{"id":1225,"date":"2022-01-20T17:43:14","date_gmt":"2022-01-20T09:43:14","guid":{"rendered":"https:\/\/damakey.com\/?p=1225"},"modified":"2023-05-24T17:47:18","modified_gmt":"2023-05-24T09:47:18","slug":"poisson-%e5%88%86%e4%bd%88","status":"publish","type":"post","link":"https:\/\/damakey.com\/?p=1225","title":{"rendered":"Poisson \u5206\u4f48"},"content":{"rendered":"

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\u5728\u300a\u70ba\u4ec0\u9ebc\u516c\u8eca\u4e00\u6b21\u4f86<\/span>3<\/span>\u73ed\u300b\u9019\u672c\u66f8\u5167\uff0c\u63d0\u5230\u300c\u8cb7\u689d\u9b5a\u90fd\u8981\u6392\u968a\u300d\uff0c\u7576\u7136\u4f86\u7684\u9867\u5ba2\u6578\u591a\u904e\u670d\u52d9\u7684\u4eba\u624b\uff0c\u7576\u7136\u5c31\u5f97\u7b49\u3002\u5728\u7279\u5b9a\u6642\u9593\u5167\u6703\u6709<\/span>x<\/span>\u4f4d\u9867\u5ba2\u51fa\u73fe\u7684\u6a5f\u7387\uff0c\u5c31\u7b26\u5408\u535c\u74e6\u677e\u5206\u5e03<\/span>(Poisson distribution)<\/span>\uff0c\u9019\u8b93\u6211\u60f3\u8d77\u5728\u670d\u52d9\u696d\u5de5\u4f5c\uff0c\u4ee4\u4eba\u982d\u75db\u7684\u670d\u52d9\u6c34\u6e96\u554f\u984c\u3002<\/span><\/p>\n

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\u5148\u8907\u7fd2\u4e00\u4e0b\u535c\u74e6\u677e\u5206\u5e03<\/span> (Poisson dustrituion)\uff0c <\/span>\u5176\u5be6\u5b83\u662f\u4e8c\u9805\u5f0f\u5206\u4f48<\/span>(binomial distraction)<\/span>\u7684\u4e00\u500b\u7279\u4f8b\u3002<\/span><\/p>\n

 <\/p>\n

\u554f\u984c\uff1a\u5047\u8a2d\u5728\u6975\u77ed\u7684\u6642\u9593\u5167<\/span>Δt<\/span>\uff0c\u5ba2\u6236\u51fa\u73fe\u7684\u6a5f\u7387<\/span>p<\/span>\u548c<\/span>Δt<\/span>\u6210\u6b63\u6bd4\uff0c<\/span>p= αΔt<\/span>\uff0c\u90a3\u9ebc\u5728\u6642\u9593<\/span>t<\/span>\u5167\u6703\u6709<\/span>x<\/span>\u4f4d\u5ba2\u6236\u51fa\u73fe\u7684\u6a5f\u7387\u70ba\u4f55\uff1f<\/span><\/p>\n

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\u6211\u53ef\u4ee5\u628a<\/span>t<\/span>\u5207\u6210<\/span>n<\/span>\u6bb5\uff0c<\/span>t=n*Δt<\/span>\uff0c<\/span>n<\/span>\u662f\u4e00\u500b\u6975\u5927\u7684\u6578\uff0c\u6240\u4ee5\u524d\u8ff0\u7684\u554f\u984c\uff0c\u5c31\u8b8a\u6210\u6211\u5011\u5728\u6642\u9593<\/span>t<\/span>\u5167\u505a<\/span>n<\/span>\u6b21\u5be6\u9a57<\/span>(<\/span>\u89c0\u5bdf<\/span>)<\/span>\uff0c\u90a3\u9ebc\u51fa\u73fe<\/span>x<\/span>\u6b21\u6210\u529f<\/span>(<\/span>\u9867\u5ba2\u51fa\u73fe<\/span>)<\/span>\u7684\u6a5f\u7387\u70ba\u4f55\uff1f<\/span><\/p>\n

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\u4ee4<\/span>λ= αt=nαΔt=np<\/span>\uff0c\u9019\u662f\u4e00\u500b\u4e8c\u9805\u5f0f\u5206\u4f48<\/span>*<\/span>\uff1a<\/span><\/p>\n

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b(x; n, p)<\/span><\/p>\n

= <\/span>[<\/span>n!\/(x!*(n-x)!)<\/span>]<\/span>*<\/span>[<\/span>p^x*(1-p)^(n-x)<\/span>]<\/span><\/p>\n

= <\/span>[<\/span>n*(n-1)*(n-2)*…(n-x+1)<\/span>]<\/span>\/x!*<\/span>[<\/span>(λ\/n)^x*(1-λ\/n)^(n-x)<\/span>]<\/span><\/p>\n

=<\/span>[<\/span>1*(1-1\/n)*(1-2\/n)*…(1-(x-1)\/n)<\/span>]<\/span>\/x!* (λ^x)*<\/span>[<\/span>(1-λ\/n)^(-n\/λ)<\/span>]<\/span>^(-λ)* (1-λ\/n)^(-x)<\/span><\/p>\n

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\u7576<\/span>t<\/span>\u4fdd\u6301\u4e0d\u8b8a<\/span>(<\/span>\u4ea6\u5373<\/span>λ <\/span>\u4e0d\u8b8a<\/span>) n—> ∞ (αΔ —> 0) <\/span>\u6642<\/span><\/p>\n

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[<\/span>1*(1-1\/n)*(1-2\/n)*…(1-(x-1)\/n)<\/span>]<\/span> <\/span>\u8da8\u8fd1\u65bc<\/span> 1<\/span><\/p>\n

[<\/span>(1-λ\/n)^(-n\/λ)<\/span>]<\/span> <\/span>\u8da8\u8fd1\u65bc<\/span> e<\/span><\/p>\n

(1-λ\/n)^(-x) <\/span>\u8da8\u8fd1\u65bc<\/span> 1<\/span><\/p>\n

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b(x; n, p)<\/span><\/p>\n

= λ^x *e^(-λ)\/x! <\/span><\/p>\n

 <\/p>\n

\u9019\u5c31\u535c\u74e6\u677e\u5206\u4f48\u7684\u6a5f\u7387\u51fd\u6578\uff0c\u53ef\u5beb\u6210<\/span>p(x;λ)= λ^x *e^(-λ)\/x! <\/span><\/p>\n

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λ= αt=nαΔt=np<\/span>\uff0c\u5c31\u662f\u5728\u6642\u9593<\/span>t<\/span>\u5167\u9810\u671f\u6703\u51fa\u73fe\u7684\u9867\u5ba2\u6578\u3002<\/span><\/p>\n

 <\/p>\n

\u4f8b\u5b50\uff1a\u5982\u679c\u5e73\u5747\u6bcf\u5206\u9418\u51fa\u73fe\u4e00\u4f4d\u9867\u5ba2\uff0c\u90a3\u9ebc\u5728\u67d0\u7279\u5b9a\u7684<\/span>1<\/span>\u5206\u9418\u5167\uff0c\u51fa\u73fe<\/span>4<\/span>\u4f4d\u9867\u5ba2\u7684\u6a5f\u7387\u662f\u591a\u5c11\uff1f<\/span>p(4;1)=1^1*e^(-1)\/4!=1\/2.71*4*3*2*1~1\/50=2% <\/span>\u6a5f\u7387\u662f\u883b\u4f4e\u7684\uff0c\u9019\u4e5f\u5408\u4e4e\u6211\u5011\u7684\u76f4\u89ba\uff0c\u65e2\u7136\u5e73\u5747\u6bcf\u5206\u9418\u624d\u6709<\/span>1<\/span>\u4f4d\u9867\u5ba2\uff0c\u90a3\u9ebc<\/span>1<\/span>\u5206\u9418\u5167\u6703\u6709<\/span>4<\/span>\u4f4d\u9867\u5ba2\u5149\u81e8\u7684\u6a5f\u7387\u7576\u7136\u4f4e\u4e86\u3002<\/span><\/p>\n

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\u90a3\u9ebc\u51fa\u73fe<\/span>2<\/span>\u4f4d\u9867\u5ba2\u7684\u6a5f\u7387\u5462\uff1f<\/span>p(2;1)= 1^1*e^(-1)\/2!~18% <\/span>\u610f\u601d\u662f\u8aaa\u5982\u679c\u63a5\u5f85\u7684\u6ac3\u53f0\u53ea\u914d\u7f6e<\/span>1<\/span>\u4f4d\u670d\u52d9\u4eba\u54e1\uff0c\u90a3\u9ebc\u5728\u7b2c<\/span>1<\/span>\u5206\u9418\u6703\u8b93<\/span>1<\/span>\u4f4d\u9867\u5ba2\u7b49\u5019\u7684\u6a5f\u7387\u662f<\/span>18%<\/span>\u3002\u516c\u53f8\u8981\u914d\u5099\u591a\u5c11\u4eba\u529b\uff0c\u5c31\u8996\u5c0d\u9867\u5ba2\u91cd\u8996\u7684\u7a0b\u5ea6\u70ba\u4f55\u4e86\u3002<\/span><\/p>\n

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\u5176\u5be6\u5c0d\u670d\u52d9\u696d\uff0c\u53ef\u4ee5\u9078\u64c7\u7684\u4e26\u4e0d\u591a\uff0c\u53ea\u80fd\u9078\u64c7\u7b49\u9867\u5ba2\uff0c\u6216\u8005\u8b93\u9867\u5ba2\u7b49\u3002<\/span><\/p>\n

 <\/p>\n

Poisson <\/span>\u5206\u4f48\uff0c\u662f\u670d\u52d9\u4e2d\u5fc3\u5728\u4f30\u7b97\u4eba\u529b\u6642\uff0c\u5c0d\u9867\u5ba2\u6578\u51fa\u73fe\u7684\u6a5f\u7387\uff0c\u6700\u6839\u672c\u7684\u6839\u64da\u3002<\/span><\/p>\n

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\u5be6\u52d9\u4e0a\u6709\u6240\u8b02\u7684<\/span>Erlang C<\/span>\u7684\u6a21\u5f0f\uff0c\u53ea\u8981\u8f38\u5165\u5e73\u5747\u55ae\u4f4d\u6642\u9593\u7684\u9867\u5ba2\u6578\u3001\u5e73\u5747\u6bcf\u4f4d\u9867\u5ba2\u7684\u670d\u52d9\u6642\u9593\u3001\u4eba\u54e1\u914d\u7f6e\uff0c\u5c31\u53ef\u4ee5\u8a08\u7b97\u51fa\u9867\u5ba2\u5e73\u5747\u7684\u7b49\u5019\u6642\u9593\u3002<\/span><\/p>\n

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\u770b\u8d77\u4f86\u5f88\u79d1\u5b78\uff0c\u5176\u5be6\u4e0d\u7136\u3002<\/span>Poisson <\/span>\u544a\u8a34\u6211\u5011\u7684\u53ea\u662f\u6a5f\u7387\uff0c\u4f46\u662f\u6a5f\u7387\u4f4e\u4e26\u4e0d\u4ee3\u8868\u5b83\u4e0d\u6703\u767c\u751f\u3002\u6240\u4ee5\uff0c\u50cf<\/span>Erlang C<\/span>\u7684\u6a21\u5f0f\uff0c\u53ea\u80fd\u662f\u4e00\u500b\u53c3\u8003\u3002\u5982\u679c\u8981\u63d0\u4f9b\u66f4\u8fc5\u901f\u7684\u670d\u52d9\uff0c\u5728\u90a3\u500b\u57fa\u6e96\u4e4b\u4e0a\uff0c\u5c31\u5fc5\u9808\u591a\u914d\u7f6e\u4e00\u4e9b\u4eba\u529b\u3002\u6709\u6642\u5019\u5c31\u96e3\u514d\u986f\u5f97\u6d6a\u8cbb\uff0c\u4f46\u662f\u90a3\u5c31\u662f\u8b93\u670d\u52d9\u4eba\u54e1\u7b49\u5f85\u9867\u5ba2\u800c\u4e0d\u662f\u9867\u5ba2\u7b49\u5f85\u670d\u52d9\u4eba\u54e1\u7684\u9078\u64c7\u3002\u5728\u898f\u5283\u7684\u6642\u5019\u505a\u5f97\u592a\u300c\u7cbe\u6e96\u300d\uff0c\u770b\u4f86\u5f88\u6709\u6548\u7387\uff0c\u5176\u5be6\u90a3\u662f\u8868\u8c61\uff0c\u56e0\u70ba\u4f9d\u7167<\/span>Poisson <\/span>\u5206\u4f48\uff0c\u610f\u5916\u5c31\u662f\u6703\u767c\u751f\uff0c\u5c31\u5f97\u8b93\u9867\u5ba2\u7b49\u4e86\u3002\u800c\u4e14\u5728\u610f\u5916\u767c\u751f\u6642\uff0c\u670d\u52d9\u6642\u9593\u4e5f\u53ef\u80fd\u62c9\u9577\uff0c\u90a3\u5c31\u662f\u96ea\u4e0a\u52a0\u971c\u4e86\u3002<\/span><\/p>\n

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\u8207\u5176\u8aaa<\/span>There is always no free lunch. <\/span>\u4e0d\u5982\u8aaa<\/span>Free lunch is always very expensive. <\/span>\u5728\u9867\u5ba2\u7684\u670d\u52d9\u4e0a\u5c31\u662f\u5982\u6b64\uff0c\u5b8c\u5168\u53d6\u5de7\u4e0d\u5f97\u3002\u53ea\u662f\uff0c\u4e0a\u982d\u7684\u8001\u95c6\u5f80\u5f80\u90fd\u4e0d\u9019\u9ebc\u60f3\u3002\u4e0b\u9762\u7684\u54e1\u5de5\u5c31\u8f9b\u82e6\u4e86\u3002<\/span><\/p>\n

 <\/p>\n

 <\/p>\n

*<\/span>\uff1a<\/span>Poisson <\/span>\u5206\u4f48\uff0c\u66f9\u5409\u4eae<\/span><\/p>\n

**<\/span>\uff1a\u300a\u70ba\u4ec0\u9ebc\u516c\u8eca\u4e00\u6b21\u4f86<\/span>3<\/span>\u73ed\u300b\uff0c\u7f85\u52c3·\u4f0a\u65af\u5a01<\/span> <\/span>\u5091\u745e\u7c73·\u6eab\u5fb3\u6f22<\/span> <\/span>\u8457\uff0c\u8521\u627f\u5fd7<\/span> <\/span>\u5370<\/span><\/p>\n

 <\/p>\n

2022\/1\/20 Poisson <\/span>\u5206\u4f48<\/span> Damakey<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

  \u5728\u300a\u70ba\u4ec0\u9ebc\u516c\u8eca\u4e00\u6b21\u4f863\u73ed\u300b\u9019\u672c\u66f8\u5167\uff0c\u63d0\u5230\u300c\u8cb7\u689d…<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"content-type":"","footnotes":""},"categories":[3],"tags":[],"class_list":["post-1225","post","type-post","status-publish","format-standard","hentry","category-3"],"_links":{"self":[{"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/posts\/1225","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/damakey.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1225"}],"version-history":[{"count":1,"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/posts\/1225\/revisions"}],"predecessor-version":[{"id":20108,"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/posts\/1225\/revisions\/20108"}],"wp:attachment":[{"href":"https:\/\/damakey.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1225"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/damakey.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1225"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/damakey.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1225"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}