{"id":10329,"date":"2025-06-12T21:35:16","date_gmt":"2025-06-12T13:35:16","guid":{"rendered":"https:\/\/www.invbus.com\/blog\/?p=10329"},"modified":"2025-06-12T21:35:17","modified_gmt":"2025-06-12T13:35:17","slug":"%e6%9c%80%e4%bc%98%e5%81%9c%e6%ad%a2%e7%90%86%e8%ae%ba-%e5%a6%82%e4%bd%95%e9%80%82%e5%8f%af%e8%80%8c%e6%ad%a2","status":"publish","type":"post","link":"https:\/\/www.invbus.com\/blog\/?p=10329","title":{"rendered":"\u6700\u4f18\u505c\u6b62\u7406\u8bba\u2014\u2014\u5982\u4f55\u9002\u201c\u53ef\u201d\u800c\u6b62"},"content":{"rendered":"\n

\u6458\u8981<\/h1>\n\n\n\n

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<\/a>\u7406\u8bba\u4ecb\u7ecd<\/h1>\n\n\n\n

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\u9762\u5bf9\u8fd9\u7c7b\u95ee\u9898\uff0cMerrill M. Flood\u57281949\u5e74\u9996\u6b21\u63d0\u51fa37%\u6cd5\u5219\u3002\u4ed6\u6240\u91c7\u53d6\u7684\u7684\u7b56\u7565\u7c7b\u4f3c\u4e8e\u4e0a\u8ff0\u82cf\u683c\u62c9\u5e95\u7684\u7b2c\u4e09\u4f4d\u5f1f\u5b50\u7684\u51b3\u7b56\uff1a\u9996\u5148\u5bf9\u524d\u9762\u4e00\u90e8\u5206\u9762\u8bd5\u8005\uff0c\u65e0\u8bba\u4f18\u79c0\u4e0e\u5426\u90fd\u76f4\u63a5\u62d2\u7edd\uff0c\u53ea\u8003\u5bdf\u76ee\u6807\uff0c\u6536\u96c6\u6570\u636e\uff0c\u7528\u4e8e\u786e\u5b9a\u201c\u6700\u57fa\u672c\u7684\u6ee1\u610f\u6807\u51c6\u201d\u3002\u7136\u540e\u5bf9\u5269\u4e0b\u7684\u4eba\u8fdb\u884c\u9762\u8bd5\uff0c\u5982\u679c\u9047\u5230\u4e00\u4f4d\u6bd4\u4e4b\u524d\u9762\u8bd5\u7684\u4eba\u90fd\u4f18\u79c0\u7684\u9762\u8bd5\u8005\uff0c\u90a3\u4e48\u5c31\u7acb\u5373\u51fa\u624b\uff0c\u76f4\u63a5\u8058\u8bf7\u8fd9\u4e2a\u4eba\uff0c\u5426\u5219\u5c31\u7ee7\u7eed\u9762\u8bd5\u3002<\/p>\n\n\n\n

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<\/a>\u7406\u8bba\u8bc1\u660e<\/h1>\n\n\n\n

\u90a3\u4e4837%\u662f\u5982\u4f55\u63a8\u5bfc\u51fa\u6765\u7684\u5462\uff1f<\/p>\n\n\n\n

\u5047\u8bbe\u4e8b\u4ef6AkAk\u4e3a\u7b2c k \u4e2a\u4eba\u88ab\u9009\u4e2d\uff0c\u4e8b\u4ef6BkBk\u4e3a\u7b2c k \u4e2a\u4eba\u662f\u6700\u4f18\u79c0\u7684\uff0c\u4e8b\u4ef6CrCr\u4e3a\u524d r \u540d\u9762\u8bd5\u8005\u7528\u4e8e\u786e\u5b9a\u6807\u51c6\u65f6\u9009\u5230\u6700\u4f18\u79c0\u5e94\u8058\u8005\uff0c\u5176\u4e2d1\u2264r<k\u2264n,r\u2208Z,k\u2208Z1\u2264r<k\u2264n,r\u2208Z,k\u2208Z\u3002\u5219\u9009\u5230\u6700\u4f18\u79c0\u7684\u5e94\u8058\u8005\u7684\u6982\u7387\uff1a<\/p>\n\n\n\n

P(Cr)=\u2211P(AkBk)=\u2211k=1nP(Bk)P(Ak|Bk)=1n\u2211k=1nP(Ak|Bk)(1)P(Cr)=\u2211P(AkBk)=\u2211k=1nP(Bk)P(Ak|Bk)=1n\u2211k=1nP(Ak|Bk)(1)<\/p>\n\n\n\n

\u56e0\u4e3a\u524d r \u4e2a\u4eba\u5fc5\u7136\u4f1a\u88ab\u62d2\u7edd\uff0c\u6240\u4ee5P(Ak|Bk)=0,k\u2264rP(Ak|Bk)=0,k\u2264r \u3002\u53c8\u56e0\u4e3a\u5f53\u7b2c k \u4e2a\u4eba\u88ab\u9009\u62e9\u65f6\uff0c\u7b2c k\u4e2a\u4eba\u524d\u6700\u4f18\u79c0\u7684\u4eba\u5fc5\u7136\u5728\u524d r \u4e2a\u4eba\u5f53\u4e2d\uff0c\u6240\u4ee5 P(Ak|Bk)=rk\u22121,k>rP(Ak|Bk)=rk\u22121,k>r \u3002\u56e0\u6b64\u5f0f\uff081\uff09\u53ef\u5316\u7b80\u4e3a\uff1a<\/p>\n\n\n\n

P(Cr)=1n\u2211k=r+1nP(Ak|Bk)=1n\u2211k=r+1nrk\u22121=rn\u2211k=r+1n1k\u22121(2)P(Cr)=1n\u2211k=r+1nP(Ak|Bk)=1n\u2211k=r+1nrk\u22121=rn\u2211k=r+1n1k\u22121(2)<\/p>\n\n\n\n

\u56e0\u4e3a\u6211\u4eec\u9700\u8981\u9009\u51fa\u6700\u4f18\u7684\u51b3\u7b56\uff0c\u6240\u4ee5\u524d r \u540d\u9762\u8bd5\u8005\u7528\u4e8e\u786e\u5b9a\u6807\u51c6\u65f6\u9009\u62e9\u5230\u6700\u4f18\u79c0\u7684\u5e94\u8058\u8005\u7684\u6982\u7387\u5fc5\u987b\u4e3a\u6700\u5927\uff0c\u5373P(Cr\u22121)\u2264P(Cr)\u2265P(Cr+1)P(Cr\u22121)\u2264P(Cr)\u2265P(Cr+1)\u3002<\/p>\n\n\n\n

\u901a\u8fc7\u8ba1\u7b97\u673a\u7a0b\u5e8f\u679a\u4e3e n \u4e3a 1~10000 \u65f6\uff0c\u9009\u53d6\u7684 r \u7684\u503c\uff0c\u5e76\u9009\u53d6\u90e8\u5206\u6570\u636e\u8bb0\u5f55\u4e8e\u4e0b\u8868\u3002<\/p>\n\n\n\n

n<\/th>r<\/th>rnrn<\/th>P(Cr)P(Cr)<\/th><\/tr><\/thead>
3<\/td>1<\/td>0.333333<\/td>0.5<\/td><\/tr>
4<\/td>1<\/td>0.25<\/td>0.458333<\/td><\/tr>
5<\/td>2<\/td>0.4<\/td>0.433333<\/td><\/tr>
10<\/td>3<\/td>0.3<\/td>0.39869<\/td><\/tr>
100<\/td>37<\/td>0.37<\/td>0.371043<\/td><\/tr>
1000<\/td>368<\/td>0.368<\/td>0.368196<\/td><\/tr>
10000<\/td>3679<\/td>0.3679<\/td>0.3679<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

\u5f53 n \u65e0\u9650\u63a5\u8fd1\u4e8e\u221e\u221e\u65f6\uff0cP(Cr)P(Cr) \u53d8\u4e3a\u79ef\u5206\u5f62\u5f0f\uff0c\u5373\u5f0f\uff082\uff09\u53d8\u4e3a\uff1a<\/p>\n\n\n\n

P(Cr)=rn\u2211k=r+1n1k\u22121\u2248rn\u222bn\u22121r1xdx=rnln(n\u22121r)\u2248rnln(nr)(3)P(Cr)=rn\u2211k=r+1n1k\u22121\u2248rn\u222brn\u221211xdx=rnln\u2061(n\u22121r)\u2248rnln\u2061(nr)(3)<\/p>\n\n\n\n

\u4ee4\u51fd\u6570f(r)=P(Cr)f(r)=P(Cr) \uff0c\u6240\u4ee5\u6211\u4eec\u9700\u8981\u6240\u6c42\u7684r0r0\u4e3af(r)f(r)\u7684\u6700\u5927\u6781\u503c\u70b9\uff0c\u5373f\u2032(r0)=0f\u2032(r0)=0 \uff0c\u6b64\u65f6\u53ef\u5f97r0=ner0=ne\u3002\u5df2\u77e51e\u22480.36788\u2248371e\u22480.36788\u224837\uff0c\u56e0\u6b64\u8ba1\u7b97\u7ed3\u679c\u7b26\u5408\u8981\u6c42\u3002<\/p>\n\n\n\n

<\/a>\u9a8c\u8bc1\u4ee3\u7801<\/h1>\n\n\n\n
1
2
3
4
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9
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24<\/td>		#include<iostream>
#include<stdio.h>
#define oo 1e9+7
using namespace std;
int main(){
freopen(“test.txt”,”w”,stdout);
for(int i=1;i<=10000;i++){
double maxp=0,temp=0;
int maxr=1;
for(int j=1;j<i;j++)temp+=1.0\/j;
for(int j=1;j<=i;j++){
double pj=1.0*j\/i;
pj*=temp;
if(pj>maxp){
maxp=pj;
maxr=j;
}
temp-=1.0\/j;
}
cout<<i<<” “<<maxr<<” “<<1.0*maxr\/i<<” “<<maxp<<endl;
}
fclose(stdout);
return 0;
}<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

<\/a>\u7ed3\u8bed<\/h1>\n\n\n\n

37%\u6cd5\u5219\u8fd8\u53ef\u4ee5\u7528\u4e8e\u751f\u6d3b\u4e2d\u6709\u5f88\u591a\u7c7b\u4f3c\u4e8e\u79d8\u4e66\u95ee\u9898\u7684\u95ee\u9898\uff0c\u50cf\u662f\u5982\u4f55\u9009\u62e9\u81ea\u5df1\u7684\u4f34\u4fa3\uff0c\u5982\u4f55\u51b3\u5b9a\u6295\u8d44\u7684\u9879\u76ee\uff0c\u5982\u4f55\u9009\u62e9\u505c\u8f66\u4f4d\u7b49\u53ef\u5316\u7b80\u4e3a\u6700\u4f18\u505c\u6b62\u95ee\u9898\u7684\u95ee\u9898\u3002\u5728\u6240\u6709\u6700\u4f18\u505c\u6b62\u95ee\u9898\u4e2d\uff0c\u6700\u5927\u7684\u96be\u70b9\u4e0d\u5728\u4e8e\u9009\u62e9\u54ea\u4e00\u79cd\u53ef\u9009\u65b9\u6848\uff0c\u800c\u662f\u786e\u5b9a\u81ea\u5df1\u9700\u8981\u8003\u8651\u591a\u5c11\u79cd\u53ef\u9009\u65b9\u6848\u3002<\/p>\n\n\n\n

\u6700\u4f18\u505c\u6b62\u95ee\u9898\u7684\u6743\u5a01\u6559\u79d1\u4e66\u5f00\u5b97\u660e\u4e49\u5730\u6307\u51fa:\u201c\u6700\u4f18\u505c\u6b62\u7406\u8bba\u5173\u6ce8\u7684\u662f\u5982\u4f55\u9009\u62e9\u65f6\u673a\u4ee5\u6267\u884c\u7279\u5b9a\u884c\u52a8\u7684\u95ee\u9898\u3002\u201d\u79d8\u4e66\u95ee\u9898\u6700\u57fa\u672c\u540c\u65f6\u4e5f\u6700\u4ee4\u4eba\u96be\u4ee5\u7f6e\u4fe1\u7684\u524d\u63d0\u6761\u4ef6\u2014\u2014\u4e25\u683c\u7684\u8fde\u7eed\u6027\uff0c\u5373\u6709\u8fdb\u65e0\u9000\u7684\u5355\u5411\u884c\u8fdb\uff0c\u6b63\u597d\u662f\u65f6\u95f4\u81ea\u8eab\u5c5e\u6027\u7684\u4e00\u4e2a\u4f53\u73b0\u3002\u751f\u6d3b\u4e2d\u6211\u4eec\u505a\u51fa\u51b3\u5b9a\u7684\u65f6\u5019\uff0c\u5f80\u5f80\u5177\u6709\u65f6\u95f4\u4e0a\u7684\u4e0d\u53ef\u633d\u56de\u6027\uff0c\u4f60\u6c38\u8fdc\u65e0\u6cd5\u56de\u5230\u8fc7\u53bb\u53bb\u91cd\u65b0\u505a\u51fa\u9009\u62e9\uff0c\u6211\u4eec\u6ca1\u6709\u4e8c\u6b21\u9009\u62e9\u673a\u4f1a\uff0c\u5982\u4f55\u9002\u53ef\u800c\u6b62\uff0c\u5728\u5408\u9002\u7684\u65f6\u95f4\u505c\u6b62\u89c2\u5bdf\u505a\u51fa\u9009\u62e9\uff0c\u8fd9\u662f\u975e\u5e38\u91cd\u8981\u7684\u95ee\u9898\u3002<\/p>\n\n\n\n

\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c37%\u6cd5\u5219\u9009\u5230\u7684\u5e76\u4e0d\u4e00\u5b9a\u662f\u6700\u4f18\u65b9\u6848\uff0c\u800c\u662f\u63a5\u8fd1\u4e8e\u6700\u4f18\u7684\u6ee1\u610f\u65b9\u6848\u3002\u8fd8\u6709\u7684\u65f6\u5019\u53ef\u80fd\u627e\u5230\u6700\u4f18\u5e76\u4e0d\u53ef\u80fd\uff0c\u6216\u8005\u4ee3\u4ef7\u6781\u5927\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u8981\u80fd\u63a5\u53d7\u201c\u6ee1\u610f\u201d\u3002\u800c\u4e14\u5f53\u6700\u4f18\u7684\u65b9\u6848\u51fa\u73b0\u5728\u524d37%\u7684\u65f6\u5019\uff0c\u4f60\u4f1a\u53d1\u73b037%\u6cd5\u5219\u4e5f\u65e0\u6cd5\u627e\u51fa\u6ee1\u610f\u7684\u65b9\u6848\u3002\u56e0\u6b64\u5728\u751f\u6d3b\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u7075\u6d3b\u7684\u5e94\u752837%\u6cd5\u5219\uff0c\u7ed3\u5408\u81ea\u5df1\u7684\u7ecf\u9a8c\uff0c\u5bfb\u627e\u5230\u90a3\u4e2a\u4ee4\u4eba\u6ee1\u610f\u7684\u201c\u53ef\u201d\u4e4b\u6240\u5728\u3002<\/p>\n\n\n\n

<\/a>\u53c2\u8003\u6587\u732e<\/h1>\n\n\n\n

[1] \u7ea6\u7ff0\u5c3c\u65af\u00b7\u5f00\u666e\u52d2 \u7b97\u6cd5\u4e4b\u7f8e\uff1a\u6307\u5bfc\u5de5\u4f5c\u4e0e\u751f\u6d3b\u7684\u7b97\u6cd5
[2] \u7ef4\u57fa\u767e\u79d1 \u79d8\u4e66\u95ee\u9898
[3] \u542c\u98ce\u4e34\u5c71 \u751f\u6d3b\u4e2d\u4e00\u5b9a\u4f1a\u7528\u5230\u7684\u6570\u5b66\u5e38\u8bc6\uff1a37%\u6cd5\u5219<\/p>\n\n\n\n

https:\/\/kxjzxc.github.io\/%E6%9C%80%E4%BC%98%E5%81%9C%E6%AD%A2%E7%90%86%E8%AE%BA%E2%80%94%E2%80%94%E5%A6%82%E4%BD%95%E9%80%82%E2%80%9C%E5%8F%AF%E2%80%9D%E8%80%8C%E6%AD%A2<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

\u6458\u8981 \u672c\u6587\u4e3b\u8981\u4ece\u6700\u4f18\u505c\u6b62\u95ee\u9898\u51fa\u53d1\uff0c\u4ecb\u7ecd\u4e86\u9ea6\u7a57\u95ee\u9898\u548c\u7ecf\u5178\u7684\u79d8\u4e66\u95ee\u9898\u4ee5\u53ca\u9762\u5bf9\u6b64\u7c7b\u95ee\u9898\u91c7\u53d6\u7684\u7b56\u7565\u3002\u672c\u6587\u8fd8\u6839\u636e\u8be5\u7b56\u7565\u63a8 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-10329","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.invbus.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/10329","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.invbus.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.invbus.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.invbus.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.invbus.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=10329"}],"version-history":[{"count":0,"href":"https:\/\/www.invbus.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/10329\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.invbus.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10329"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.invbus.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10329"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.invbus.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10329"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}