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u662f\u7edf\u8ba1\u5b66\u5bb6\u603b\u7ed3\u51fa\u7684\u81ea\u7136\u73b0\u8c61\uff0c\u662f\u6982\u7387\u7edf\u8ba1\u7684\u57fa\u77f3\u3002\u5f88\u591a\u5b9a\u7406\u548c\u63a8\u8bba\u90fd\u662f\u57fa\u4e8e\u5b83\u4eec\u4e4b\u4e0a\u7684\u7814\u7a76\u3002<\/strong><\/p>\n<h2><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/?size=100&id=eEKZ8J8UAmTx&format=png&color=000000\" style=\"height:50px;display:inline\"> \u7edf\u8ba1\u5b66<\/h2>\n<hr \/>\n<hr \/>\n<p>\u7edf\u8ba1\u5b66\u65e8\u5728\u6839\u636e\u6570\u636e\u6837\u672c\u63a8\u6d4b\u603b\u60c5\u51b5\u3002\u5927\u90e8\u5206\u7edf\u8ba1\u5206\u6790\u90fd\u57fa\u4e8e\u6982\u7387\uff0c\u6240\u4ee5\u8fd9\u4e24\u65b9\u9762\u7684\u5185\u5bb9\u901a\u5e38\u517c\u800c\u6709\u4e4b\u3002<\/p>\n<p>\u7edf\u8ba1\u63a8\u65ad\u662f\u4f9d\u636e\u4ece\u603b\u4f53\u4e2d\u62bd\u53d6\u7684\u4e00\u4e2a\u7b80\u5355\u968f\u673a\u6837\u672c\u5bf9\u603b\u4f53\u8fdb\u884c\u5206\u6790\u548c\u5224\u65ad\u3002<\/p>\n<p>\u7edf\u8ba1\u63a8\u65ad\u7684\u57fa\u672c\u95ee\u9898\u53ef\u4ee5\u5206\u4e3a\u4e24\u5927\u7c7b\uff1a<\/p>\n<ul>\n<li>\u4e00\u7c7b\u662f\u53c2\u6570\u4f30\u8ba1\u95ee\u9898\n<ul>\n<li>\u70b9\u4f30\u8ba1<\/li>\n<li>\u533a\u95f4\u4f30\u8ba1<\/li>\n<\/ul>\n<\/li>\n<li>\u4e00\u7c7b\u662f\u5047\u8bbe\u68c0\u9a8c\u95ee\u9898\n<ul>\n<li>\u5047\u8bbe\u6027\u68c0\u9a8c<\/li>\n<li>\u76f8\u5173\u6027\u5206\u6790<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h3><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/?size=100&id=OjxLQPQNpbaT&format=png&color=000000\" style=\"height:50px;display:inline\"> \u70b9\u4f30\u8ba1<\/h3>\n<hr \/>\n<p>\u53c2\u6570\u662f\u6307\u603b\u4f53\u5206\u5e03\u4e2d\u7684\u672a\u77e5\u53c2\u6570\u3002\u4f8b\u5982, \u5728\u6b63\u6001\u603b\u4f53 $N\\left(\\mu, \\sigma^2\\right)$ \u4e2d, $\\mu, \\sigma^2$ \u672a\u77e5, $\\mu$ \u4e0e $\\sigma^2$ \u5c31\u662f\u53c2\u6570; \u82e5\u5728\u6307\u6570\u5206\u5e03 $E(\\lambda)$ \u7684\u603b\u4f53\u4e2d, $\\lambda$ \u672a\u77e5, \u5219 $\\lambda$ \u662f\u53c2\u6570\u3002<\/p>\n<ul>\n<li>\n<p>\u6240\u8c13\u53c2\u6570\u4f30\u8ba1\u5c31\u662f\u7531\u6837\u672c\u503c\u5bf9\u603b\u4f53\u7684\u672a\u77e5\u53c2\u6570\u4f5c\u51fa\u4f30\u8ba1\u3002<\/p>\n<\/li>\n<li>\n<p>\u70b9\u4f30\u8ba1\uff1a\u70b9\u4f30\u8ba1\u662f\u901a\u8fc7\u6837\u672c\u6570\u636e\u8ba1\u7b97\u4e00\u4e2a\u5355\u4e00\u503c\u6765\u4f30\u8ba1\u603b\u4f53\u7684\u672a\u77e5\u53c2\u6570\u3002<\/p>\n<\/li>\n<li>\n<p>\u533a\u95f4\u4f30\u8ba1\uff1a\u533a\u95f4\u4f30\u8ba1\u662f\u901a\u8fc7\u6837\u672c\u6570\u636e\u8ba1\u7b97\u4e00\u4e2a\u8303\u56f4\uff0c\u7528\u4e8e\u5305\u542b\u603b\u4f53\u672a\u77e5\u53c2\u6570\u7684\u53ef\u80fd\u503c\uff0c\u5e76\u63d0\u4f9b\u4e00\u4e2a\u7f6e\u4fe1\u6c34\u5e73\u3002<\/p>\n<\/li>\n<li>\n<p>\u6784\u9020\u70b9\u4f30\u8ba1\u7684\u4e00\u4e2a\u7ecf\u5178\u5e38\u7528\u65b9\u6cd5\u662f<strong>\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u6cd5<\/strong><\/p>\n<\/li>\n<\/ul>\n<p>\u5728\u7edf\u8ba1\u5b66\u4e2d\uff0c\u6211\u4eec\u7ecf\u5e38\u9700\u8981\u4ece\u6837\u672c\u6570\u636e\u63a8\u65ad\u603b\u4f53\u53c2\u6570\u3002\u6781\u5927\u4f3c\u7136\u4f30\u8ba1\u662f\u4e00\u79cd\u901a\u8fc7\u6700\u5927\u5316\u4f3c\u7136\u51fd\u6570\u6765\u4f30\u8ba1\u672a\u77e5\u53c2\u6570\u7684\u65b9\u6cd5\u3002\u5176\u6838\u5fc3\u601d\u60f3\u662f\u5bfb\u627e\u4f7f\u6837\u672c\u6570\u636e\u6700\u53ef\u80fd\u51fa\u73b0\u7684\u53c2\u6570\u503c\u3002<\/p>\n<p><strong>\u6781\u5927\u4f3c\u7136\u4f30\u8ba1\u7684\u76ee\u6807\u662f\u901a\u8fc7\u6837\u672c\u6570\u636e\u627e\u5230\u4e00\u7ec4\u53c2\u6570\uff0c\u4f7f\u5f97\u5728\u8fd9\u7ec4\u53c2\u6570\u4e0b\uff0c\u6570\u636e\u51fa\u73b0\u7684\u6982\u7387\u6700\u5927\u3002<\/strong><\/p>\n<p>\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u6837\u672c $X=\\left(X_1, X_2, \\ldots, X_n\\right)$ \uff0c\u5176\u89c2\u6d4b\u503c\u6765\u81ea\u4e00\u4e2a\u672a\u77e5\u53c2\u6570\u4e3a $\\theta$ \u7684\u6982\u7387\u5206\u5e03 $f(x ; \\theta)$ \u3002\u6781\u5927\u4f3c\u7136\u4f30\u8ba1\u7684\u6b65\u9aa4\u5982\u4e0b\uff1a<\/p>\n<ol>\n<li>\n<p>\u6784\u5efa\u4f3c\u7136\u51fd\u6570\uff1a\u4f3c\u7136\u51fd\u6570 $L(\\theta)$ \u662f\u7ed9\u5b9a\u6837\u672c\u6570\u636e $X$ \u7684\u60c5\u51b5\u4e0b\uff0c\u53c2\u6570 $\\theta$ \u7684\u6982\u7387\u3002\u5bf9\u4e8e\u72ec\u7acb\u540c\u5206\u5e03\u7684\u6837\u672c\uff0c\u4f3c\u7136\u51fd\u6570\u53ef\u4ee5\u8868\u793a\u4e3a:<br \/>\n$$<br \/>\nL(\\theta)=\\prod_{i=1}^n f\\left(x_i ; \\theta\\right)<br \/>\n$$<\/p>\n<\/li>\n<li>\n<p>\u6784\u5efa\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\uff1a\u7531\u4e8e\u4f3c\u7136\u51fd\u6570\u662f\u591a\u4e2a\u6982\u7387\u7684\u4e58\u79ef\uff0c\u8ba1\u7b97\u4e0a\u53ef\u80fd\u4f1a\u51fa\u73b0\u6570\u503c\u4e0b\u6ea2\u7684\u95ee\u9898\u3002\u56e0\u6b64\uff0c\u901a\u5e38\u4f7f\u7528\u5bf9\u6570\u4f3c\u7136\u51fd\u6570 $l(\\theta)$ :<br \/>\n$$<br \/>\nl(\\theta)=\\log L(\\theta)=\\sum_{i=1}^n \\log f\\left(x_i ; \\theta\\right)<br \/>\n$$<\/p>\n<\/li>\n<li>\n<p>\u6c42\u89e3\u6700\u5927\u5316\u95ee\u9898: \u627e\u5230\u4f7f\u5bf9\u6570\u4f3c\u7136\u51fd\u6570 $l(\\theta)$ \u6700\u5927\u7684\u53c2\u6570 $\\theta$ \uff0c\u5373:<br \/>\n$$<br \/>\n\\hat{\\theta}=\\arg \\max _\\theta l(\\theta)<br \/>\n$$<\/p>\n<\/li>\n<\/ol>\n<p>\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6c42\u89e3\u6781\u5927\u4f3c\u7136\u4f30\u8ba1\u7684\u95ee\u9898\u901a\u5e38\u6d89\u53ca\u5230\u4ee5\u4e0b\u6b65\u9aa4\uff1a<\/p>\n<p><strong>step 1<\/strong> \u9009\u62e9\u5408\u9002\u7684\u5047\u5b9a\u5206\u5e03\u6216\u6a21\u578b<\/p>\n<p>\u9996\u5148\uff0c\u6211\u4eec\u9700\u8981\u9009\u62e9\u4e00\u4e2a\u6982\u7387\u5206\u5e03\u6a21\u578b $f(x; \\theta)$\uff0c\u8fd9\u4e2a\u6a21\u578b\u5e94\u80fd\u591f\u5408\u7406\u5730\u63cf\u8ff0\u6211\u4eec\u6570\u636e\u7684\u5206\u5e03\u3002\u5e38\u89c1\u7684\u6a21\u578b\u5305\u62ec:<\/p>\n<ul>\n<li>\u6b63\u6001\u5206\u5e03: \u9002\u7528\u4e8e\u6570\u636e\u5448\u73b0\u51fa\u5bf9\u79f0\u7684\u949f\u5f62\u66f2\u7ebf\u7684\u60c5\u51b5\u3002<\/li>\n<li>\u6307\u6570\u5206\u5e03: \u5e38\u7528\u4e8e\u63cf\u8ff0\u5bff\u547d\u6570\u636e\u6216\u65f6\u95f4\u95f4\u9694\u6570\u636e\u3002<\/li>\n<li>\u6cca\u677e\u5206\u5e03: \u9002\u7528\u4e8e\u8ba1\u6570\u6570\u636e\uff0c\u6bd4\u5982\u5355\u4f4d\u65f6\u95f4\u5185\u7684\u4e8b\u4ef6\u53d1\u751f\u6b21\u6570\u3002<\/li>\n<\/ul>\n<p>\u5047\u5b9a\u7684\u9009\u62e9\u5f88\u5927\u7a0b\u5ea6\u4e0a\u53d6\u51b3\u4e8e\u6570\u636e\u7684\u7279\u6027\u548c\u9886\u57df\u77e5\u8bc6\u3002<\/p>\n<p>\u5982\u679c\u5b9e\u5728\u6ca1\u529e\u6cd5\u786e\u5b9a\u5047\u5b9a\u5206\u5e03\uff0c\u5b9e\u9645\u4e0a\u4e5f\u53ef\u4ee5\u901a\u8fc7\u4e0d\u540c\u7684\u6a21\u578b\u6765\u62df\u5408$f$\u3002\u6839\u636e\u5177\u4f53\u7684\u5e94\u7528\u573a\u666f\u548c\u6570\u636e\u7684\u7279\u70b9\uff0c\u6211\u4eec\u53ef\u4ee5\u9009\u62e9\u4ee5\u4e0b\u51e0\u79cd\u5e38\u89c1\u7684\u6a21\u578b\u6765\u63cf\u8ff0 $f(x ; \\theta)$\u3002<\/p>\n<ul>\n<li>\u7ebf\u6027\u6a21\u578b<\/li>\n<li>\u795e\u7ecf\u7f51\u7edc<\/li>\n<li>\u6df7\u5408\u6a21\u578b<\/li>\n<li>\u7b49\u7b49...<\/li>\n<\/ul>\n<p><strong>step 2<\/strong> \u6784\u5efa\u5bf9\u6570\u4f3c\u7136\u51fd\u6570<\/p>\n<p>\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u4e86\u4e00\u4e2a\u5408\u9002\u7684\u6982\u7387\u5206\u5e03 $f(x ; \\theta)$\uff0c\u7136\u540e\u6211\u4eec\u57fa\u4e8e\u8fd9\u4e2a\u5206\u5e03\u6784\u5efa\u4f3c\u7136\u51fd\u6570\uff1a<br \/>\n$$<br \/>\nL(\\theta)=\\prod_{i=1}^n f\\left(x_i ; \\theta\\right)<br \/>\n$$<\/p>\n<p>\u8fd9\u4e2a\u4f3c\u7136\u51fd\u6570 $L(\\theta)$ \u8868\u793a\u5728\u7ed9\u5b9a\u53c2\u6570 $\\theta$ \u4e0b\uff0c\u89c2\u6d4b\u5230\u6837\u672c\u6570\u636e $X$ \u7684\u53ef\u80fd\u6027\u3002<\/p>\n<p>\u7531\u4e8e\u76f4\u63a5\u4f7f\u7528\u4f3c\u7136\u51fd\u6570\u8fdb\u884c\u8ba1\u7b97\u53ef\u80fd\u4f1a\u9762\u4e34\u6570\u503c\u7a33\u5b9a\u6027\u95ee\u9898\uff08\u5982\u4e0b\u6ea2\uff09\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u5bf9\u6570\u4f3c\u7136\u51fd\u6570 $l(\\theta)$ \u8fdb\u884c\u8ba1\u7b97\uff0c\u901a\u8fc7\u5bf9\u6570\u53d8\u6362\uff0c\u539f\u6765\u7684\u4e58\u79ef\u5f62\u5f0f\u53d8\u4e3a\u52a0\u6cd5\u5f62\u5f0f\uff0c\u6781\u5927\u5730\u7b80\u5316\u4e86\u8ba1\u7b97\u3002<\/p>\n<p><strong>step 3<\/strong> \u6c42\u89e3\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u7684\u6781\u503c<\/p>\n<ul>\n<li>\n<p>\u4e3a\u4e86\u627e\u5230\u53c2\u6570 $\\theta$ \u4f7f\u5f97\u5bf9\u6570\u4f3c\u7136\u51fd\u6570 $l(\\theta)$ \u6700\u5927\u5316\uff0c\u6211\u4eec\u901a\u5e38\u9700\u8981\u5bf9\u5176\u8fdb\u884c\u6c42\u5bfc\uff0c\u7136\u540e\u89e3\u51fa\u5bfc\u6570\u7b49\u4e8e\u96f6\u7684\u70b9\uff0c\u5373\u627e\u5230 $l(\\theta)$ \u7684\u9a7b\u70b9\u3002<\/p>\n<\/li>\n<li>\n<p>\u6211\u4eec\u901a\u8fc7\u5bf9 $\\theta$ \u6c42\u504f\u5bfc\u6570 $\\frac{\\partial l(\\theta)}{\\partial \\theta}$ \uff0c\u5e76\u4ee4\u5176\u7b49\u4e8e\u96f6\uff0c\u5f97\u5230\u4e00\u7ec4\u65b9\u7a0b\u3002\u901a\u8fc7\u89e3\u8fd9\u4e9b\u65b9\u7a0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u53c2\u6570 $\\theta$ \u7684\u4f30\u8ba1\u503c $\\hat{\\theta}$ \u3002<\/p>\n<\/li>\n<li>\n<p>\u5bf9\u4e8e\u7b80\u5355\u7684\u5206\u5e03\uff0c\u89e3\u6790\u6c42\u89e3\u5bfc\u6570\u65b9\u7a0b\u53ef\u80fd\u662f\u53ef\u884c\u7684\u3002\u7136\u800c\uff0c\u5728\u8bb8\u591a\u60c5\u51b5\u4e0b\uff0c\u5c24\u5176\u662f\u6a21\u578b\u6bd4\u8f83\u590d\u6742\u6216\u8005\u53c2\u6570\u7ef4\u5ea6\u8f83\u9ad8\u65f6\uff0c\u89e3\u6790\u6c42\u89e3\u53d8\u5f97\u975e\u5e38\u56f0\u96be\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u901a\u5e38\u91c7\u7528\u6570\u503c\u4f18\u5316\u65b9\u6cd5\u6765\u6c42\u89e3\u8fd9\u4e9b\u65b9\u7a0b\u3002<\/p>\n<\/li>\n<li>\n<p>\u6b64\u65f6\uff0c\u4f1a\u5c06\u6c42\u89e3\u6781\u5927\u4f3c\u7136\u95ee\u9898\uff0c\u7b49\u4ef7\u8f6c\u6362\u6210\u6c42\u89e3\u8d1f\u5bf9\u6570\u4f3c\u7136\u7684\u6781\u5c0f\u503c\u95ee\u9898\uff0c\u5373\u4f18\u5316\u95ee\u9898\u3002\u8d1f\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u5f62\u5f0f\u5982\u4e0b:<br \/>\n$$<br \/>\n-l(\\theta)=-\\sum_{i=1}^n \\log f\\left(x_i ; \\theta\\right)<br \/>\n$$<br \/>\n\u901a\u8fc7\u6700\u5c0f\u5316\u8fd9\u4e2a\u8d1f\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\uff0c\u6211\u4eec\u53ef\u4ee5\u627e\u5230\u4f7f\u6837\u672c\u6570\u636e\u51fa\u73b0\u6982\u7387\u6700\u5927\u7684\u53c2\u6570 $\\theta$\u3002 \u5e38\u7528\u7684\u4f18\u5316\u7b97\u6cd5\u5305\u62ec\uff1a\u68af\u5ea6\u4e0b\u964d\u6cd5\uff0c\u725b\u987f\u6cd5\uff0cEM\u7b97\u6cd5\u7b49\uff0c\u6211\u4eec\u5c06\u5728\u5176\u4ed6\u7ae0\u8282\u91cc\u8be6\u7ec6\u4ecb\u7ecd<\/p>\n<\/li>\n<\/ul>\n<h3>\u5bf9\u6570\u8fb9\u9645\u4f3c\u7136\uff08Log Marginal Likelihood\uff09<\/h3>\n<p>\u5b83\u8868\u793a\u5728\u7ed9\u5b9a\u6a21\u578b\u53c2\u6570\u4e0b\u89c2\u6d4b\u6570\u636e\u7684\u603b\u4f53\u6982\u7387\u3002\u4e3a\u4e86\u66f4\u597d\u5730\u7406\u89e3\u5bf9\u6570\u8fb9\u9645\u4f3c\u7136\uff0c\u53ef\u4ee5\u5c06\u5176\u4e0e\u6781\u5927\u4f3c\u7136\u4f30\u8ba1\u8fdb\u884c\u5bf9\u6bd4\u548c\u89e3\u91ca\u3002<\/p>\n<p>\u8fb9\u9645\u4f3c\u7136\u662f\u901a\u8fc7\u5bf9\u53c2\u6570\u7684\u8054\u5408\u5206\u5e03\u79ef\u5206\u5f97\u5230\u7684\uff0c\u5176\u76ee\u6807\u662f\u8ba1\u7b97\u89c2\u6d4b\u6570\u636e $X$ \u7684\u6574\u4f53\u6982\u7387 $P(X)$ \u3002<\/p>\n<p>\u5bf9\u6570\u8fb9\u9645\u4f3c\u7136\u5b9a\u4e49\u4e3a:<br \/>\n$$<br \/>\n\\log P(X)=\\log \\int P(X \\mid \\theta) P(\\theta) d \\theta<br \/>\n$$<\/p>\n<p>\u8fd9\u4e2a\u79ef\u5206\u8003\u8651\u4e86\u6240\u6709\u53ef\u80fd\u7684\u53c2\u6570\u503c\u53ca\u5176\u5148\u9a8c\u6982\u7387\uff0c\u8868\u793a\u5728\u6a21\u578b\u4e0b\u89c2\u6d4b\u5230\u6570\u636e $X$ \u7684\u603b\u6982\u7387\u3002<\/p>\n<p><strong>\u6781\u5927\u4f3c\u7136\u4f30\u8ba1 vs. \u5bf9\u6570\u8fb9\u9645\u4f3c\u7136<\/strong><\/p>\n<ul>\n<li>\n<p>\u6781\u5927\u4f3c\u7136\u4f30\u8ba1 (MLE)\uff1a\u6700\u5927\u5316\u7ed9\u5b9a\u53c2\u6570\u4e0b\u89c2\u6d4b\u6570\u636e\u7684\u4f3c\u7136\u3002\u5b83\u4e0d\u8003\u8651\u53c2\u6570\u7684\u5148\u9a8c\u5206\u5e03\uff0c\u53ea\u5173\u6ce8\u627e\u5230\u6700\u53ef\u80fd\u7684\u53c2\u6570\u503c:<br \/>\n$$<br \/>\n\\hat{\\theta}_{\\mathrm{MLE}}=\\arg \\max _\\theta \\log P(X \\mid \\theta)<br \/>\n$$<\/p>\n<\/li>\n<li>\n<p>\u5bf9\u6570\u8fb9\u9645\u4f3c\u7136\uff1a\u8003\u8651\u4e86\u53c2\u6570\u7684\u5148\u9a8c\u5206\u5e03\uff0c\u901a\u8fc7\u5bf9\u6240\u6709\u53ef\u80fd\u7684\u53c2\u6570\u503c\u8fdb\u884c\u79ef\u5206\u8ba1\u7b97\u89c2\u6d4b\u6570\u636e\u7684\u6574\u4f53\u6982\u7387:<br \/>\n$$<br \/>\n\\log P(X)=\\log \\int P(X \\mid \\theta) P(\\theta) d \\theta<br \/>\n$$<\/p>\n<\/li>\n<\/ul>\n<p>\u4ece\u8ba1\u7b97\u590d\u6742\u6027\u6765\u8bf4\uff1a<\/p>\n<ul>\n<li>\u6781\u5927\u4f3c\u7136\u4f30\u8ba1\uff1a\u901a\u5e38\u901a\u8fc7\u4f18\u5316\u7b97\u6cd5\u76f4\u63a5\u627e\u5230\u4f7f\u5bf9\u6570\u4f3c\u7136\u6700\u5927\u7684\u53c2\u6570\u503c\uff0c\u8ba1\u7b97\u76f8\u5bf9\u7b80\u5355\u3002<\/li>\n<li>\u5bf9\u6570\u8fb9\u9645\u4f3c\u7136\uff1a\u9700\u8981\u5bf9\u53c2\u6570\u7a7a\u95f4\u8fdb\u884c\u79ef\u5206\uff0c\u8ba1\u7b97\u590d\u6742\u4e14\u901a\u5e38\u4e0d\u53ef\u884c\uff0c\u7279\u522b\u662f\u5728\u9ad8\u7ef4\u53c2\u6570\u7a7a\u95f4\u4e2d\u3002<\/li>\n<\/ul>\n<h3>\u8499\u7279\u5361\u6d1b\u91c7\u6837\uff08Monte Carlo Sampling\uff09<\/h3>\n<p>\u7531\u4e8e\u5bf9\u6570\u8fb9\u9645\u4f3c\u7136\u6d89\u53ca\u5bf9\u9ad8\u7ef4\u53c2\u6570\u7a7a\u95f4\u8fdb\u884c\u79ef\u5206\uff0c\u76f4\u63a5\u8ba1\u7b97\u901a\u5e38\u662f\u4e0d\u53ef\u884c\u7684\u3002\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e00\u95ee\u9898\uff0c\u53ef\u4ee5\u5f15\u5165\u8499\u7279\u5361\u6d1b\u91c7\u6837\u65b9\u6cd5\u3002\u8499\u7279\u5361\u6d1b\u91c7\u6837\u662f\u4e00\u79cd\u901a\u8fc7\u968f\u673a\u6837\u672c\u4f30\u8ba1\u590d\u6742\u79ef\u5206\u7684\u6570\u503c\u65b9\u6cd5\uff0c\u5e7f\u6cdb\u5e94\u7528\u4e8e\u7edf\u8ba1\u5b66\u548c\u673a\u5668\u5b66\u4e60\u4e2d\u3002<\/p>\n<p><strong>\u8499\u7279\u5361\u6d1b\u91c7\u6837\u7684\u57fa\u672c\u6b65\u9aa4\u5982\u4e0b<\/strong><\/p>\n<p>\u539f\u6765\u7684\u79ef\u5206\u53ef\u4ee5\u5199\u6210\u671f\u671b\u7684\u5f62\u5f0f $\\int p(x \\mid z ; \\theta) p(z) \\mathrm{d} z=\\mathbb{E}_{z \\sim p(z)}[p(x \\mid z ; \\theta)]$, \u7136\u540e\u5229\u7528\u671f\u671b\u6cd5\u6c42\u79ef\u5206, \u6b65\u9aa4\u5982\u4e0b\u3002<\/p>\n<p>(1) \u4ece $p(z)$ \u4e2d\u591a\u6b21\u91c7\u6837 $z_1, z_2, \\cdots, z_m$ \uff1b<\/p>\n<p>(2) \u6839\u636e $p(x \\mid z ; \\theta)$ \u8ba1\u7b97 $x_1, x_2, \\cdots, x_m$<\/p>\n<p>(3) \u6c42 $x$ \u7684\u5747\u503c\u3002\u7528\u6570\u5b66\u8868\u8fbe\u4e3a:<\/p>\n<p>$$<br \/>\n\\int p(x \\mid z ; \\theta) p(z) \\mathrm{d} z=\\mathbb{E}_{z \\sim p(z)}[p(x \\mid z ; \\theta)] \\approx \\frac{1}{m} \\sum_{j=1}^m p\\left(x_j \\mid z_j ; \\theta\\right)<br \/>\n$$<\/p>\n<p>\u901a\u8fc7\u5bf9 $z$ \u591a\u6b21\u91c7\u6837\uff0c\u53ef\u4ee5\u8ba1\u7b97 $\\nabla_\\theta L(\\theta ; X)$ \u7684\u8fd1\u4f3c\u503c\u3002<\/p>\n<p><strong>\u7b80\u5355\u6765\u8bf4, \u8499\u7279\u5361\u6d1b\u91c7\u6837\u5c31\u662f\u901a\u8fc7\u6837\u672c\u7684\u5747\u503c\u6765\u8fd1\u4f3c\u603b\u4f53\u7684\u79ef\u5206\u3002<\/strong><\/p>\n<h3><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/dusk\/64\/000000\/cheap-2.png\" style=\"height:50px;display:inline\">  \u793a\u4f8b\uff1a\u7ebf\u6027\u56de\u5f52\u6a21\u578b\u7684\u6781\u5927\u4f3c\u7136\u6c42\u89e3<\/h3>\n<hr \/>\n<p>\u5047\u8bbe\u6211\u4eec\u7684\u7ebf\u6027\u6a21\u578b\u662f:<br \/>\n$$<br \/>\ny_i=\\theta^T x_i+\\epsilon_i<br \/>\n$$<\/p>\n<p>\u5176\u4e2d\uff0c$\\theta$ \u662f\u6211\u4eec\u9700\u8981\u4f30\u8ba1\u7684\u53c2\u6570\u5411\u91cf\uff0c$x_i$ \u662f\u81ea\u53d8\u91cf\uff0c$y_i$ \u662f\u56e0\u53d8\u91cf\uff0c$\\epsilon_i$ \u662f\u8bef\u5dee\u9879\u3002\u5047\u8bbe\u8bef\u5dee\u9879 $\\epsilon_i$ \u72ec\u7acb\u540c\u5206\u5e03\u4e14\u670d\u4ece\u5747\u503c\u4e3a\u96f6\u3001\u65b9\u5dee\u4e3a $\\sigma^2$ \u7684\u6b63\u6001\u5206\u5e03\uff0c\u5373\uff1a<br \/>\n$$<br \/>\n\\epsilon_i \\sim \\mathcal{N}\\left(0, \\sigma^2\\right)<br \/>\n$$<\/p>\n<p>\u8fd9\u610f\u5473\u7740\uff0c\u5bf9\u4e8e\u6bcf\u4e00\u4e2a\u89c2\u6d4b\u503c $y_i$\uff0c\u6761\u4ef6\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3a:<br \/>\n$$<br \/>\nf\\left(y_i \\mid x_i ; \\theta, \\sigma^2\\right)=\\frac{1}{\\sqrt{2 \\pi \\sigma^2}} \\exp \\left(-\\frac{\\left(y_i-\\theta^T x_i\\right)^2}{2 \\sigma^2}\\right)<br \/>\n$$<\/p>\n<p>\u8fd9\u662f\u56e0\u4e3a\u6211\u4eec\u5047\u8bbe\u8bef\u5dee\u9879 $\\epsilon_i$ \u670d\u4ece\u6b63\u6001\u5206\u5e03\uff0c\u800c $y_i$ \u662f $\\theta^T x_i$ \u548c\u8bef\u5dee\u9879 $\\epsilon_i$ \u7684\u7ebf\u6027\u7ec4\u5408\u3002\u7531\u4e8e\u6b63\u6001\u5206\u5e03\u7684\u7ebf\u6027\u7ec4\u5408\u4ecd\u7136\u662f\u6b63\u6001\u5206\u5e03\uff0c\u56e0\u6b64 $y_i$ \u5728\u7ed9\u5b9a $x_i$ \u7684\u60c5\u51b5\u4e0b\uff0c\u670d\u4ece\u5747\u503c\u4e3a $\\theta^T x_i$\u3001\u65b9\u5dee\u4e3a $\\sigma^2$ \u7684\u6b63\u6001\u5206\u5e03\u3002\u53c2\u8003\u4e0b\u8ff0\u6b63\u6001\u5206\u5e03\u7684\u516c\u5f0f\uff0c\u53ef\u6784\u5efa\u4e0a\u5f0f\uff1a<br \/>\n$$<br \/>\nf(x)=\\frac{1}{\\sqrt{2 \\pi} \\sigma} \\exp \\left(-\\frac{(x-\\mu)^2}{2 \\sigma^2}\\right)<br \/>\n$$<\/p>\n<p>\u7531\u4e8e\u4e58\u6cd5\u5f62\u5f0f\u7684\u4f3c\u7136\u51fd\u6570\u5728\u8ba1\u7b97\u4e0a\u53ef\u80fd\u4f1a\u9047\u5230\u6570\u503c\u4e0b\u6ea2\u7684\u95ee\u9898\uff0c\u6211\u4eec\u901a\u5e38\u91c7\u7528\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u3002\u5bf9\u6570\u4f3c\u7136\u51fd\u6570 $l(\\theta, \\sigma^2)$ \u662f\u4f3c\u7136\u51fd\u6570 $L(\\theta, \\sigma^2)$ \u7684\u5bf9\u6570\uff1a<br \/>\n$$<br \/>\nl\\left(\\theta, \\sigma^2\\right)=\\log \\left(\\prod_{i=1}^n \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} \\exp \\left(-\\frac{\\left(y_i-\\theta^T x_i\\right)^2}{2 \\sigma^2}\\right)\\right)<br \/>\n$$<\/p>\n<p>$$<br \/>\nl\\left(\\theta, \\sigma^2\\right)=\\sum_{i=1}^n \\log \\left(\\frac{1}{\\sqrt{2 \\pi \\sigma^2}} \\exp \\left(-\\frac{\\left(y_i-\\theta^T x_i\\right)^2}{2 \\sigma^2}\\right)\\right)<br \/>\n$$<\/p>\n<p>\u5229\u7528\u5bf9\u6570\u7684\u6027\u8d28 $\\log(ab) = \\log a + \\log b$ \u548c $\\log(a^b) = b \\log a$\uff0c\u5bf9\u4e0a\u5f0f\u8fdb\u884c\u7b80\u5316\uff1a<\/p>\n<p>$$<br \/>\nl\\left(\\theta, \\sigma^2\\right)=\\sum_{i=1}^n\\left(\\log \\frac{1}{\\sqrt{2 \\pi \\sigma^2}}+\\log \\exp \\left(-\\frac{\\left(y_i-\\theta^T x_i\\right)^2}{2 \\sigma^2}\\right)\\right)<br \/>\n$$<\/p>\n<p>\u9996\u5148\uff0c\u5904\u7406\u5e38\u6570\u9879\uff1a<br \/>\n$$<br \/>\n\\log \\frac{1}{\\sqrt{2 \\pi \\sigma^2}}=-\\frac{1}{2} \\log \\left(2 \\pi \\sigma^2\\right)<br \/>\n$$<\/p>\n<p>\u63a5\u4e0b\u6765\uff0c\u5904\u7406\u6307\u6570\u9879\u7684\u5bf9\u6570:<br \/>\n$$<br \/>\n\\log \\exp \\left(-\\frac{\\left(y_i-\\theta^T x_i\\right)^2}{2 \\sigma^2}\\right)=-\\frac{\\left(y_i-\\theta^T x_i\\right)^2}{2 \\sigma^2}<br \/>\n$$<\/p>\n<p>\u5c06\u8fd9\u4e24\u90e8\u5206\u7ed3\u5408\uff0c\u6211\u4eec\u5f97\u5230:<br \/>\n$$<br \/>\nl\\left(\\theta, \\sigma^2\\right)=\\sum_{i=1}^n\\left(-\\frac{1}{2} \\log \\left(2 \\pi \\sigma^2\\right)-\\frac{\\left(y_i-\\theta^T x_i\\right)^2}{2 \\sigma^2}\\right)<br \/>\n$$<\/p>\n<ul>\n<li>\n<p>\u6ce8\u610f\uff0c\u4e0a\u8ff0\u63a8\u5bfc\u7684\u6700\u540e\u7ed3\u679c\u4e2d\uff0c\u5e26\u6709\u53c2\u6570 $\\theta^T$ \u7684\u9879\uff0c\u5b9e\u9645\u4e0a\u5c31\u662fMSE\uff01<\/p>\n<\/li>\n<li>\n<p>\u5728\u540e\u7eed\u7684\u5b66\u4e60\u4e2d\uff0cMSE\u7ecf\u5e38\u4f5c\u4e3a\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u7684\u635f\u5931\u51fd\u6570\uff0c\u6765\u8861\u91cf\u6a21\u578b\u7684\u9884\u6d4b\u7ed3\u679c\u548c\u771f\u503c\uff0c\u5176\u672c\u8d28\u539f\u7406\u5b9e\u9645\u4e0a\u5c31\u662f\u6781\u5927\u4f3c\u7136\u4f30\u8ba1\uff01<\/p>\n<\/li>\n<\/ul>\n<h3><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/dusk\/64\/000000\/cheap-2.png\" style=\"height:50px;display:inline\">  \u793a\u4f8b\uff1a\u751f\u6210\u6a21\u578b\u7684\u5bf9\u6570\u8fb9\u754c\u4f3c\u7136\u6c42\u89e3<\/h3>\n<hr \/>\n<p>\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u7b80\u5355\u7684\u751f\u6210\u6a21\u578b\uff0c\u7528\u4e8e\u751f\u6210\u6570\u636e $X$ \u3002\u751f\u6210\u6a21\u578b\u7684\u53c2\u6570\u4e3a $\\theta$ \uff0c\u6211\u4eec\u5e0c\u671b\u4f30\u8ba1\u8fd9\u4e9b\u53c2\u6570\u4f7f\u5f97\u6a21\u578b\u80fd\u591f\u5f88\u597d\u5730\u751f\u6210\u89c2\u6d4b\u6570\u636e\u3002 <\/p>\n<p><strong>\u751f\u6210\u6a21\u578b\u7684\u5b9a\u4e49<\/strong><\/p>\n<p>\u5047\u8bbe\u89c2\u6d4b\u6570\u636e $X$ \u662f\u7531\u6f5c\u5728\u53d8\u91cf $Z$ \u751f\u6210\u7684\u3002\u6a21\u578b\u53c2\u6570 $\\theta$ \u63a7\u5236\u751f\u6210\u8fc7\u7a0b\u3002\u5047\u8bbe\u6f5c\u5728\u53d8\u91cf $Z$ \u670d\u4ece\u5747\u503c\u4e3a $\\mu$ \u65b9\u5dee\u4e3a $\\sigma^2$ \u7684\u6b63\u6001\u5206\u5e03:<br \/>\n$$<br \/>\nZ \\sim \\mathcal{N}\\left(\\mu, \\sigma^2\\right)<br \/>\n$$<\/p>\n<p>\u89c2\u6d4b\u6570\u636e $X$ \u5728\u7ed9\u5b9a\u6f5c\u5728\u53d8\u91cf $Z$ \u548c\u53c2\u6570 $\\theta$ \u7684\u60c5\u51b5\u4e0b\uff0c\u670d\u4ece\u5747\u503c\u4e3a $f(Z ; \\theta)$ \u3001\u65b9\u5dee\u4e3a $\\sigma^2$ \u7684\u6b63\u6001\u5206\u5e03:<br \/>\n$$<br \/>\nX \\mid Z, \\theta \\sim \\mathcal{N}\\left(f(Z ; \\theta), \\sigma^2\\right)<br \/>\n$$<\/p>\n<p><strong>\u8fb9\u9645\u4f3c\u7136\u548c\u5bf9\u6570\u8fb9\u9645\u4f3c\u7136<\/strong><\/p>\n<p>\u6211\u4eec\u611f\u5174\u8da3\u7684\u662f\u89c2\u6d4b\u6570\u636e $X$ \u7684\u8fb9\u9645\u4f3c\u7136 $P(X)$ \uff0c\u5b83\u53ef\u4ee5\u901a\u8fc7\u5bf9\u6f5c\u5728\u53d8\u91cf $Z$ \u7684\u8054\u5408\u5206\u5e03\u8fdb\u884c\u79ef\u5206\u5f97\u5230:<br \/>\n$$<br \/>\nP(X)=\\int P(X \\mid Z, \\theta) P(Z) d Z<br \/>\n$$<\/p>\n<p>\u5bf9\u6570\u8fb9\u9645\u4f3c\u7136\uff08Log Marginal Likelihood\uff09\u5219\u662f\u8fb9\u9645\u4f3c\u7136\u7684\u5bf9\u6570:<br \/>\n$$<br \/>\n\\log P(X)=\\log \\int P(X \\mid Z, \\theta) P(Z) d Z<br \/>\n$$<\/p>\n<p><strong>\u5bf9\u6570\u8fb9\u9645\u4f3c\u7136\u7684\u63a8\u5bfc<\/strong><\/p>\n<ol>\n<li>\u8054\u5408\u5206\u5e03:<\/li>\n<\/ol>\n<p>\u9996\u5148\uff0c\u6211\u4eec\u9700\u8981\u77e5\u9053\u89c2\u6d4b\u6570\u636e $X$ \u548c\u6f5c\u5728\u53d8\u91cf $Z$ \u7684\u8054\u5408\u5206\u5e03 $P(X, Z \\mid \\theta)$ :<br \/>\n$$<br \/>\nP(X, Z \\mid \\theta)=P(X \\mid Z, \\theta) P(Z)<br \/>\n$$<\/p>\n<p>\u7531\u4e8e $Z$ \u670d\u4ece\u6b63\u6001\u5206\u5e03 $\\mathcal{N}\\left(\\mu, \\sigma^2\\right)$ \uff0c\u6211\u4eec\u6709:<br \/>\n$$<br \/>\nP(Z)=\\frac{1}{\\sqrt{2 \\pi \\sigma^2}} \\exp \\left(-\\frac{(Z-\\mu)^2}{2 \\sigma^2}\\right)<br \/>\n$$<\/p>\n<p>\u540c\u65f6\uff0c\u7531\u4e8e $X \\mid Z, \\theta$ \u670d\u4ece\u5747\u503c\u4e3a $f(Z ; \\theta)$ \u3001\u65b9\u5dee\u4e3a $\\sigma^2$ \u7684\u6b63\u6001\u5206\u5e03\uff0c\u6211\u4eec\u6709:<br \/>\n$$<br \/>\nP(X \\mid Z, \\theta)=\\frac{1}{\\sqrt{2 \\pi \\sigma^2}} \\exp \\left(-\\frac{(X-f(Z ; \\theta))^2}{2 \\sigma^2}\\right)<br \/>\n$$<\/p>\n<p>\u56e0\u6b64\uff0c\u8054\u5408\u5206\u5e03\u53ef\u4ee5\u5199\u6210:<br \/>\n$$<br \/>\nP(X, Z \\mid \\theta)=\\left(\\frac{1}{\\sqrt{2 \\pi \\sigma^2}}\\right)^2 \\exp \\left(-\\frac{(X-f(Z ; \\theta))^2+(Z-\\mu)^2}{2 \\sigma^2}\\right)<br \/>\n$$<\/p>\n<ol start=\"2\">\n<li>\u5bf9\u6570\u8fb9\u9645\u4f3c\u7136:<\/li>\n<\/ol>\n<p>\u4e3a\u4e86\u5f97\u5230\u8fb9\u9645\u4f3c\u7136 $P(X)$ \uff0c\u6211\u4eec\u9700\u8981\u5bf9\u6f5c\u5728\u53d8\u91cf $Z$ \u79ef\u5206:<br \/>\n$$<br \/>\nP(X)=\\int P(X \\mid Z, \\theta) P(Z) d Z=\\int\\left(\\frac{1}{\\sqrt{2 \\pi \\sigma^2}}\\right)^2 \\exp \\left(-\\frac{(X-f(Z ; \\theta))^2+(Z-\\mu)^2}{2 \\sigma^2}\\right)<br \/>\n$$<\/p>\n<p>\u5bf9\u6570\u8fb9\u9645\u4f3c\u7136\u662f\u8fb9\u9645\u4f3c\u7136\u7684\u5bf9\u6570:<br \/>\n$$<br \/>\n\\log P(X)=\\log \\int\\left(\\frac{1}{\\sqrt{2 \\pi \\sigma^2}}\\right)^2 \\exp \\left(-\\frac{(X-f(Z ; \\theta))^2+(Z-\\mu)^2}{2 \\sigma^2}\\right) d Z<br \/>\n$$<\/p>\n<p><strong>\u8fd9\u4e2a\u79ef\u5206\u901a\u5e38\u662f\u590d\u6742\u4e14\u96be\u4ee5\u89e3\u6790\u6c42\u89e3\u7684\u3002<\/strong><\/p>\n<p><strong>\u7531\u4e8e\u76f4\u63a5\u8ba1\u7b97\u5bf9\u6570\u8fb9\u9645\u4f3c\u7136\u901a\u5e38\u4e0d\u53ef\u884c\uff0c\u6211\u4eec\u4f7f\u7528\u53d8\u5206\u63a8\u65ad\u65b9\u6cd5\u8fdb\u884c\u8fd1\u4f3c\u3002\u8be6\u89c1\u4fe1\u606f\u8bba\u6559\u7a0b<\/strong><\/p>\n<h3><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/?size=100&id=58824&format=png&color=000000\" style=\"height:50px;display:inline\">  \u533a\u95f4\u4f30\u8ba1<\/h3>\n<hr \/>\n<ul>\n<li>\n<p>\u70b9\u4f30\u8ba1\u503c\u7ecf\u5e38\u6709\u5dee\u5f02\u3002\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u6709\u4e86\u533a\u95f4\u4f30\u8ba1\u7684\u505a\u6cd5\u3002\u901a\u4fd7\u5730\u8bb2\uff1a\u533a\u95f4\u4f30\u8ba1\u662f\u5728\u70b9\u4f30\u8ba1\u7684\u57fa\u7840\u4e0a\uff0c\u7ed9\u4e00\u4e2a\u5408\u7406\u53d6\u503c\u8303\u56f4\u3002<\/p>\n<\/li>\n<li>\n<p>\u6bd4\u5982\uff1a\u62bd\u6837\u9e21\u817f\u7684\u5e73\u5747\u91cd\u91cf\u4e3a150\u514b\uff0c\u662f\u4e00\u4e2a\u70b9\u4f30\u8ba1\u503c\u3002\u62bd\u6837\u9e21\u817f\u7684\u5e73\u5747\u91cd\u91cf\u4e3a145\u514b\u5230155\u514b\u4e4b\u95f4\uff0c\u662f\u4e00\u4e2a\u533a\u95f4\u4f30\u8ba1\u3002<\/p>\n<ul>\n<li>\u5176\u4e2d\uff0c145\u5230155\u79f0\u4e3a\u7f6e\u4fe1\u533a\u95f4\u3002\u8fd9\u5f88\u7b26\u5408\u4eba\u4eec\u7684\u5e38\u89c4\u7406\u89e3\uff1a\u4e1c\u897f\u5f88\u96be100%\u51c6\u786e\uff0c\u6709\u4e2a\u8303\u56f4\u4e5f\u662f\u53ef\u4ee5\u7406\u89e3\u7684\u3002<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u4f46\u8fd9\u4e2a\u8303\u56f4\u6709\u591a\u5927\u53ef\u4fe1\u5ea6\u5462\uff1f<\/p>\n<ul>\n<li>\u901a\u5e38\u7528\u7f6e\u4fe1\u6c34\u5e73\u6765\u8861\u91cf\uff0c\u5373\uff1a\u201c\u6709\u591a\u5927\u628a\u63e1\uff0c\u771f\u5b9e\u503c\u5728\u7f6e\u4fe1\u533a\u95f4\u5185\u201d\u3002\u4e00\u822c\u7528 $(1-\\alpha)$ \u8868\u793a\u3002\u5982\u679c $\\alpha$ \u53d6 0.05 , \u5219\u7f6e\u4fe1\u6c34\u5e73\u4e3a 0.95 , \u5373 $95 \\%$ \u7684\u628a\u63e1\u3002 $\\alpha$ \u6307\u7684\u662f\u663e\u8457\u6027\u6c34\u5e73\u3002<\/li>\n<li>\u7f6e\u4fe1\u533a\u95f4\u4e0e\u7f6e\u4fe1\u6c34\u5e73\u8fde\u8d77\u6765\uff0c\u5b8c\u6574\u7684\u8868\u8fbe\u4e3a\uff1a\u201c\u670995%\uff08\u7f6e\u4fe1\u6c34\u5e73\uff09\u7684\u628a\u63e1\uff0c\u9e21\u817f\u5e73\u5747\u91cd\u91cf\u5728145\u81f3155\u514b\u4e4b\u95f4\uff08\u7f6e\u4fe1\u533a\u95f4\uff09\u3002\u201d<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u6709\u5c0f\u4f19\u4f34\u4f1a\u597d\u5947\uff0c\u4e3a\u4ec0\u4e48\u7f6e\u4fe1\u6c34\u5e73\u4e0d\u662f100%\uff01<\/p>\n<ul>\n<li>\u901a\u4fd7\u5730\u8bf4\uff0c\u5f53\u7f6e\u4fe1\u6c34\u5e73\u592a\u9ad8\u65f6\uff0c\u7f6e\u4fe1\u533a\u95f4\u4f1a\u53d8\u5f97\u975e\u5e38\u5927\uff0c\u4ece\u800c\u4ea7\u751f\u4e00\u4e9b\u6b63\u786e\u4f46\u65e0\u7528\u7684\u7ed3\u8bba\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h3><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/?size=100&id=49487&format=png&color=000000\" style=\"height:50px;display:inline\">  \u5047\u8bbe\u6027\u68c0\u9a8c<\/h3>\n<ul>\n<li>\n<p>\u5047\u8bbe\u68c0\u9a8c\u7684\u76ee\u7684\u4e0e\u53c2\u6570\u4f30\u8ba1\u7684\u76ee\u7684\u76f8\u540c\uff0c\u90fd\u662f\u6839\u636e\u6837\u672c\u6c42\u603b\u4f53\u7684\u53c2\u6570\u3002<\/p>\n<\/li>\n<li>\n<p>\u4f46\u662f\u601d\u60f3\u6b63\u597d\u76f8\u53cd\uff1a\u53ef\u4ee5\u628a\u53c2\u6570\u4f30\u8ba1\u770b\u4f5c\u6b63\u63a8\uff0c\u5373\u6839\u636e\u6837\u672c\u63a8\u6d4b\u603b\u4f53\uff1b\u800c\u5047\u8bbe\u68c0\u9a8c\u662f\u53cd\u8bc1\uff0c\u5373\u5148\u5728\u603b\u4f53\u4e0a\u4f5c\u67d0\u9879\u5047\u8bbe\uff0c\u7528\u4ece\u603b\u4f53\u4e2d\u968f\u673a\u62bd\u53d6\u7684\u4e00\u4e2a\u6837\u672c\u6765\u68c0\u9a8c\u6b64\u9879\u5047\u8bbe\u662f\u5426\u6210\u7acb\u3002<\/p>\n<\/li>\n<li>\n<p>\u5047\u8bbe\u68c0\u9a8c\u53ef\u5206\u4e3a\u4e24\u7c7b\uff1a<\/p>\n<ul>\n<li>\u4e00\u7c7b\u662f\u603b\u4f53\u5206\u5e03\u5f62\u5f0f\u5df2\u77e5\uff0c\u4e3a\u4e86\u63a8\u65ad\u603b\u4f53\u7684\u67d0\u4e9b\u6027\u8d28\uff0c\u5bf9\u5176\u53c2\u6570\u4f5c\u67d0\u79cd\u5047\u8bbe\uff0c\u4e00\u822c\u5bf9\u6570\u5b57\u7279\u5f81\u4f5c\u5047\u8bbe\uff0c\u7528\u6837\u672c\u6765\u68c0\u9a8c\u6b64\u9879\u5047\u8bbe\u662f\u5426\u6210\u7acb\uff0c\u79f0\u6b64\u7c7b\u5047\u8bbe\u4e3a\u53c2\u6570\u5047\u8bbe\u68c0\u9a8c\u3002<\/li>\n<li>\u53e6\u4e00\u7c7b\u662f\u603b\u4f53\u5f62\u5f0f\u672a\u77e5\uff0c\u5bf9\u603b\u4f53\u5206\u5e03\u4f5c\u67d0\u79cd\u5047\u8bbe\u3002\u4f8b\u5982\uff0c\u5047\u8bbe\u603b\u4f53\u670d\u4ece\u6cca\u677e\u5206\u5e03\uff0c\u7528\u6837\u672c\u6765\u68c0\u9a8c\u5047\u8bbe\u662f\u5426\u6210\u7acb\uff0c\u79f0\u6b64\u7c7b\u68c0\u9a8c\u4e3a\u5206\u5e03\u5047\u8bbe\u68c0\u9a8c\u3002<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u5047\u8bbe\u68c0\u9a8c\u4f9d\u636e\u7684\u662f\u5c0f\u6982\u7387\u601d\u60f3\uff0c\u5373\u5c0f\u6982\u7387\u4e8b\u4ef6\u5728\u4e00\u6b21\u8bd5\u9a8c\u4e2d\u57fa\u672c\u4e0a\u4e0d\u4f1a\u53d1\u751f\u3002<\/p>\n<\/li>\n<li>\n<p>\u5982\u679c\u6837\u672c\u6570\u636e\u62d2\u7edd\u8be5\u5047\u8bbe\uff0c\u90a3\u4e48\u8bf4\u660e\u8be5\u5047\u8bbe\u68c0\u9a8c\u7ed3\u679c\u5177\u6709\u7edf\u8ba1\u663e\u8457\u6027\u3002\u4e00\u9879\u68c0\u9a8c\u7ed3\u679c\u5728\u7edf\u8ba1\u4e0a\u662f\u201c\u663e\u8457\u7684\u201d\uff0c\u610f\u601d\u662f\u6307\u6837\u672c\u548c\u603b\u4f53\u4e4b\u95f4\u7684\u5dee\u522b\u4e0d\u662f\u7531\u4e8e\u62bd\u6837\u8bef\u5dee\u6216\u5076\u7136\u800c\u9020\u6210\u7684\uff0c\u800c\u662f\u8bbe\u7acb\u7684\u5047\u8bbe\u9519\u8bef\u3002<\/p>\n<\/li>\n<li>\n<p>\u5176\u5b9e\u8fd9\u4e2a\u601d\u60f3\u524d\u4eba\u65e9\u5c31\u6709\u8fc7\u603b\u7ed3\uff1a\u4e8b\u51fa\u53cd\u5e38\u5fc5\u4e3a\u5996\u3002<\/p>\n<\/li>\n<li>\n<p>\u5047\u8bbe\u68c0\u9a8c\u7684\u5e38\u89c1\u672f\u8bed:<\/p>\n<ul>\n<li>\n<p>\uff081\uff09\u96f6\u5047\u8bbe\uff08null hypothesis\uff09\uff1a\u662f\u8bd5\u9a8c\u8005\u60f3\u6536\u96c6\u8bc1\u636e\u4e88\u4ee5\u53cd\u5bf9\u7684\u5047\u8bbe\uff0c\u4e5f\u79f0\u4e3a\u539f\u5047\u8bbe\uff0c\u901a\u5e38\u8bb0\u4e3a$H_0$\u3002\u4f8b\u5982\uff1a\u96f6\u5047\u8bbe\u662f\u68c0\u9a8c\u201c\u6837\u672c\u7684\u5747\u503c\u4e0d\u7b49\u4e8e\u603b\u4f53\u5747\u503c\u201d\u8fd9\u4e00\u89c2\u70b9\u662f\u5426\u6210\u7acb\u3002<\/p>\n<\/li>\n<li>\n<p>\uff082\uff09\u5907\u62e9\u5047\u8bbe\uff08alternative hypothesis\uff09\uff1a\u662f\u8bd5\u9a8c\u8005\u60f3\u6536\u96c6\u8bc1\u636e\u4e88\u4ee5\u652f\u6301\u7684\u5047\u8bbe\uff0c\u901a\u5e38\u8bb0\u4e3a$H_1$\u3002\u4f8b\u5982\uff1a\u5907\u62e9\u5047\u8bbe\u662f\u68c0\u9a8c\u201c\u6837\u672c\u7684\u5747\u503c\u7b49\u4e8e\u603b\u4f53\u5747\u503c\u201d\u8fd9\u4e00\u89c2\u70b9\u662f\u5426\u6210\u7acb\u3002<\/p>\n<\/li>\n<li>\n<p>\uff083\uff09\u53cc\u5c3e\u68c0\u9a8c\uff08two-tailed test\uff09\uff1a\u5982\u679c\u5907\u62e9\u5047\u8bbe\u6ca1\u6709\u7279\u5b9a\u7684\u65b9\u5411\u6027\uff0c\u5e76\u542b\u6709\u7b26\u53f7\u201c=\/\u201d\uff0c\u8fd9\u6837\u7684\u68c0\u9a8c\u79f0\u4e3a\u53cc\u5c3e\u68c0\u9a8c\u3002\u4f8b\u5982\u4e0a\u9762\u7ed9\u51fa\u7684\u96f6\u5047\u8bbe\u548c\u5907\u62e9\u5047\u8bbe\u7684\u4f8b\u5b50\u3002<\/p>\n<\/li>\n<li>\n<p>\uff084\uff09\u5355\u5c3e\u68c0\u9a8c\uff08one-tailed test\uff09\uff1a\u5982\u679c\u5907\u62e9\u5047\u8bbe\u5177\u6709\u7279\u5b9a\u7684\u65b9\u5411\u6027\uff0c\u5e76\u542b\u6709\u7b26\u53f7\u201c&gt;\u201d\u6216\u201c&lt;\u201d\uff0c\u8fd9\u6837\u7684\u68c0\u9a8c\u79f0\u4e3a\u5355\u5c3e\u68c0\u9a8c\u3002\u5355\u5c3e\u68c0\u9a8c\u5206\u4e3a\u5de6\u5c3e\uff08lower tail\uff09\u548c\u53f3\u5c3e\uff08upper tail\uff09\u3002 \u4f8b\u5982\uff1a\u96f6\u5047\u8bbe\u662f\u68c0\u9a8c\u201c\u6837\u672c\u7684\u5747\u503c\u5c0f\u4e8e\u7b49\u4e8e\u603b\u4f53\u5747\u503c\u201d\uff0c\u5907\u62e9\u5047\u8bbe\u662f\u68c0\u9a8c\u201c\u6837\u672c\u7684\u5747\u503c\u5927\u4e8e\u603b\u4f53\u5747\u503c\u201d\u3002<\/p>\n<\/li>\n<li>\n<p>\uff085\uff09\u7b2cI\u7c7b\u9519\u8bef\uff08\u5f03\u771f\u9519\u8bef\uff09\uff1a\u610f\u601d\u662f\u96f6\u5047\u8bbe\u4e3a\u771f\u65f6\u9519\u8bef\u5730\u62d2\u7edd\u4e86\u96f6\u5047\u8bbe\u3002\u72af\u7b2cI\u7c7b\u9519\u8bef\u7684\u6700\u5927\u6982\u7387\u8bb0\u4e3a $\\alpha$\u3002<\/p>\n<\/li>\n<li>\n<p>\uff086\uff09\u7b2cII\u7c7b\u9519\u8bef\uff08\u53d6\u4f2a\u9519\u8bef\uff09\uff1a\u610f\u601d\u662f\u96f6\u5047\u8bbe\u4e3a\u5047\u65f6\u9519\u8bef\u5730\u63a5\u53d7\u4e86\u96f6\u5047\u8bbe\u3002\u72af\u7b2cII\u7c7b\u9519\u8bef\u7684\u6700\u5927\u6982\u7387\u8bb0\u4e3a $\\beta$\u3002<\/p>\n<\/li>\n<li>\n<p>\uff087\uff09\u68c0\u9a8c\u7edf\u8ba1\u91cf\uff08test statistic\uff09\uff1a\u7528\u4e8e\u5047\u8bbe\u68c0\u9a8c\u8ba1\u7b97\u7684\u7edf\u8ba1\u91cf\u3002\u4f8b\u5982\uff1aZ\u503c\u3001t\u503c\u3001f\u503c\u548c\u5361\u65b9\u503c\u3002 <\/p>\n<\/li>\n<li>\n<p>\uff088\uff09\u663e\u8457\u6027\u6c34\u5e73\uff08level of significance\uff09\uff1a\u5f53\u96f6\u5047\u8bbe\u4e3a\u771f\u65f6\uff0c\u9519\u8bef\u62d2\u7edd\u96f6\u5047\u8bbe\u7684\u4e34\u754c\u6982\u7387\uff0c\u5373\u72af\u7b2c\u4e00\u7c7b\u9519\u8bef\u7684\u6700\u5927\u6982\u7387\uff0c\u7528$\\alpha$\u8868\u793a\u3002\u663e\u8457\u6027\u6c34\u5e73\u4e00\u822c\u6839\u636e\u6b63\u6001\u5206\u5e03\u7684\u7ecf\u9a8c\u6cd5\u5219\uff0868%\u300195%\u300199%\uff09\u8fdb\u884c\u9009\u53d6\uff0c\u4f8b\u5982\uff1a\u57285%\uff081-95%\uff09\u7684\u663e\u8457\u6027\u6c34\u5e73\u4e0b\uff0c\u6837\u672c\u6570\u636e\u62d2\u7edd\u539f\u5047\u8bbe\u3002<\/p>\n<\/li>\n<li>\n<p>\uff089\uff09\u7f6e\u4fe1\u5ea6\uff08confidence level\uff09\uff1a\u7f6e\u4fe1\u533a\u95f4\u5305\u542b\u603b\u4f53\u53c2\u6570\u7684\u786e\u4fe1\u7a0b\u5ea6\uff0c\u5373$1-\\alpha$\u3002\u4f8b\u5982\uff1a95%\u7684\u7f6e\u4fe1\u5ea6\u8868\u660e\uff0c\u670995%\u7684\u786e\u4fe1\u5ea6\u76f8\u4fe1\u7f6e\u4fe1\u533a\u95f4\u5305\u542b\u603b\u4f53\u53c2\u6570\u3002<\/p>\n<\/li>\n<li>\n<p>\uff0810\uff09\u7f6e\u4fe1\u533a\u95f4\uff08confidence interval\uff09\uff1a\u5305\u542b\u603b\u4f53\u53c2\u6570\u7684\u968f\u673a\u533a\u95f4\u3002<\/p>\n<\/li>\n<li>\n<p>\uff0811\uff09\u529f\u6548\uff08power\uff09\uff1a\u6b63\u786e\u62d2\u7edd\u96f6\u5047\u8bbe\u7684\u6982\u7387\uff08$1-\\beta$\uff09\uff0c\u5373\u4e0d\u72af\u4e8c\u7c7b\u9519\u8bef\u7684\u6982\u7387\u3002<\/p>\n<\/li>\n<li>\n<p>\uff0812\uff09\u4e34\u754c\u503c\uff08critical value\uff09\uff1a\u4e0e\u68c0\u9a8c\u7edf\u8ba1\u91cf\u7684\u5177\u4f53\u503c\u8fdb\u884c\u6bd4\u8f83\u7684\u503c\u3002\u662f\u5728\u6982\u7387\u5bc6\u5ea6\u5206\u5e03\u56fe\u4e0a\u7684\u5206\u4f4d\u6570\u3002\u8fd9\u4e2a\u5206\u4f4d\u6570\u5728\u5b9e\u9645\u8ba1\u7b97\u4e2d\u6bd4\u8f83\u9ebb\u70e6\uff0c\u5b83\u9700\u8981\u5bf9\u6570\u636e\u5206\u5e03\u7684\u5bc6\u5ea6\u51fd\u6570\u79ef\u5206\u6765\u83b7\u5f97\u3002<\/p>\n<\/li>\n<li>\n<p>\uff0813\uff09\u4e34\u754c\u533a\u57df\uff08critical region\uff09\uff1a\u62d2\u7edd\u539f\u5047\u8bbe\u7684\u68c0\u9a8c\u7edf\u8ba1\u91cf\u7684\u53d6\u503c\u8303\u56f4\uff0c\u4e5f\u79f0\u4e3a\u62d2\u7edd\u57df\uff08rejection region\uff09\uff0c\u662f\u7531\u4e00\u7ec4\u4e34\u754c\u503c\u7ec4\u6210\u7684\u533a\u57df\u3002\u5982\u679c\u68c0\u9a8c\u7edf\u8ba1\u91cf\u5728\u62d2\u7edd\u57df\u5185\uff0c\u90a3\u4e48\u62d2\u7edd\u539f\u5047\u8bbe\u3002<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u5047\u8bbe\u68c0\u9a8c\u7684\u4e00\u822c\u6b65\u9aa4<\/p>\n<ul>\n<li>\uff081\uff09\u5b9a\u4e49\u603b\u4f53\u3002<\/li>\n<li>\uff082\uff09\u786e\u5b9a\u539f\u5047\u8bbe\u548c\u5907\u62e9\u5047\u8bbe\u3002<\/li>\n<li>\uff083\uff09\u9009\u62e9\u68c0\u9a8c\u7edf\u8ba1\u91cf\uff08\u7814\u7a76\u7684\u662f\u7edf\u8ba1\u91cf\uff1a\u503c\u3001\u503c\u3001\u503c\u548c\u5361\u65b9\u503c\uff09\u3002<\/li>\n<li>\uff084\uff09\u9009\u62e9\u663e\u8457\u6027\u6c34\u5e73\uff08\u4e00\u822c\u7ea6\u5b9a\u4fd7\u6210\u7684\u5b9a\u4e49\u4e3a0.05\uff09\u3002<\/li>\n<li>\uff085\uff09\u4ece\u603b\u4f53\u8fdb\u884c\u62bd\u6837\uff0c\u5f97\u5230\u4e00\u5b9a\u7684\u6570\u636e\u3002<\/li>\n<li>\uff086\uff09\u6839\u636e\u6837\u672c\u6570\u636e\u8ba1\u7b97\u68c0\u9a8c\u7edf\u8ba1\u91cf\u7684\u5177\u4f53\u503c\u3002<\/li>\n<li>\uff087\uff09\u4f9d\u636e\u6240\u6784\u9020\u7684\u68c0\u9a8c\u7edf\u8ba1\u91cf\u7684\u62bd\u6837\u5206\u5e03\u548c\u663e\u8457\u6027\u6c34\u5e73\uff0c\u786e\u5b9a\u4e34\u754c\u503c\u548c\u62d2\u7edd\u57df\u3002<\/li>\n<li>\uff088\uff09\u6bd4\u8f83\u68c0\u9a8c\u7edf\u8ba1\u91cf\u7684\u503c\u4e0e\u4e34\u754c\u503c\uff0c\u5982\u679c\u68c0\u9a8c\u7edf\u8ba1\u91cf\u7684\u503c\u5728\u62d2\u7edd\u57df\u5185\uff0c\u5219\u62d2\u7edd\u539f\u5047\u8bbe\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>\u4f8b\uff1a<\/p>\n<p>\u67d0\u8336\u53f6\u5382\u7528\u81ea\u52a8\u5305\u88c5\u673a\u5c06\u8336\u53f6\u88c5\u888b\u3002\u6bcf\u888b\u7684\u6807\u51c6\u8d28\u91cf\u89c4\u5b9a\u4e3a100 g\u3002\u6bcf\u5929\u5f00\u5de5\u65f6\uff0c\u9700\u8981\u68c0\u9a8c\u4e00\u4e0b\u5305\u88c5\u673a\u5de5\u4f5c\u662f\u5426\u6b63\u5e38\u3002\u6839\u636e\u4ee5\u5f80\u7684\u7ecf\u9a8c\u77e5\u9053\uff0c\u7528\u81ea\u52a8\u5305\u88c5\u673a\u88c5\u888b\u8d28\u91cf\u670d\u4ece\u6b63\u6001\u5206\u5e03\uff0c\u88c5\u888b\u8d28\u91cf\u7684\u6807\u51c6\u5dee$\\sigma=1.15(\\mathrm{~g})$\u3002<\/p>\n<p>\u67d0\u65e5\u5f00\u5de5\u540e\uff0c\u62bd\u6d4b\u4e869\u888b\uff0c\u5176\u8d28\u91cf\u5982\u4e0b(\u5355\u4f4d\uff1ag)\uff1a<\/p>\n<p>99.3\uff0c98.7\uff0c100.5\uff0c101.2\uff0c98.3\uff0c99.7\uff0c99.5\uff0c102.1\uff0c100.5\u3002<\/p>\n<p>\u8bd5\u95ee\u6b64\u5305\u88c5\u673a\u5de5\u4f5c\u662f\u5426\u6b63\u5e38\uff1f<\/p>\n<ul>\n<li>\n<p>\u89e3\u6cd5\u5982\u4e0b\uff1a<\/p>\n<ul>\n<li>\u8bbe\u8336\u53f6\u88c5\u888b\u8d28\u91cf\u4e3a $X \\mathrm{~g}, X \\sim N\\left(\\mu, 1.15^2\\right)$ \u3002<\/li>\n<li>\u73b0\u5728\u7684\u95ee\u9898\u662f\u8336\u53f6\u888b\u7684\u5e73\u5747\u8d28\u91cf\u662f\u5426\u4e3a $100 \\mathrm{~g}$<\/li>\n<li>\u5373\u539f\u5047\u8bbe $\\mu=100$, \u8bb0\u4f5c $\\mathrm{H} 0: \\mu=100$, <\/li>\n<li>\u8bb0\u5907\u62e9\u5047\u8bbe $\\mathrm{H} 1: \\mu \\neq 0$ \u3002<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>\u5982\u679c\u8fd9\u4e2a\u5047\u8bbe $\\mathrm{H} 0$ \u6210\u7acb, \u5219 $X \\sim N\\left(100,1.15^2\\right)$ \u3002<\/p>\n<ul>\n<li>\n<p>\u53d6\u7edf\u8ba1\u91cfz\u503c:<br \/>\n$$<br \/>\nU=\\frac{\\bar{X}-100}{1.15 \/ \\sqrt{9}}<br \/>\n$$<\/p>\n<\/li>\n<li>\n<p>\u6839\u636e\u4e2d\u5fc3\u6cd5\u5219\u548c $\\mathrm{z}$ \u503c\u7684\u5b9a\u4e49, \u8fd9\u4e2a\u7edf\u8ba1\u91cf\u670d\u4ece\u6807\u51c6\u6b63\u6001\u5206\u5e03, \u5373:<br \/>\n$$<br \/>\nU=\\frac{\\bar{X}-100}{1.15 \/ \\sqrt{9}} \\sim N(0,1)<br \/>\n$$<\/p>\n<\/li>\n<li>\n<p>\u5b9a\u4e49\u4e00\u4e2a\u9009\u62e9\u663e\u8457\u6027\u6c34\u5e73, \u6bd4\u5982 $\\alpha=0.05$, \u5f53\u4e8b\u4ef6\u7684\u53d1\u751f\u6982\u7387\u5c0f\u4e8e\u8fd9\u4e2a\u503c\u65f6, \u5219\u4e8b\u4ef6\u662f\u4e00\u4e2a\u5c0f\u6982\u7387\u4e8b\u4ef6\u3002<\/p>\n<\/li>\n<li>\n<p>\u6839\u636e\u6807\u51c6\u6b63\u6001\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u8868\u67e5\u5f97 $u_{0.025}=1.96$, \u53c8 $\\bar{x}=99.98$,\u5f97\u7edf\u8ba1\u91cf $U$ \u7684\u89c2\u6d4b\u503c:<br \/>\n$$<br \/>\nu=\\frac{\\bar{x}-100}{1.15 \/ \\sqrt{9}}=-0.052<br \/>\n$$<\/p>\n<\/li>\n<\/ul>\n<p>\u7531\u4e8e $|u|=0.052&lt;1.96$, \u6240\u4ee5\u5c0f\u6982\u7387\u4e8b\u4ef6$\\lbrace\\left|\\frac{\\bar{X}-100}{1.15 \/ \\sqrt{9}}\\right| \\geq u_{0.025}\\rbrace$  \u6ca1\u6709\u53d1\u751f, \u56e0\u6b64\u53ef\u8ba4\u4e3a\u539f\u6765\u7684\u5047\u8bbe $\\mathrm{H} 0$ \u6210\u7acb, \u5373: $\\mu=100$ \u3002<\/p>\n<p><strong>P\u503c\u7684\u5b9a\u4e49<\/strong><\/p>\n<p>\u5728\u7edf\u8ba1\u5047\u8bbe\u68c0\u9a8c\u4e2d\uff0cP\u503c\uff08\u6216\u6982\u7387\u503c\uff09\u662f\u7528\u4e8e\u8861\u91cf\u89c2\u5bdf\u5230\u7684\u6570\u636e\u5728\u539f\u5047\u8bbe\u6210\u7acb\u65f6\u51fa\u73b0\u7684\u6982\u7387\u3002\u5177\u4f53\u6765\u8bf4\uff0cP\u503c\u8868\u793a\u5728\u539f\u5047\u8bbe\u4e3a\u771f\u65f6\uff0c\u89c2\u5bdf\u5230\u6bd4\u5f53\u524d\u7ed3\u679c\u66f4\u6781\u7aef\uff08\u66f4\u504f\u79bb\u539f\u5047\u8bbe\uff09\u7684\u7ed3\u679c\u7684\u6982\u7387\u3002<\/p>\n<p><strong>P\u503c\u7684\u89e3\u91ca<\/strong><\/p>\n<p>\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff1a<\/p>\n<ol>\n<li>\u539f\u5047\u8bbe $\\left(\\mathrm{H}_0\\right): \\mu=100$ (\u5373\u5305\u88c5\u673a\u88c5\u888b\u7684\u5e73\u5747\u8d28\u91cf\u4e3a 100 \u514b)\u3002<\/li>\n<li>\u5907\u62e9\u5047\u8bbe $\\left(\\mathrm{H}_1\\right): \\mu \\neq 100$ (\u5373\u5305\u88c5\u673a\u88c5\u888b\u7684\u5e73\u5747\u8d28\u91cf\u4e0d\u7b49\u4e8e 100 \u514b)\u3002<\/li>\n<\/ol>\n<p>\u6211\u4eec\u5df2\u7ecf\u8ba1\u7b97\u4e86\u6837\u672c\u5747\u503c $\\bar{x}=99.98$ \u548c\u7edf\u8ba1\u91cf $U$ \u7684\u89c2\u6d4b\u503c:<br \/>\n$$<br \/>\nu=\\frac{\\bar{x}-100}{1.15 \/ \\sqrt{9}}=-0.052<br \/>\n$$<\/p>\n<p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u9700\u8981\u786e\u5b9a P\u503c\u3002 \u8ba1\u7b97 P\u503c\u7684\u5177\u4f53\u6b65\u9aa4\u5982\u4e0b\uff1a<\/p>\n<ol>\n<li>\u8ba1\u7b97\u89c2\u6d4b\u503c $u=-0.052$ \u5bf9\u5e94\u7684\u7d2f\u8ba1\u6982\u7387\u3002<\/li>\n<li>\u67e5\u627e\u6807\u51c6\u6b63\u6001\u5206\u5e03\u8868\uff0c\u6216\u8005\u4f7f\u7528\u7edf\u8ba1\u8f6f\u4ef6\/\u8ba1\u7b97\u5668\uff0c\u627e\u5230 $u=-0.052$ \u7684\u7d2f\u79ef\u6982\u7387\u3002<\/li>\n<\/ol>\n<p>\u5047\u8bbe\u6211\u4eec\u4f7f\u7528\u7edf\u8ba1\u8f6f\u4ef6\u6216\u8868\u683c\uff0c\u6211\u4eec\u53ef\u4ee5\u627e\u5230:<br \/>\n$$<br \/>\nP(Z \\leq-0.052) \\approx 0.4807<br \/>\n$$<\/p>\n<p>\u7531\u4e8e\u8fd9\u662f\u4e00\u4e2a\u53cc\u5c3e\u68c0\u9a8c\uff0c\u6211\u4eec\u8fd8\u9700\u8981\u8003\u8651\u6b63\u5c3e\u90e8\u5206:<br \/>\n$$<br \/>\nP(Z \\geq 0.052) \\approx 0.4807<br \/>\n$$<\/p>\n<p>\u56e0\u6b64\uff0c\u603b\u7684 P\u503c\u4e3a:<br \/>\n$$<br \/>\nP=2 \\times 0.4807=0.9614<br \/>\n$$<\/p>\n<p>\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c \u5047\u8bbe\u5305\u88c5\u673a\u5de5\u4f5c\u6b63\u5e38\uff0c\u5373\u5e73\u5747\u88c5\u888b\u8d28\u91cf\u786e\u5b9e\u662f 100 \u514b\u3002<\/p>\n<ul>\n<li>\n<p>P \u503c\u8868\u793a\u7684\u6982\u7387\uff1aP \u503c 0.9614 \u8868\u793a\uff0c\u5982\u679c\u5305\u88c5\u673a\u7684\u5e73\u5747\u88c5\u888b\u8d28\u91cf\u786e\u5b9e\u662f 100 \u514b\uff0c\u90a3\u4e48\u6211\u4eec\u4f1a\u89c2\u5bdf\u5230\u50cf 99.98 \u514b\u8fd9\u6837\u6216\u8005\u6bd4\u8fd9\u66f4\u504f\u79bb 100 \u514b\u7684\u6837\u672c\u5747\u503c\u7684\u6982\u7387\u662f 96.14%\u3002<\/p>\n<\/li>\n<li>\n<p>\u901a\u5e38\u6211\u4eec\u9009\u62e9\u7684\u663e\u8457\u6027\u6c34\u5e73\u662f $\\alpha=0.05$ (\u5373 $5 \\%$ )\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u5e0c\u671b\u6709\u5c0f\u4e8e $5 \\%$\u7684\u6982\u7387\u624d\u4f1a\u62d2\u7edd\u539f\u5047\u8bbe\uff0c\u56e0\u4e3a\u8fd9\u6837\u7684\u6982\u7387\u88ab\u8ba4\u4e3a\u662f\u201c\u663e\u8457\u201d\u7684\u5c0f\u6982\u7387\u4e8b\u4ef6\u3002<\/p>\n<\/li>\n<li>\n<p>\u5bf9\u6bd4 P \u503c\u548c\u663e\u8457\u6027\u6c34\u5e73: $\\mathrm{P}$ \u503c 0.9614 \u8fdc\u5927\u4e8e 0.05 , \u8fd9\u8bf4\u660e\u89c2\u5bdf\u5230\u6837\u672c\u5747\u503c 99.98 \u514b\u6216\u66f4\u6781\u7aef\u503c\u7684\u60c5\u51b5\u975e\u5e38\u5e38\u89c1 (\u6709 $96.14 \\%$ \u7684\u6982\u7387)\u3002<\/p>\n<\/li>\n<li>\n<p>\u53ef\u4ee5\u628a P \u503c\u60f3\u8c61\u6210\u4e00\u79cd\u201c\u60ca\u8bb6\u5ea6\u201d\u7684\u6d4b\u91cf\u3002\u5982\u679c P \u503c\u5f88\u5c0f\uff08\u6bd4\u5982\u5c0f\u4e8e 0.05\uff09\uff0c\u610f\u5473\u7740\u89c2\u5bdf\u5230\u7684\u7ed3\u679c\u8ba9\u6211\u4eec\u975e\u5e38\u60ca\u8bb6\uff0c\u539f\u5047\u8bbe\u53ef\u80fd\u4e0d\u6210\u7acb\u3002\u4f46\u5728\u8fd9\u91cc\uff0cP \u503c\u5f88\u5927\uff0c\u8868\u793a\u6211\u4eec\u5e76\u4e0d\u60ca\u8bb6\u4e8e\u8fd9\u6837\u7684\u7ed3\u679c\uff0c\u6240\u4ee5\u6211\u4eec\u76f8\u4fe1\u5305\u88c5\u673a\u7684\u5e73\u5747\u88c5\u888b\u8d28\u91cf\u4ecd\u7136\u662f 100 \u514b\u3002<\/p>\n<\/li>\n<\/ul>\n<h3><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/cotton\/64\/000000\/combo-chart.png\" style=\"height:50px;display:inline\"> \u76f8\u5173\u6027\u5206\u6790<\/h3>\n<hr \/>\n<p>\u76f8\u5173\u6027\u5173\u7cfb(Correlational Relationship)\u63cf\u8ff0\u7684\u662f\u4e24\u4e2a\u53d8\u91cf\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\u7684\u5f3a\u5ea6\u548c\u65b9\u5411\uff0c\u4f46\u4e0d\u6d89\u53ca\u56e0\u679c\u5173\u7cfb\u3002<\/p>\n<p>\u4f8b\u5982\uff0c\u5047\u8bbe\u6709\u5173\u4e8e\u4eba\u7c7b\u7684\u8eab\u9ad8\u548c\u978b\u7801\u7684\u6570\u636e\uff0c\u867d\u7136\u53ef\u80fd\u53d1\u73b0\u8fd9\u4e24\u8005\u4e4b\u95f4\u6709\u6b63\u76f8\u5173\uff0c\u4f46\u8fd9\u5e76\u4e0d\u610f\u5473\u7740\u8eab\u9ad8\u51b3\u5b9a\u4e86\u978b\u7801\uff0c\u6216\u978b\u7801\u51b3\u5b9a\u4e86\u8eab\u9ad8\u3002<\/p>\n<ul>\n<li>\u76f8\u5173\u5173\u7cfb\u7684\u7c7b\u578b\uff1a<\/li>\n<\/ul>\n<p>\u2460\u6839\u636e\u6d89\u53ca\u53d8\u91cf\u7684\u4e2a\u6570\u4e0d\u540c\u5206\u4e3a\uff1a\u5355\u76f8\u5173\u548c\u590d\u76f8\u5173\u3002<\/p>\n<p>\u2461\u6839\u636e\u53d8\u5316\u65b9\u5411\u4e0d\u540c\u5206\u4e3a\uff1a\u6b63\u76f8\u5173\u548c\u8d1f\u76f8\u5173\u3002<\/p>\n<p>\u2462\u6839\u636e\u76f8\u5173\u7a0b\u5ea6\u4e0d\u540c\u5206\u4e3a\uff1a\u5b8c\u5168\u76f8\u5173\uff0c\u4e0d\u5b8c\u5168\u76f8\u5173\u548c\u65e0\u76f8\u5173\u3002<\/p>\n<p>\u2463\u6839\u636e\u53d8\u5316\u5f62\u5f0f\u4e0d\u540c\u5206\u4e3a\uff1a\u7ebf\u6027\u76f8\u5173\u548c\u975e\u7ebf\u6027\u76f8\u5173\u3002<\/p>\n<pre><code class=\"language-python\">import matplotlib.pyplot as plt\nimport numpy as np\nimport pandas as pd\nimport seaborn as sns\n\n# \u8bbe\u7f6e\u6570\u636e\nnp.random.seed(42)\nx = np.linspace(0, 10, 100)\ny1 = 2 * x + np.random.normal(0, 1, 100)  # \u6b63\u76f8\u5173\ny2 = -2 * x + np.random.normal(0, 1, 100)  # \u8d1f\u76f8\u5173\ny3 = np.sin(x) + np.random.normal(0, 0.1, 100)  # \u975e\u7ebf\u6027\u76f8\u5173\ny4 = np.random.normal(0, 1, 100)  # \u65e0\u76f8\u5173\n\n# \u521b\u5efa\u4e00\u4e2aDataFrame\ndata = pd.DataFrame({\n    &#039;x&#039;: x,\n    &#039;Positive Linear&#039;: y1,\n    &#039;Negative Linear&#039;: y2,\n    &#039;Non-linear&#039;: y3,\n    &#039;No Correlation&#039;: y4\n})\n\n# \u7ed8\u5236\u56fe\u5f62\nfig, axs = plt.subplots(2, 2, figsize=(6, 5))\n\n# \u6b63\u76f8\u5173\nsns.scatterplot(x=&#039;x&#039;, y=&#039;Positive Linear&#039;, data=data, ax=axs[0, 0])\naxs[0, 0].set_title(&#039;Positive Linear Correlation&#039;)\n\n# \u8d1f\u76f8\u5173\nsns.scatterplot(x=&#039;x&#039;, y=&#039;Negative Linear&#039;, data=data, ax=axs[0, 1])\naxs[0, 1].set_title(&#039;Negative Linear Correlation&#039;)\n\n# \u975e\u7ebf\u6027\u76f8\u5173\nsns.scatterplot(x=&#039;x&#039;, y=&#039;Non-linear&#039;, data=data, ax=axs[1, 0])\naxs[1, 0].set_title(&#039;Non-linear Correlation&#039;)\n\n# \u65e0\u76f8\u5173\nsns.scatterplot(x=&#039;x&#039;, y=&#039;No Correlation&#039;, data=data, ax=axs[1, 1])\naxs[1, 1].set_title(&#039;No Correlation&#039;)\n\nplt.tight_layout()\nplt.show()\n<\/code><\/pre>\n<p align=\"center\">\n  <img decoding=\"async\" src=\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_54_0.png\" style=\"height:400px\">\n<\/p>\n<p><strong>\u76ae\u5c14\u900a\u76f8\u5173\u7cfb\u6570\uff08Pearson Correlation Coefficient\uff09<\/strong><\/p>\n<p>Pearson correlation coefficient \u7528\u4e8e\u603b\u4f53\uff08population\uff09\u65f6\u8bb0\u4f5c $\\rho$ \uff08population correlation coefficient), \u7ed9\u5b9a\u4e24\u4e2a\u968f\u673a\u53d8\u91cf X,Y, $\\rho$ \u7684\u516c\u5f0f\u4e3a:<\/p>\n<p>$$<br \/>\n\\rho_{X, Y}=\\frac{\\operatorname{cov}(X, Y)}{\\sigma_X \\sigma_Y}<br \/>\n$$<\/p>\n<p>\u5176\u4e2d: $\\operatorname{cov}(X, Y)$ \u662f $X, Y$ \u7684\u534f\u65b9\u5dee; $\\sigma_X$  \u662f $X$ \u7684\u6807\u51c6\u5dee; $\\sigma_Y$ \u662f $Y$ \u7684\u6807\u51c6\u5dee\u3002<br \/>\n\u7528\u4e8e\u6837\u672c\uff08sample\uff09\u65f6\u8bb0\u4f5c $r$ (sample correlation coefficient, \u7ed9\u5b9a\u4e24\u4e2a\u968f\u673a\u53d8\u91cf $X, Y$, $r$ \u7684\u516c\u5f0f\u4e3a:<\/p>\n<p>$$<br \/>\nr=\\frac{\\sum_{i=1}^n\\left(X_i-\\bar{X}\\right)\\left(Y_i-\\bar{Y}\\right)}{\\sqrt{\\sum_{i=1}^n\\left(X_i-\\bar{X}\\right)^2} \\sqrt{\\sum_{i=1}^n\\left(Y_i-\\bar{Y}\\right)^2}}<br \/>\n$$<\/p>\n<p>\u5176\u4e2d: $n$ \u662f\u6837\u672c\u6570\u91cf; $X_i, Y_i$ \u662f\u53d8\u91cf $X, Y$ \u5bf9\u5e94\u7684 $i$ \u70b9\u89c2\u6d4b\u503c; $\\bar{X}$ \u662f $X$ \u6837\u672c\u5e73\u5747\u6570, $\\bar{Y}$ \u662f $Y$ \u6837\u672c\u5e73\u5747\u6570\u3002<\/p>\n<p>\u8fd9\u91cc\u89e3\u91ca\u4e00\u4e0b\u4ec0\u4e48\u53eb\u505a\u534f\u65b9\u5dee\uff1a\u7edf\u8ba1\u5b66\u4e0a\u7528\u65b9\u5dee\u548c\u6807\u51c6\u5dee\u6765\u5ea6\u91cf\u6570\u636e\u7684\u79bb\u6563\u7a0b\u5ea6\uff0c\u4f46\u662f\u65b9\u5dee\u548c\u6807\u51c6\u5dee\u662f\u7528\u6765\u63cf\u8ff0\u4e00\u7ef4\u6570\u636e\u7684\uff08\u6216\u8005\u8bf4\u662f\u591a\u7ef4\u6570\u636e\u7684\u4e00\u4e2a\u7ef4\u5ea6\uff09\uff0c\u73b0\u5b9e\u751f\u6d3b\u4e2d\u5e38\u5e38\u4f1a\u78b0\u5230\u591a\u7ef4\u6570\u636e\uff0c\u56e0\u6b64\u4eba\u4eec\u53d1\u660e\u4e86\u534f\u65b9\u5dee\uff08covariance\uff09\uff0c\u7528\u6765\u5ea6\u91cf\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u4eff\u7167\u65b9\u5dee\u7684\u516c\u5f0f\u6765\u5b9a\u4e49\u534f\u65b9\u5dee\uff1a\uff08\u8fd9\u91cc\u6307\u6837\u672c\u65b9\u5dee\u548c\u6837\u672c\u534f\u65b9\u5dee\uff09\u3002<\/p>\n<p>\u65b9\u5dee\uff1a<br \/>\n$$<br \/>\ns^2=\\frac{1}{n-1} \\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2<br \/>\n$$<\/p>\n<p>\u534f\u65b9\u5dee:<br \/>\n$$<br \/>\n\\operatorname{cov}(X, Y)=\\frac{1}{n-1} \\sum\\left(x_i-\\bar{x}\\right)\\left(y_i-\\bar{y}\\right)<br \/>\n$$<\/p>\n<ul>\n<li>\n<p>\u56e0\u4e3a\u8fd9\u91cc\u662f\u8ba1\u7b97\u6837\u672c\u7684\u65b9\u5dee\u548c\u534f\u65b9\u5dee, \u56e0\u6b64\u7528 $n-1$ \u3002\u4e4b\u6240\u4ee5\u9664\u4ee5 $n-1$ \u800c\u4e0d\u662f\u9664\u4ee5 $n$, \u662f\u56e0\u4e3a\u8fd9\u6837\u80fd\u4f7f\u6211\u4eec\u4ee5\u8f83\u5c0f\u7684\u6837\u672c\u96c6\u66f4\u597d\u5730\u903c\u8fd1\u603b\u4f53, \u5373\u7edf\u8ba1\u4e0a\u6240\u8c13\u7684\u201c\u65e0\u504f\u4f30\u8ba1\u201d\u3002<\/p>\n<\/li>\n<li>\n<p>\u534f\u65b9\u5dee\u5982\u679c\u4e3a\u6b63\u503c, \u8bf4\u660e\u4e24\u4e2a\u53d8\u91cf\u7684\u53d8\u5316\u8d8b\u52bf\u4e00\u81f4; <\/p>\n<\/li>\n<li>\n<p>\u5982\u679c\u4e3a\u8d1f\u503c, \u8bf4\u660e\u4e24\u4e2a\u53d8\u91cf\u7684\u53d8\u5316\u8d8b\u52bf\u76f8\u53cd; <\/p>\n<\/li>\n<li>\n<p>\u5982\u679c\u4e3a 0 , \u5219\u4e24\u4e2a\u53d8\u91cf\u4e4b\u95f4\u4e0d\u76f8\u5173 (\u6ce8: \u534f\u65b9\u5dee\u4e3a 0 \u4e0d\u4ee3\u8868\u8fd9\u4e24\u4e2a\u53d8\u91cf\u76f8\u4e92\u72ec\u7acb\u3002\u4e0d\u76f8\u5173\u662f\u6307\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\u4e4b\u95f4\u6ca1\u6709\u8fd1\u4f3c\u7684\u7ebf\u6027\u5173\u7cfb, \u800c\u72ec\u7acb\u662f\u6307\u4e24\u4e2a\u53d8\u91cf\u4e4b\u95f4\u6ca1\u6709\u4efb\u4f55\u5173\u7cfb)\u3002<\/p>\n<\/li>\n<li>\n<p>\u4f46\u662f\u534f\u65b9\u5dee\u4e5f\u53ea\u80fd\u5904\u7406\u4e8c\u7ef4\u5173\u7cfb, \u5982\u679c\u6709 $n$ \u4e2a\u53d8\u91cf $X_1, X_2, \\cdots X_n$, \u90a3\u600e\u4e48\u8868\u793a\u8fd9\u4e9b\u53d8\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u5462? <\/p>\n<ul>\n<li>\u89e3\u51b3\u529e\u6cd5\u5c31\u662f\u628a\u5b83\u4eec\u4e24\u4e24\u4e4b\u95f4\u7684\u534f\u65b9\u5dee\u7ec4\u6210\u534f\u65b9\u5dee\u77e9\u9635 (covariance matrix)\u3002<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u6700\u540e\u5f3a\u8c03\u4e00\u4e0b $\\mathrm{p}$ \u7684\u610f\u4e49: <\/p>\n<ul>\n<li>$\\mathrm{p}$ \u7684\u53d6\u503c\u5728-1 \u4e0e 1 \u4e4b\u95f4\u3002\u53d6\u503c\u4e3a 1 \u65f6, \u8868\u793a\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\u4e4b\u95f4\u5448\u5b8c\u5168\u6b63\u76f8\u5173\u5173\u7cfb; <\/li>\n<li>\u53d6\u503c\u4e3a -1 \u65f6, \u8868\u793a\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\u4e4b\u95f4\u5448\u5b8c\u5168\u8d1f\u76f8\u5173\u5173\u7cfb; <\/li>\n<li>\u53d6\u503c\u4e3a 0 \u65f6,\u8868\u793a\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\u4e4b\u95f4\u7ebf\u6027\u65e0\u5173\u3002<\/li>\n<li>\u4e0d\u540c $\\mathrm{p}$ \u53d6\u503c\u4e0b\u7684\u6563\u70b9\u56fe\u6848\u4f8b\u5982\u4e0b\u56fe\u6240\u793a\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<pre><code class=\"language-python\"># Generate example data for different Pearson correlation coefficients\nnp.random.seed(0)\n\n# Pearson correlation coefficients: 1, 0.5, 0, -0.5, -1\ncorrelations = [1, 0.5, 0, -0.5, -1]\ndata = {}\n\nfor p in correlations:\n    x = np.linspace(0, 10, 100)\n    if p == 1:\n        y = x\n    elif p == -1:\n        y = -x\n    elif p == 0:\n        y = np.random.uniform(0, 10, 100)\n    else:\n        y = p * x + np.random.normal(0, 1, 100) * np.sqrt(1 - p**2)\n    data[p] = (x, y)\n\n# Improved visualization\nfig, axes = plt.subplots(1, 5, figsize=(25, 5), sharex=True, sharey=True)\nfig.suptitle(&#039;Scatter Plots for Different Pearson Correlation Coefficients&#039;, fontsize=16)\n\nfor i, p in enumerate(correlations):\n    x, y = data[p]\n    axes[i].scatter(x, y, color=&#039;blue&#039;)\n    axes[i].set_title(f&#039;p = {p}&#039;)\n    axes[i].set_xlabel(&#039;x&#039;)\n    if i == 0:\n        axes[i].set_ylabel(&#039;y&#039;)\n\nplt.tight_layout(rect=[0, 0, 1, 0.95])\nplt.show()\n<\/code><\/pre>\n<p align=\"center\">\n  <img decoding=\"async\" src=\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_56_0.png\" style=\"height:300px\">\n<\/p>\n<h1>\u70ed\u529b\u56fe<\/h1>\n<h3><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/?size=100&id=pC7GuQD4E3kg&format=png&color=000000\" style=\"height:50px;display:inline\"> \u70ed\u529b\u56fe\uff08Heatmap\uff09<\/h3>\n<hr \/>\n<p>\u70ed\u529b\u56fe\u4f7f\u7528\u989c\u8272\u68af\u5ea6\u6765\u8868\u793a\u6570\u636e\u7684\u4e0d\u540c\u503c\u3002\u901a\u5e38\uff0c\u989c\u8272\u4ece\u51b7\u8272\uff08\u5982\u84dd\u8272\uff09\u5230\u6696\u8272\uff08\u5982\u7ea2\u8272\uff09\u9010\u6e10\u53d8\u5316\uff0c\u4ee5\u663e\u793a\u6570\u636e\u503c\u4ece\u4f4e\u5230\u9ad8\u7684\u53d8\u5316\u3002\uff1a\u5728\u7edf\u8ba1\u5206\u6790\u4e2d\uff0c\u70ed\u529b\u56fe\u53ef\u4ee5\u7528\u6765\u5c55\u793a\u53d8\u91cf\u4e4b\u95f4\u7684\u76f8\u5173\u6027\u3002\u5177\u4f53\u6765\u8bf4\uff1a<\/p>\n<ol>\n<li>\n<p>\u8ba1\u7b97\u76f8\u5173\u7cfb\u6570\u77e9\u9635\uff1a\u9996\u5148\uff0c\u8ba1\u7b97\u591a\u4e2a\u53d8\u91cf\u4e4b\u95f4\u7684\u76ae\u5c14\u68ee\u76f8\u5173\u7cfb\u6570\uff0c\u5f97\u5230\u4e00\u4e2a\u76f8\u5173\u7cfb\u6570\u77e9\u9635\u3002\u77e9\u9635\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u8868\u793a\u4e24\u4e2a\u53d8\u91cf\u4e4b\u95f4\u7684\u76f8\u5173\u7cfb\u6570\u3002<\/p>\n<\/li>\n<li>\n<p>\u989c\u8272\u7f16\u7801\uff1a\u5c06\u76f8\u5173\u7cfb\u6570\u77e9\u9635\u4e2d\u7684\u503c\u901a\u8fc7\u989c\u8272\u7f16\u7801\u663e\u793a\u5728\u70ed\u529b\u56fe\u4e2d\u3002\u901a\u5e38\uff0c\u6b63\u76f8\u5173\u7528\u6696\u8272\u8868\u793a\uff08\u5982\u7ea2\u8272\uff09\uff0c\u8d1f\u76f8\u5173\u7528\u51b7\u8272\u8868\u793a\uff08\u5982\u84dd\u8272\uff09\uff0c\u65e0\u76f8\u5173\u6216\u4f4e\u76f8\u5173\u7528\u4e2d\u6027\u8272\u8868\u793a\uff08\u5982\u767d\u8272\u6216\u7070\u8272\uff09\u3002<\/p>\n<\/li>\n<li>\n<p>\u89e3\u91ca\u70ed\u529b\u56fe\uff1a\u901a\u8fc7\u89c2\u5bdf\u70ed\u529b\u56fe\uff0c\u53ef\u4ee5\u76f4\u89c2\u5730\u770b\u51fa\u53d8\u91cf\u4e4b\u95f4\u7684\u76f8\u5173\u6027\u5f3a\u5ea6\u548c\u65b9\u5411\u3002\u4f8b\u5982\uff0c\u989c\u8272\u8d8a\u63a5\u8fd1\u7ea2\u8272\uff0c\u8868\u793a\u53d8\u91cf\u4e4b\u95f4\u6b63\u76f8\u5173\u6027\u8d8a\u5f3a\uff1b\u989c\u8272\u8d8a\u63a5\u8fd1\u84dd\u8272\uff0c\u8868\u793a\u53d8\u91cf\u4e4b\u95f4\u8d1f\u76f8\u5173\u6027\u8d8a\u5f3a\u3002<\/p>\n<\/li>\n<\/ol>\n<p>\u5bf9\u4e8e\u673a\u5668\u5b66\u4e60\u548c\u6570\u636e\u6316\u6398\u4efb\u52a1\uff0c\u53ef\u4ee5\u6839\u636e\u70ed\u529b\u56fe\u7ed3\u679c\u8fdb\u884c<strong>\u7279\u5f81\u9009\u62e9\u548c\u964d\u7ef4<\/strong>\uff0c\u4ee5\u63d0\u9ad8\u6a21\u578b\u6027\u80fd\u548c\u51cf\u5c11\u8ba1\u7b97\u590d\u6742\u5ea6\u3002\u4f8b\u5982\uff1a<\/p>\n<ul>\n<li>\u9009\u62e9\u4e0e\u76ee\u6807\u53d8\u91cf\u9ad8\u5ea6\u76f8\u5173\u7684\u7279\u5f81\u4f5c\u4e3a\u6a21\u578b\u8f93\u5165\u3002<\/li>\n<li>\u5220\u9664\u9ad8\u5ea6\u76f8\u5173\uff08\u5171\u7ebf\u6027\uff09\u7684\u7279\u5f81\uff0c\u4ee5\u907f\u514d\u591a\u91cd\u5171\u7ebf\u6027\u95ee\u9898\u3002<\/li>\n<\/ul>\n<p>\u9664\u4e86\u53ef\u89c6\u5316\u7279\u5f81\u76f8\u5173\u6027\u4ee5\u5916\uff0c\u70ed\u529b\u56fe\u4e5f\u901a\u5e38\u7528\u6765\u5c55\u793a\u6570\u636e\u7684\u5f3a\u5ea6\u3001\u5bc6\u5ea6\u6216\u503c\u7684\u5206\u5e03\u3002\u5b83\u901a\u8fc7\u989c\u8272\u7684\u53d8\u5316\u6765\u8868\u793a\u6570\u636e\u503c\u7684\u5927\u5c0f\u6216\u5bc6\u5ea6\uff0c\u4ece\u800c\u4f7f\u4eba\u4eec\u80fd\u591f\u76f4\u89c2\u5730\u8bc6\u522b\u51fa\u6570\u636e\u7684\u6a21\u5f0f\u548c\u8d8b\u52bf\u3002\u4f8b\u5982\uff1a<\/p>\n<ul>\n<li>\u5728\u5730\u7406\u4fe1\u606f\u7cfb\u7edf\u4e2d\uff0c\u70ed\u529b\u56fe\u53ef\u4ee5\u7528\u6765\u5c55\u793a\u5730\u7406\u533a\u57df\u5185\u7684\u67d0\u4e9b\u73b0\u8c61\u7684\u5bc6\u5ea6\uff0c\u5982\u72af\u7f6a\u7387\u3001\u623f\u4ef7\u3001\u4ea4\u901a\u6d41\u91cf\u7b49\u3002<\/li>\n<li>\u5546\u4e1a\u5206\u6790\uff1a\u4f01\u4e1a\u53ef\u4ee5\u4f7f\u7528\u70ed\u529b\u56fe\u6765\u5206\u6790\u9500\u552e\u6570\u636e\u3001\u5e02\u573a\u6d3b\u52a8\u7684\u6548\u679c\u3001\u5ba2\u6237\u884c\u4e3a\u7b49\uff0c\u4ece\u800c\u53d1\u73b0\u6f5c\u5728\u7684\u5546\u4e1a\u673a\u4f1a\u548c\u95ee\u9898\u3002<\/li>\n<li>\u5728\u7f51\u7ad9\u5206\u6790\u4e2d\uff0c\u70ed\u529b\u56fe\u53ef\u4ee5\u663e\u793a\u7528\u6237\u5728\u7f51\u9875\u4e0a\u7684\u70b9\u51fb\u70ed\u533a\u548c\u6d4f\u89c8\u70ed\u533a\uff0c\u5e2e\u52a9\u7f51\u7ad9\u4f18\u5316\u3002<\/li>\n<li>\u7b49\u7b49<\/li>\n<\/ul>\n<pre><code class=\"language-python\">import seaborn as sns\nimport matplotlib.pyplot as plt\nimport numpy as np\n\n# \u751f\u6210\u793a\u4f8b\u6570\u636e\ndata = np.random.rand(10, 12)\n\n# \u521b\u5efa\u70ed\u529b\u56fe\nsns.heatmap(data, annot=True, cmap=&#039;coolwarm&#039;)\n\n# \u663e\u793a\u56fe\u8868\nplt.show()\n<\/code><\/pre>\n<p align=\"center\">\n  <img decoding=\"async\" src=\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_59_0-1.png\" style=\"height:300px\">\n<\/p>\n<h2><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/dusk\/64\/000000\/prize.png\" style=\"height:50px;display:inline\"> Credits<\/h2>\n<hr \/>\n<ul>\n<li>Icons made by <a href=\"https:\/\/www.flaticon.com\/authors\/becris\" title=\"Becris\">Becris<\/a> from <a href=\"https:\/\/www.flaticon.com\/\" title=\"Flaticon\">www.flaticon.com<\/a><\/li>\n<li>Icons from <a href=\"https:\/\/icons8.com\/\">Icons8.com<\/a> - <a href=\"https:\/\/icons8.com\">https:\/\/icons8.com<\/a><\/li>\n<li>Datasets from <a href=\"https:\/\/www.kaggle.com\/\">Kaggle<\/a> - <a href=\"https:\/\/www.kaggle.com\/\">https:\/\/www.kaggle.com\/<\/a><\/li>\n<li>Examples and code snippets were taken from <a href=\"http:\/\/shop.oreilly.com\/product\/0636920052289.do\">&quot;Hands-On Machine Learning with Scikit-Learn and TensorFlow&quot;<\/a><\/li>\n<li><a href=\"https:\/\/taldatech.github.io\">Tal Daniel<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Deep Learning Math \u6982\u7387\u4e0e\u7edf\u8ba1\uff08Probability and Statistics\uff09 \u6982\u7387 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":1646,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[14],"tags":[],"class_list":["post-1476","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-14"],"_links":{"self":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/1476","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1476"}],"version-history":[{"count":31,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/1476\/revisions"}],"predecessor-version":[{"id":1686,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/1476\/revisions\/1686"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/media\/1646"}],"wp:attachment":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1476"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1476"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1476"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}