{"id":1469,"date":"2018-11-13T19:49:15","date_gmt":"2018-11-13T11:49:15","guid":{"rendered":"http:\/\/ykyi.net\/?p=1469"},"modified":"2018-11-13T19:49:15","modified_gmt":"2018-11-13T11:49:15","slug":"%e4%bc%af%e5%8a%aa%e5%88%a9binomial%e5%88%86%e5%b8%83%e5%92%8c%e4%bc%af%e5%8a%aa%e5%88%a9%e8%af%95%e9%aa%8c%e7%9a%84%e5%8c%ba%e5%88%ab%ef%bc%8c%e8%b4%9f%e4%bc%af%e5%8a%aa%e5%88%a9%e5%88%86%e5%b8%83","status":"publish","type":"post","link":"https:\/\/ykyi.net\/?p=1469","title":{"rendered":"\u4f2f\u52aa\u5229(Binomial)\u5206\u5e03\u548c\u4f2f\u52aa\u5229\u8bd5\u9a8c\u7684\u533a\u522b\uff0c\u8d1f\u4f2f\u52aa\u5229\u5206\u5e03\u53ca\u51e0\u4f55\u5206\u5e03,\u8d85\u51e0\u4f55\u5206\u5e03"},"content":{"rendered":"<p>\u6211\u53c8\u91cd\u65b0\u628a\u6982\u7387\uff0c\u7edf\u8ba1\u4e66\u62ff\u8d77\u6765\u4e86\u3002\u8fd9\u6b21\u662f\u771f\u7684\uff0c\u8981\u628a\u673a\u5668\u5b66\u4e60\uff08\u7edf\u8ba1\u5b66\u4e60)\u5b66\u597d\u3002\u663e\u7136\uff0c\u5b66\u597d\u673a\u5668\u5b66\u4e60\u5e76\u4e0d\u662f\u4e00\u4ef6\u5bb9\u6613\u7684\u4e8b\u60c5\u3002\u9700\u8981\u8981\u624e\u5b9e\u7684\u6570\u5b66\u57fa\u7840\u3002\u597d\u5728\u6211\u7684\u6570\u5b66\u50a8\u5907\u8fd8\u52c9\u5f3a\u53ef\u4ee5\uff0c\u4f46\u8fdc\u7b97\u4e0d\u4e0a\u201c\u57fa\u7840\u624e\u5b9e\u201d\u3002\u6240\u4ee5\uff0c\u6211\u6709\u4e86\u4e00\u4e2a\u957f\u671f\u8fdc\u666f\u89c4\u5212\uff0c\u91cd\u65b0\u5f00\u59cb\u5b66\u4e60\u5427\uff01<\/p>\n<p>\u4eca\u5929\u770b\u4e86\u4f2f\u52aa\u529b\u5206\u5e03\uff0c\u8d1f\u4f2f\u52aa\u5229\u5206\u5e03\uff0c\u8d85\u51e0\u4f55\u5206\u5e03\uff0c\u51e0\u4f55\u5206\u5e03\u8fd9\u51e0\u4e2a\u77e5\u8bc6\u70b9\u3002\u4e0b\u9762\u6211\u4e00\u8fb9\u5199\u56de\u987e\u4eca\u5929\u5b66\u5230\u7684\uff0c\u4e00\u8fb9\u5199\u6587\u7ae0\u3002<\/p>\n<h4>\u4f2f\u52aa\u5229\u5206\u5e03\u548c\u4f2f\u52aa\u5229\u8bd5\u9a8c\u7684\u533a\u522b<\/h4>\n<p>\u4f2f\u52aa\u5229\u8bd5\u9a8c(Binomial Experiment)\u6307\u7684\u4ec5\u4ec5\u662f&#8221;\u4e00\u6b21\u8bd5\u9a8c&#8221;\uff0c\u6216\u8005\u6210\u529f\uff0c\u6216\u8005\u5931\u8d25\u3002\u4f2f\u5974\u5229\u5206\u5e03\u662f\u4e00\u4e2a\u5e38\u7528\u5206\u5e03\uff0c\u8fd9\u662f\u5b8c\u5168\u4e0d\u540c\u7684\u4e24\u4e2a\u6982\u5ff5\u3002<br \/>\n\u6bcf\u6b21\u4f2f\u52aa\u5229\u8bd5\u9a8c\u7684\u6210\u529f\u6982\u7387\u8bb0\u4e3ap\uff0c\u53c8\u628a\u91cd\u590d\u5f88\u591a\u6b21\u4f2f\u52aa\u5229\u8bd5\u9a8c\u79f0\u4e3a\u4f2f\u52aa\u5229\u8fc7\u7a0b(Binomial Process)\u3002\u4e00\u4e2a\u91cd\u590d\u4e86n\u6b21\u4f2f\u52aa\u5229\u8bd5\u9a8c\u7684\u4f2f\u52aa\u5229\u8fc7\u7a0b\u4e2d\uff0c\u6210\u529f\u7684\u8bd5\u9a8c\u6b21\u6570\u8bb0\u4e3aX\uff0c\u8fd9\u4e2aX\u5219\u662f\u4f2f\u52aa\u5229\u968f\u673a\u53d8\u91cf\u3002X\u7684\u6982\u7387\u5206\u5e03\u88ab\u79f0\u4e3a\u4f2f\u52aa\u5229\u5206\u5e03\u3002<br \/>\n\u4f2f\u52aa\u5229\u5206\u5e03\uff1a<\/p>\n<p><span class=\"katex math multi-line\">b(x;n,p) = {n \\choose x} p^k (1-p)^{ n-k}<\/span><\/p>\n<p>\u5176\u4e2d\uff0cn\u8868\u793a\u8bd5\u9a8c\u6b21\u6570\uff0cp\u8868\u793a\u6bcf\u6b21\u72ec\u7acb\u8bd5\u9a8c\u6210\u529f\u7684\u6982\u7387\uff0cx\u8868\u793a\u6210\u529f\u6b21\u6570\u3002<\/p>\n<h3>\u4ec0\u4e48\u662f\u8d1f\u4f2f\u52aa\u5229\u5206\u5e03(Negative Binomial Distribution)<\/h3>\n<p>\u4f2f\u52aa\u5229\u5206\u5e03\u548c\u8d1f\u4f2f\u52aa\u5229\u5206\u5e03\u7684\u5927\u591a\u6570\u8bbe\u5b9a\u90fd\u662f\u4e00\u6837\u7684\uff0c\u90fd\u4e0d\u65ad\u91cd\u590d\u4f2f\u52aa\u5229\u8bd5\u9a8c\uff0c\u8981\u4e48\u6210\u529f\u8981\u4e48\u5931\u8d25\u3002<br \/>\n\u4e0d\u540c\u70b9\u662f\uff1a\u4f2f\u52aa\u5229\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cfX\uff0c\u8868\u793a\u7684\u662f\u5728\u524dn\u6b21\u4f2f\u52aa\u5229\u8bd5\u9a8c\u4e2d\u6210\u529f\u4e86x\u6b21\uff1b\u800c\u8d1f\u4f2f\u52aa\u5229\u7684\u968f\u673a\u53d8\u91cfx\uff0c\u8868\u793a\u7684\u662f\u8981\u8fbe\u6210k\u6b21\u6210\u529f\u8bd5\u9a8c\uff0c\u9700\u8981\u8bd5\u9a8cx\u6b21\uff08\u5373\u7b2cx\u6b21\u8bd5\u9a8c\u6210\u529f\uff0c\u524dx-1\u6b21\u8bd5\u9a8c\u6210\u529f\u4e86k-1\u6b21\uff09\u3002\u4f2f\u52aa\u5229\u5206\u5e03\u9650\u5b9a\u4e86\u8bd5\u9a8c\u6b21\u6570\u8bb0\u4e3an\uff0c\u8d1f\u4f2f\u52aa\u5229\u9650\u5b9a\u4e86\u6210\u529f\u6b21\u6570\u8bb0\u4e3ak\u3002<br \/>\n\u8d1f\u4f2f\u52aa\u5229\u5206\u5e03\uff1a<\/p>\n<p><span class=\"katex math multi-line\">b^*(x;k, p) = {{x-1} \\choose {k-1}}p^kq^x-k<\/span><\/p>\n<h3>\u51e0\u4f55\u5206\u5e03\u548c\u8d85\u51e0\u4f55\u5206\u5e03<\/h3>\n<p>\u518d\u4ee5\u4f2f\u52aa\u5229\u5206\u5e03\u4e3a\u8ba8\u8bba\u7684\u57fa\u7840\uff0c\u4ec5\u66f4\u6539\u4e00\u5904\uff0c\u5373\u628a\u4f2f\u52aa\u5229\u8fc7\u7a0b\u4e2d\u7684\u653e\u56de\u62bd\u6837(with replacement)\u6539\u6210\u4e0d\u653e\u56de\u62bd\u6837(without replacement)\uff0c\u5373\u53ef\u5f97\u5230\u8d85\u51e0\u4f55\u5206\u5e03\u3002<br \/>\n\u4ee5\u8d1f\u4f2f\u52aa\u5229\u5206\u5e03\u4e3a\u8ba8\u8bba\u7684\u57fa\u7840\uff0c\u4ec5\u66f4\u6539\u4e00\u5904\uff0c\u5373k\u8bbe\u4e3a1\uff0c\u8868\u793a\u6210\u529f\u4e00\u6b21\u3002\u5373\u4e0d\u65ad\u91cd\u590d\u4f2f\u52aa\u5229\u8bd5\u9a8c\uff0c\u76f4\u5230\u7b2c\u4e00\u6b21\u6210\u529f\u5c31\u7ed3\u675f\uff0c\u5f97\u5230\u4e86\u51e0\u4f55\u5206\u5e03\u3002<\/p>\n<h3>\u4f2f\u52aa\u5229\u5206\u5e03\uff0c\u8d1f\u4f2f\u52aa\u5229\u5206\u5e03\u7684\u547d\u540d<\/h3>\n<ul>\n<li>\u4e3a\u4ec0\u4e48\u53eb\u4f2f\u52aa\u5229\u5206\u5e03<br \/>\n\u56e0\u4e3a\u4f2f\u52aa\u5229\u5c55\u5f00\u5f0f<span class=\"katex math multi-line\">(q+p)^n<\/span>\u7684n+1\u4e2a\u9879\u5bf9\u5e94\u4f2f\u52aa\u5229\u5206\u5e03\u53d6\u5404\u4e2a\u503c\u65f6\u7684\u60c5\u51b5\u3002<\/li>\n<li>\u4e3a\u4ec0\u4e48\u53eb\u8d1f\u4f2f\u52aa\u5229\u5206\u5e03<br \/>\n\u56e0\u4e3a\u5c55\u5f00\u5f0f<span class=\"katex math multi-line\">p^k(1-p)^(-k)<\/span>\u4e2d\u7684\u6bcf\u4e2a\u9879\u5bf9\u5e94<span class=\"katex math multi-line\">b^*(x;k,p)<\/span>\u53d6x=k, k+1, k+2, &#8230;\u65f6\u7684\u60c5\u51b5\u3002<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>\u6211\u53c8\u91cd\u65b0\u628a\u6982\u7387\uff0c\u7edf\u8ba1\u4e66\u62ff\u8d77\u6765\u4e86\u3002\u8fd9\u6b21\u662f\u771f\u7684\uff0c\u8981\u628a\u673a\u5668\u5b66\u4e60\uff08\u7edf\u8ba1\u5b66\u4e60)\u5b66\u597d\u3002\u663e\u7136\uff0c\u5b66\u597d\u673a\u5668\u5b66\u4e60\u5e76\u4e0d\u662f\u4e00\u4ef6\u5bb9\u6613\u7684\u4e8b\u60c5\u3002\u9700\u8981\u8981\u624e\u5b9e\u7684\u6570\u5b66\u57fa\u7840\u3002\u597d\u5728\u6211\u7684\u6570\u5b66\u50a8\u5907\u8fd8\u52c9\u5f3a\u53ef\u4ee5\uff0c\u4f46\u8fdc\u7b97\u4e0d\u4e0a\u201c\u57fa\u7840\u624e\u5b9e\u201d\u3002\u6240\u4ee5\uff0c\u6211\u6709\u4e86\u4e00\u4e2a\u957f\u671f\u8fdc\u666f\u89c4\u5212\uff0c\u91cd\u65b0\u5f00\u59cb\u5b66\u4e60\u5427\uff01 \u4eca\u5929\u770b\u4e86\u4f2f\u52aa\u529b\u5206\u5e03\uff0c\u8d1f\u4f2f\u52aa\u5229\u5206\u5e03\uff0c\u8d85\u51e0\u4f55\u5206\u5e03\uff0c\u51e0\u4f55\u5206\u5e03\u8fd9\u51e0\u4e2a\u77e5\u8bc6\u70b9\u3002\u4e0b\u9762\u6211\u4e00\u8fb9\u5199\u56de\u987e\u4eca\u5929\u5b66\u5230\u7684\uff0c\u4e00\u8fb9\u5199\u6587\u7ae0\u3002 \u4f2f\u52aa\u5229\u5206\u5e03\u548c\u4f2f\u52aa\u5229\u8bd5\u9a8c\u7684\u533a\u522b \u4f2f\u52aa\u5229\u8bd5\u9a8c(Binomial Experiment)\u6307\u7684\u4ec5\u4ec5\u662f&#8221;\u4e00\u6b21\u8bd5\u9a8c&#8221;\uff0c\u6216\u8005\u6210\u529f\uff0c\u6216\u8005\u5931\u8d25\u3002\u4f2f\u5974\u5229\u5206\u5e03\u662f\u4e00\u4e2a\u5e38\u7528\u5206\u5e03\uff0c\u8fd9\u662f\u5b8c\u5168\u4e0d\u540c\u7684\u4e24\u4e2a\u6982\u5ff5\u3002 \u6bcf\u6b21\u4f2f\u52aa\u5229\u8bd5\u9a8c\u7684\u6210\u529f\u6982\u7387\u8bb0\u4e3ap\uff0c\u53c8\u628a\u91cd\u590d\u5f88\u591a\u6b21\u4f2f\u52aa\u5229\u8bd5\u9a8c\u79f0\u4e3a\u4f2f\u52aa\u5229\u8fc7\u7a0b(Binomial Process)\u3002\u4e00\u4e2a\u91cd\u590d\u4e86n\u6b21\u4f2f\u52aa\u5229\u8bd5\u9a8c\u7684\u4f2f\u52aa\u5229\u8fc7\u7a0b\u4e2d\uff0c\u6210\u529f\u7684\u8bd5\u9a8c\u6b21\u6570\u8bb0\u4e3aX\uff0c\u8fd9\u4e2aX\u5219\u662f\u4f2f\u52aa\u5229\u968f\u673a\u53d8\u91cf\u3002X\u7684\u6982\u7387\u5206\u5e03\u88ab\u79f0\u4e3a\u4f2f\u52aa\u5229\u5206\u5e03\u3002 \u4f2f\u52aa\u5229\u5206\u5e03\uff1a b(x;n,p) = {n \\choose x} p^k (1-p)^{ n-k} \u5176\u4e2d\uff0cn\u8868\u793a\u8bd5\u9a8c\u6b21\u6570\uff0cp\u8868\u793a\u6bcf\u6b21\u72ec\u7acb\u8bd5\u9a8c\u6210\u529f\u7684\u6982\u7387\uff0cx\u8868\u793a\u6210\u529f\u6b21\u6570\u3002 \u4ec0\u4e48\u662f\u8d1f\u4f2f\u52aa\u5229\u5206\u5e03(Negative Binomial Distribution) \u4f2f\u52aa\u5229\u5206\u5e03\u548c\u8d1f\u4f2f\u52aa\u5229\u5206\u5e03\u7684\u5927\u591a\u6570\u8bbe\u5b9a\u90fd\u662f\u4e00\u6837\u7684\uff0c\u90fd\u4e0d\u65ad\u91cd\u590d\u4f2f\u52aa\u5229\u8bd5\u9a8c\uff0c\u8981\u4e48\u6210\u529f\u8981\u4e48\u5931\u8d25\u3002 \u4e0d\u540c\u70b9\u662f\uff1a\u4f2f\u52aa\u5229\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cfX\uff0c\u8868\u793a\u7684\u662f\u5728\u524dn\u6b21\u4f2f\u52aa\u5229\u8bd5\u9a8c\u4e2d\u6210\u529f\u4e86x\u6b21\uff1b\u800c\u8d1f\u4f2f\u52aa\u5229\u7684\u968f\u673a\u53d8\u91cfx\uff0c\u8868\u793a\u7684\u662f\u8981\u8fbe\u6210k\u6b21\u6210\u529f\u8bd5\u9a8c\uff0c\u9700\u8981\u8bd5\u9a8cx\u6b21\uff08\u5373\u7b2cx\u6b21\u8bd5\u9a8c\u6210\u529f\uff0c\u524dx-1\u6b21\u8bd5\u9a8c\u6210\u529f\u4e86k-1\u6b21\uff09\u3002\u4f2f\u52aa\u5229\u5206\u5e03\u9650\u5b9a\u4e86\u8bd5\u9a8c\u6b21\u6570\u8bb0\u4e3an\uff0c\u8d1f\u4f2f\u52aa\u5229\u9650\u5b9a\u4e86\u6210\u529f\u6b21\u6570\u8bb0\u4e3ak\u3002 \u8d1f\u4f2f\u52aa\u5229\u5206\u5e03\uff1a b^*(x;k, p) = {{x-1} \\choose {k-1}}p^kq^x-k \u51e0\u4f55\u5206\u5e03\u548c\u8d85\u51e0\u4f55\u5206\u5e03 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