{"id":73955,"date":"2026-02-08T21:17:20","date_gmt":"2026-02-08T13:17:20","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/73955.html"},"modified":"2026-02-08T21:17:20","modified_gmt":"2026-02-08T13:17:20","slug":"%e8%b4%9d%e5%8f%b6%e6%96%af%e5%ae%9a%e7%90%86-%e5%8e%86%e5%8f%b2%e6%ba%af%e6%ba%90%e3%80%81%e7%90%86%e8%ae%ba%e6%8e%a8%e5%af%bc%e4%b8%8e%e7%bb%9f%e8%ae%a1%e5%ba%94%e7%94%a8","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/73955.html","title":{"rendered":"\u8d1d\u53f6\u65af\u5b9a\u7406 | \u5386\u53f2\u6eaf\u6e90\u3001\u7406\u8bba\u63a8\u5bfc\u4e0e\u7edf\u8ba1\u5e94\u7528"},"content":{"rendered":"

\u6ce8：\u672c\u6587\u4e3a \u201c\u8d1d\u53f6\u65af\u5b9a\u7406\u201d \u76f8\u5173\u5408\u8f91\u3002 \u56fe\u7247\u6e05\u6670\u5ea6\u53d7\u5f15\u6587\u539f\u56fe\u6240\u9650\u3002 \u7565\u4f5c\u91cd\u6392，\u5982\u6709\u5185\u5bb9\u5f02\u5e38，\u8bf7\u770b\u539f\u6587\u3002<\/p>\n

\n

\u8d1d\u53f6\u65af\u4e0e\u8d1d\u6c0f\u5b9a\u7406 (1)<\/h2>\n

Thomas Bayes and Bayes\u2019 Theorem (1)<\/h2>\n

Posted on 2014\/11\/01<\/p>\n

\u53f0\u5317\u5e02\u7acb\u7b2c\u4e00\u5973\u5b50\u9ad8\u7ea7\u4e2d\u5b66 \u82cf\u4fca\u9e3f \u8001\u5e08<\/p>\n

\u8d1d\u6c0f\u5b9a\u7406（Bayes\u2019 Theorem）\u51fa\u73b0\u5728\u9ad8\u4e2d\u6570\u5b66\u7684\u6982\u7387\u5355\u5143\u4e2d，\u88ab\u5f52\u4e3a\u6761\u4ef6\u6982\u7387\u7684\u76f8\u5173\u8bae\u9898，\u5176\u5b9a\u7406\u8868\u8ff0\u5982\u4e0b：<\/p>\n

\u8bbe <\/p>\n

{ <\/p>\n

A <\/p>\n

1 <\/p>\n

, <\/p>\n

A <\/p>\n

2 <\/p>\n

, <\/p>\n

\u22ef
\n \u2009 <\/p>\n

, <\/p>\n

A <\/p>\n

n <\/p>\n

} <\/p>\n

\\\\{A_1,A_2,\\\\cdots,A_n\\\\} <\/p>\n

<\/span><\/span>{<\/span>A<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>A<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u22ef<\/span><\/span><\/span>,<\/span><\/span>A<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span> \u4e3a\u6837\u672c\u7a7a\u95f4 <\/p>\n

S <\/p>\n

S <\/p>\n

<\/span><\/span>S<\/span><\/span><\/span><\/span><\/span> \u7684\u4e00\u7ec4\u5206\u5272， <\/p>\n

B <\/p>\n

B <\/p>\n

<\/span><\/span>B<\/span><\/span><\/span><\/span><\/span> \u4e3a <\/p>\n

S <\/p>\n

S <\/p>\n

<\/span><\/span>S<\/span><\/span><\/span><\/span><\/span> \u7684\u4efb\u4e00\u4e8b\u4ef6，\u82e5 <\/p>\n

P <\/p>\n

( <\/p>\n

B <\/p>\n

) <\/p>\n

> <\/p>\n

0 <\/p>\n

P(B)>0 <\/p>\n

<\/span><\/span>P<\/span>(<\/span>B<\/span>)<\/span><\/span>><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，\u5219\u5728\u4e8b\u4ef6 <\/p>\n

B <\/p>\n

B <\/p>\n

<\/span><\/span>B<\/span><\/span><\/span><\/span><\/span> \u53d1\u751f\u7684\u6761\u4ef6\u4e0b，\u4e8b\u4ef6 <\/p>\n

A <\/p>\n

k <\/p>\n

A_k <\/p>\n

<\/span><\/span>A<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u53d1\u751f\u7684\u6982\u7387\u4e3a <\/p>\n

P <\/p>\n

( <\/p>\n

A <\/p>\n

k <\/p>\n

\u2223 <\/p>\n

B <\/p>\n

) <\/p>\n

= <\/p>\n

P <\/p>\n

( <\/p>\n

A <\/p>\n

k <\/p>\n

) <\/p>\n

P <\/p>\n

( <\/p>\n

B <\/p>\n

\u2223 <\/p>\n

A <\/p>\n

k <\/p>\n

) <\/p>\n

\u2211 <\/p>\n

i <\/p>\n

= <\/p>\n

1 <\/p>\n

n <\/p>\n

P <\/p>\n

( <\/p>\n

A <\/p>\n

i <\/p>\n

) <\/p>\n

P <\/p>\n

( <\/p>\n

B <\/p>\n

\u2223 <\/p>\n

A <\/p>\n

i <\/p>\n

) <\/p>\n

, <\/p>\n

\u00a0 <\/p>\n

1 <\/p>\n

\u2264 <\/p>\n

k <\/p>\n

\u2264 <\/p>\n

n <\/p>\n

\\\\displaystyle P\\\\left( A_k \\\\mid B \\\\right) = \\\\frac{P\\\\left( A_k \\\\right)P\\\\left( B \\\\mid A_k \\\\right)}{\\\\sum\\\\limits_{i = 1}^n P\\\\left( A_i \\\\right)P\\\\left( B \\\\mid A_i \\\\right)},\\\\ 1\\\\le k\\\\le n <\/p>\n

<\/span><\/span>P<\/span><\/span>(<\/span>A<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span>B<\/span>)<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>(<\/span>A<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span>P<\/span><\/span>(<\/span>B<\/span><\/span>\u2223<\/span><\/span>A<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>(<\/span>A<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span>P<\/span><\/span>(<\/span>B<\/span><\/span>\u2223<\/span><\/span>A<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span>\u00a0<\/span><\/span>1<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>k<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6559\u6750\u4e2d\u4ea6\u5217\u4e3e\u4e86\u8be5\u5b9a\u7406\u7684\u76f8\u5173\u5e94\u7528，\u5982\u8d28\u91cf\u68c0\u9a8c\u3001\u533b\u5b66\u68c0\u6d4b\u7b49\u3002\u4f46\u5173\u4e8e\u8d1d\u53f6\u65af\u7684\u751f\u5e73\u80cc\u666f\u3001\u63a8\u52a8\u8be5\u5b9a\u7406\u8bde\u751f\u7684\u7814\u7a76\u95ee\u9898，\u4ee5\u53ca\u8be5\u5b9a\u7406\u63d0\u51fa\u521d\u671f\u662f\u5426\u83b7\u5f97\u6570\u5b66\u5bb6\u4e0e\u7edf\u8ba1\u5b66\u5bb6\u7684\u8ba4\u53ef，\u5747\u4e3a\u591a\u6570\u4eba\u6240\u672a\u77e5\u3002\u4e0a\u8ff0\u95ee\u9898\u6784\u6210\u672c\u6587\u7684\u64b0\u5199\u7f18\u8d77\u3002\u672c\u6587\u9996\u5148\u9610\u8ff0\u6258\u9a6c\u65af\u00b7\u8d1d\u53f6\u65af（Thomas Bayes, 1702-1761）\u7684\u751f\u5e73\u80cc\u666f，\u8d1d\u6c0f\u5b9a\u7406\u5373\u7531\u5176\u9996\u6b21\u63d0\u51fa\u3002<\/p>\n

\u6258\u9a6c\u65af\u00b7\u8d1d\u53f6\u65af\u51fa\u8eab\u4e8e\u82f1\u56fd\u65b0\u6559\u5bb6\u5ead，\u4e3a\u5bb6\u4e2d\u4e03\u540d\u5b50\u5973\u4e2d\u7684\u957f\u5b50\u3002\u5176\u7236\u7ea6\u4e66\u4e9a\u00b7\u8d1d\u53f6\u65af（Joshua Bayes）\u4e3a\u82f1\u56fd\u9996\u6279\u53d7\u4efb\u7684\u516d\u4f4d\u65b0\u6559\u7267\u5e08\u4e4b\u4e00\u3002\u5173\u4e8e\u8d1d\u53f6\u65af\u7684\u5e7c\u5e74\u6559\u80b2，\u76ee\u524d\u53ef\u8003\u53f2\u6599\u8f83\u5c11\u3002\u90e8\u5206\u5386\u53f2\u5b66\u8005\u63a8\u6d4b\u5176\u66fe\u53d7\u68e3\u7f8e\u5f17（Abraham de Moivre, 1667-1754）\u7684\u5b66\u672f\u6307\u5bfc，\u4ea6\u6709\u5b66\u8005\u8ba4\u4e3a\u5176\u63a5\u53d7\u8fc7\u7267\u5e08\u804c\u4e1a\u7684\u57f9\u517b\u8bad\u7ec3\u3002<\/p>\n

1719 \u5e74，\u8d1d\u53f6\u65af\u8003\u5165\u82cf\u683c\u5170\u7231\u4e01\u5821\u5927\u5b66，\u4e3b\u4fee\u903b\u8f91\u5b66\u4e0e\u795e\u5b66\u3002\u5728\u8be5\u5386\u53f2\u65f6\u671f，\u65b0\u6559\u6559\u5f92\u65e0\u6cd5\u83b7\u5f97\u725b\u6d25\u5927\u5b66\u4e0e\u5251\u6865\u5927\u5b66\u7684\u5165\u5b66\u8d44\u683c\u3002\u5c3d\u7ba1\u76ee\u524d\u672a\u53d1\u73b0\u5176\u5728\u7231\u4e01\u5821\u5927\u5b66\u4fee\u4e60\u6570\u5b66\u7684\u76f8\u5173\u8bb0\u5f55，\u4f46\u8d1d\u53f6\u65af\u66fe\u4e8e1736 \u5e74\u64b0\u5199\u8bba\u6587\u3008\u6d41\u6570\u5b66\u8bf4\u7684\u4ecb\u7ecd，\u4ee5\u53ca\u5bf9\u300a\u5206\u6790\u5b66\u5bb6\u300b\u4f5c\u8005\u7684\u53cd\u5bf9\u63d0\u51fa\u6570\u5b66\u5bb6\u7684\u9632\u5fa1\u3009（An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of The Analyst），\u4ee5\u53cd\u9a73\u8d1d\u514b\u83b1\u4e3b\u6559（George Berkeley, 1685-1753）\u5728\u300a\u5206\u6790\u5b66\u5bb6\u300b（The Analyst, 1734）\u4e2d\u5bf9\u5fae\u79ef\u5206\u903b\u8f91\u57fa\u7840\u7684\u8d28\u7591\u3002\u8be5\u8bba\u6587\u5f00\u7bc7\u5373\u6307\u51fa：<\/p>\n

\u6211\u65e9\u5c31\u8ba4\u4e3a\u6d41\u6570\u6cd5\u7684\u57fa\u672c\u539f\u7406\u548c\u89c4\u5219，\u9700\u8981\u66f4\u4e3a\u5168\u9762\u4e14\u6e05\u695a\u7684\u89e3\u91ca\u548c\u8bc1\u660e\u3002<\/p>\n

\u8be5\u8868\u8ff0\u76f4\u63a5\u4f53\u73b0\u51fa\u8d1d\u53f6\u65af\u5bf9\u6d41\u6570\u6cd5\u7684\u719f\u7ec3\u638c\u63e1，\u540c\u65f6\u5176\u5bf9\u8d1d\u514b\u83b1\u57fa\u4e8e\u5b97\u6559\u89c6\u89d2\u63d0\u51fa\u7684\u76f8\u5173\u6279\u8bc4\u6301\u53cd\u5bf9\u6001\u5ea6\u3002\u8d1d\u53f6\u65af\u540e\u7eed\u88ab\u4efb\u547d\u4e3a\u7267\u5e08，\u62c5\u4efb\u5176\u7236\u7684\u52a9\u7406\u795e\u804c\u4eba\u5458\u30021733 \u5e74\u4e4b\u524d，\u5176\u51fa\u4efbTunbridge Wells（\u8ddd\u4f26\u656635 \u82f1\u91cc）\u5f53\u5730\u6559\u5802\u7684\u7267\u5e08，\u76f4\u81f31752 \u5e74\u5378\u4efb\u8be5\u804c\u52a1，\u4e14\u5378\u4efb\u540e\u4ecd\u5b9a\u5c45\u4e8eTunbridge Wells\u3002<\/p>\n

\"58765_p1\" \u6258\u9a6c\u65af\u00b7\u8d1d\u53f6\u65af<\/p>\n

\u5c3d\u7ba1\u5f7c\u65f6\u8d1d\u53f6\u65af\u5c1a\u672a\u516c\u5f00\u53d1\u8868\u4efb\u4f55\u6570\u5b66\u76f8\u5173\u8457\u4f5c，\u4f46\u5176\u4e8e1742 \u5e74\u5f53\u9009\u82f1\u56fd\u7687\u5bb6\u5b66\u4f1a\u9662\u58eb\u3002\u8be5\u73b0\u8c61\u4e0e\u5f53\u65f6\u7684\u5b66\u672f\u98ce\u6c14\u76f8\u5173：\u5b66\u8005\u6781\u5c11\u4ee5\u672c\u540d\u5728\u751f\u524d\u53d1\u8868\u7814\u7a76\u6210\u679c，\u524d\u8ff01736 \u5e74\u7684\u6570\u5b66\u76f8\u5173\u8bba\u6587\u5373\u4e3a\u533f\u540d\u51fa\u7248\u3002\u8d1d\u53f6\u65af\u901d\u4e16\u540e，\u5176\u5173\u4e8e <\/p>\n

log <\/p>\n

\u2061 <\/p>\n

z <\/p>\n

! <\/p>\n

\\\\log z! <\/p>\n

<\/span><\/span>log<\/span><\/span><\/span>z<\/span>!<\/span><\/span><\/span><\/span><\/span>（\u8be5\u8868\u8fbe\u5f0f\u7531\u65af\u7279\u6797\u4e0e\u68e3\u7f8e\u5f17\u63d0\u51fa）\u7684\u6e10\u8fd1\u7ea7\u6570\u655b\u6563\u6027\u7684\u6570\u5b66\u7814\u7a76\u6210\u679c\u65b9\u5f97\u4ee5\u51fa\u7248\u3002<\/p>\n

\u8d1d\u53f6\u65af\u5728\u6982\u7387\u8bba\u9886\u57df\u7684\u7814\u7a76\u6210\u679c\u5177\u6709\u72ec\u7279\u7684\u5b66\u672f\u4ef7\u503c，\u5176\u4ee3\u8868\u6027\u8bba\u6587\u4e3a\u3008\u300a\u6982\u7387\u8bba\u300b\u4e2d\u4e00\u4e2a\u95ee\u9898\u7684\u89e3\u51b3\u3009（An Essay towards Solving a Problem in the Doctrine of Chances）；\u5176\u4e2d\u300a\u6982\u7387\u8bba\u300b（The Doctrine of Chances, 1718）\u4e3a\u68e3\u7f8e\u5f17\u64b0\u5199\u7684\u6982\u7387\u8bba\u9886\u57df\u7ecf\u5178\u8457\u4f5c\u3002\u8be5\u8bba\u6587\u540c\u6837\u4e8e\u8d1d\u53f6\u65af\u901d\u4e16\u540e\u88ab\u5176\u53cb\u4eba\u7406\u67e5\u5fb7\u00b7\u666e\u83b1\u65af（Richard Price, 1723-1791）\u53d1\u73b0，\u7ecf\u666e\u83b1\u65af\u6574\u7406\u4e0e\u6ce8\u91ca\u540e，\u4e8e1764 \u5e74\u88ab\u9001\u4ea4\u82f1\u56fd\u7687\u5bb6\u5b66\u4f1a，\u53d1\u8868\u4e8e\u300a\u81ea\u7136\u79d1\u5b66\u4f1a\u62a5\u300b（The Philosophical Transactions of the Royal Society）\u3002<\/p>\n

\u8be5\u8bba\u6587\u4e2d\u5305\u542b\u4e86\u73b0\u4eca\u5e7f\u4e3a\u719f\u77e5\u7684\u6761\u4ef6\u6982\u7387\u76f8\u5173\u8bba\u8ff0，\u4ee5\u53ca\u88ab\u547d\u540d\u4e3a\u8d1d\u6c0f\u5b9a\u7406\u7684\u547d\u9898\u3002\u5173\u4e8e\u8d1d\u53f6\u65af\u5f00\u5c55\u8be5\u9879\u7814\u7a76\u7684\u95ee\u9898\u7f18\u8d77\u4e0e\u5177\u4f53\u8bba\u8bc1\u8fc7\u7a0b，\u5c06\u5728\u3008\u8d1d\u53f6\u65af\u4e0e\u8d1d\u6c0f\u5b9a\u7406 (2)\u3009\u4e2d\u5c55\u5f00\u8be6\u7ec6\u9610\u8ff0\u3002<\/p>\n

\n

\u8d1d\u53f6\u65af\u4e0e\u8d1d\u6c0f\u5b9a\u7406 (2)<\/h2>\n

Thomas Bayes and Bayes\u2019 Theorem (2)<\/h2>\n

Posted on 2014\/11\/01<\/p>\n

\u53f0\u5317\u5e02\u7acb\u7b2c\u4e00\u5973\u5b50\u9ad8\u7ea7\u4e2d\u5b66 \u82cf\u4fca\u9e3f \u8001\u5e08<\/p>\n

\u5728\u9610\u8ff0\u8d1d\u53f6\u65af\u7684\u8bba\u6587\u3008\u300a\u6982\u7387\u8bba\u300b\u4e2d\u4e00\u4e2a\u95ee\u9898\u7684\u89e3\u51b3\u3009（An Essay towards Solving a Problem in the Doctrine of Chances）\u7684\u5177\u4f53\u5185\u5bb9\u524d，\u5148\u68b3\u7406\u8be5\u5386\u53f2\u9636\u6bb5\u6982\u7387\u8bba\u7684\u53d1\u5c55\u8109\u7edc\u3002\u6cd5\u56fd\u8d35\u65cf\u6885\u96f7\u9a91\u58eb（Chevalier de M\u00e9r\u00e9）\u5c31\u9ab0\u5b50\u63b7\u70b9\u3001\u8d4c\u91d1\u5206\u914d\u7b49\u95ee\u9898\u5411\u5e15\u65af\u5361\u5c14（Blaise Pascal, 1623-1662）\u8bf7\u6559，\u8be5\u95ee\u9898\u5f15\u53d1\u5e15\u65af\u5361\u5c14\u4e0e\u8d39\u9a6c（Pierre de Fermat, 1601-1665）\u901a\u8fc7\u4e66\u4fe1\u5c55\u5f00\u8ba8\u8bba\u5e76\u7ed9\u51fa\u89e3\u7b54，\u8be5\u7814\u7a76\u5de5\u4f5c\u5960\u5b9a\u4e86\u6982\u7387\u8bba\u7684\u7406\u8bba\u57fa\u7840\u3002<\/p>\n

1655 \u5e74，\u8377\u5170\u6570\u5b66\u5bb6\u60e0\u66f4\u65af（Christiaan Huygens, 1629-1695）\u5230\u8bbf\u5df4\u9ece，\u4e86\u89e3\u5230\u5e15\u65af\u5361\u5c14\u4e0e\u8d39\u9a6c\u7684\u76f8\u5173\u8ba8\u8bba\u95ee\u9898\u540e\u4ea7\u751f\u7814\u7a76\u5174\u8da3，\u5e76\u5bf9\u8be5\u7c7b\u95ee\u9898\u5c55\u5f00\u5ef6\u4f38\u7814\u7a76，\u4e8e1657 \u5e74\u51fa\u7248\u4e13\u8457\u300a\u8bba\u6982\u7387\u535a\u5f08\u7684\u8ba1\u7b97\u300b（On the Calculations in Games of Chance）\u3002\u76f4\u81f318 \u4e16\u7eaa\u521d，\u8be5\u4e66\u59cb\u7ec8\u4e3a\u6982\u7387\u8bba\u9886\u57df\u7684\u5165\u95e8\u8457\u4f5c\u3002<\/p>\n

\u96c5\u5404\u5e03\u00b7\u4f2f\u52aa\u5229（Jakob I. Bernoulli, 1654-1705）\u5728\u60e0\u66f4\u65af\u7684\u7814\u7a76\u57fa\u7840\u4e0a，\u7ed3\u5408\u5f53\u65f6\u4fdd\u9669\u98ce\u9669\u8bc4\u4f30\u7b49\u5b9e\u9645\u5e94\u7528\u9700\u6c42，\u63a2\u7a76\u6982\u7387\u8bba\u4e0e\u5404\u7c7b\u5b9e\u9645\u95ee\u9898\u7684\u7ed3\u5408\u8def\u5f84\u3002\u5176\u901d\u4e168 \u5e74\u540e，\u9057\u4f5c\u300a\u731c\u5ea6\u672f\u300b（Ars Conjectandi, 1713）\u6b63\u5f0f\u51fa\u7248，\u6210\u4e3a\u6982\u7387\u8bba\u9886\u57df\u7684\u7ecf\u5178\u8457\u4f5c\u3002\u73b0\u4eca\u6982\u7387\u8bba\u4e2d\u7684\u5927\u6570\u5b9a\u5f8b\u3001\u4e8c\u9879\u5206\u5e03\u7b49\u6982\u5ff5，\u5747\u9996\u6b21\u7cfb\u7edf\u5448\u73b0\u4e8e\u8be5\u8457\u4f5c\u4e2d\u3002\u4f2f\u52aa\u5229\u63d0\u51fa，\u5bf9\u4e8e\u4efb\u610f\u7ed9\u5b9a\u7684\u8bef\u5dee\u8303\u56f4，\u5f53\u8bd5\u9a8c\u6b21\u6570\u8db3\u591f\u5927\u65f6，\u4e8b\u4ef6\u53d1\u751f\u7684\u5b9e\u9645\u6b21\u6570\u4e0e\u8bd5\u9a8c\u603b\u6b21\u6570\u7684\u6bd4\u503c，\u4e0e\u8be5\u4e8b\u4ef6\u53d1\u751f\u7684\u7406\u8bba\u6982\u7387 <\/p>\n

p <\/p>\n

p <\/p>\n

<\/span><\/span>p<\/span><\/span><\/span><\/span><\/span> \u4e4b\u95f4\u7684\u504f\u5dee\u5c06\u843d\u5165\u8be5\u8bef\u5dee\u8303\u56f4\u5185\u3002\u57fa\u4e8e\u8be5\u7ed3\u8bba\u53ef\u5bf9\u6240\u9700\u7684\u89c2\u6d4b\u6b21\u6570\u8fdb\u884c\u4f30\u7b97，\u4f46\u4f2f\u52aa\u5229\u6240\u63a8\u5bfc\u7684\u4e8c\u9879\u5206\u5e03\u8fd1\u4f3c\u516c\u5f0f\u7cbe\u5ea6\u4e0d\u8db3，\u65e0\u6cd5\u5e94\u7528\u4e8e\u5b9e\u9645\u573a\u666f\u7684\u8ba1\u7b97\u3002<\/p>\n

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X <\/p>\n

X <\/p>\n

<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u4e3a <\/p>\n

n <\/p>\n

n <\/p>\n

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x <\/p>\n

x <\/p>\n

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r <\/p>\n

, <\/p>\n

s <\/p>\n

r,s <\/p>\n

<\/span><\/span>r<\/span>,<\/span><\/span>s<\/span><\/span><\/span><\/span><\/span> \u4e3a\u4e24\u4e2a\u6307\u5b9a\u6982\u7387\u503c\u4e14\u6ee1\u8db3 <\/p>\n

r <\/p>\n

< <\/p>\n

s <\/p>\n

r<s <\/p>\n

<\/span><\/span>r<\/span><\/span><<\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span><\/span>，\u5219\u8d1d\u53f6\u65af\u7684\u7814\u7a76\u95ee\u9898\u53ef\u8868\u793a\u4e3a： <\/p>\n

P <\/p>\n

( <\/p>\n

r <\/p>\n

< <\/p>\n

x <\/p>\n

< <\/p>\n

s <\/p>\n

\u2223 <\/p>\n

X <\/p>\n

) <\/p>\n

= <\/p>\n

? <\/p>\n

P(r<x<s \\\\mid X)=? <\/p>\n

<\/span><\/span>P<\/span>(<\/span>r<\/span><\/span><<\/span><\/span><\/span><\/span>x<\/span><\/span><<\/span><\/span><\/span><\/span>s<\/span><\/span>\u2223<\/span><\/span><\/span><\/span>X<\/span>)<\/span><\/span>=<\/span><\/span><\/span>?<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

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