{"id":73317,"date":"2016-07-28T07:16:41","date_gmt":"2016-07-27T23:16:41","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=73317"},"modified":"2021-10-06T16:04:03","modified_gmt":"2021-10-06T08:04:03","slug":"%e4%bc%af%e5%8a%aa%e5%8a%9b%e8%a9%a6%e9%a9%97%e8%88%87%e4%ba%8c%e9%a0%85%e5%88%86%e5%b8%83","status":"publish","type":"post","link":"http:\/\/localhost\/%e4%bc%af%e5%8a%aa%e5%8a%9b%e8%a9%a6%e9%a9%97%e8%88%87%e4%ba%8c%e9%a0%85%e5%88%86%e5%b8%83\/","title":{"rendered":"\u4f2f\u52aa\u529b\u8a66\u9a57\u8207\u4e8c\u9805\u5206\u5e03"},"content":{"rendered":"

\u4f2f\u52aa\u529b\u8a66\u9a57\u8207\u4e8c\u9805\u5206\u5e03 (Bernoulli Trial and Binomial Distribution)<\/strong><\/span>
\n\u570b\u7acb\u6210\u529f\u5927\u5b78\u7d71\u8a08\u7cfb\/\u6771\u5433\u5927\u5b78\u8ca1\u52d9\u5de5\u7a0b\u8207\u7cbe\u7b97\u6578\u5b78\u7cfb\u5c08\u4efb\u7d71\u8a08\u52a9\u6559 \u675c\u67cf\u6bc5<\/strong><\/span><\/p>\n

\u5728\u751f\u6d3b\u4e2d\uff0c\u6709\u5f88\u591a\u7684\u4e8b\u60c5\u90fd\u53ea\u6709\u5169\u7a2e\u7d50\u679c (outcome)\uff0c\u4f8b\u5982\u8003\u8a66\u662f\u5426\u53ca\u683c\u3001\u660e\u5929\u662f\u5426\u4e0b\u96e8\u3001\u4e1f\u64f2\u9285\u677f\u4e26\u89c0\u5bdf\u5176\u7d50\u679c\u3002\u7576\u4e00\u500b\u8a66\u9a57\u53ea\u6709\u5169\u7a2e\u53ef\u80fd\u7d50\u679c\uff08\u6210\u529f\u8207\u5931\u6557\uff09\uff0c\u4e14\u5169\u500b\u7d50\u679c\u51fa\u73fe\u4e4b\u6a5f\u7387\u70ba\u56fa\u5b9a\uff08\u82e5\u6210\u529f\u6a5f\u7387\u70ba \\(p\\)\uff0c\u5247\u5931\u6557\u6a5f\u7387\u70ba \\(1-p\\)\uff09\uff0c\u6211\u5011\u7a31\u9019\u6a23\u7684\u8a66\u9a57\u70ba\u4f2f\u52aa\u529b\u8a66\u9a57 (Bernoulli trial)\u3002\u7576\u6211\u5011\u91cd\u8907\u9032\u884c\u591a\u6b21\u76f8\u540c\u7684\u4f2f\u52aa\u529b\u8a66\u9a57\uff08\u5982\u4e1f\u64f2\u4e00\u76f8\u540c\u786c\u5e63\u6578\u6b21\uff09\uff0c\u4e14\u5df2\u77e5\u9019\u4e9b\u8a66\u9a57\u4e4b\u9593\u7684\u7d50\u679c\u4e92\u76f8\u7368\u7acb\uff08\u5373\u9019\u6b21\u8a66\u9a57\u7684\u7d50\u679c\u4e0d\u5f71\u97ff\u4e0b\u6b21\u8a66\u9a57\u7684\u7d50\u679c\uff09\uff0c\u5247\u7a31\u70ba\u4e8c\u9805\u5be6\u9a57 (binomial experiment)\u3002<\/p>\n

\u8209\u4f8b\u4f86\u8aaa\uff0c\u672c\u5b63\u7f70\u7403\u547d\u4e2d\u7387\u7d04\u70ba\u4e5d\u6210\u7684 NBA \u7403\u661f Stephen Curry \u9023\u7f70 \\(3\\) \u7403\uff08\u5c0d\u904b\u52d5\u54e1\u4f86\u8aaa\uff0c\u9023\u7f70 \\(3\\) \u7403\u4e26\u4e0d\u6703\u6709\u9ad4\u529b\u4e0a\u7684\u8017\u640d\uff09\u5373\u662f\u4e00\u500b\u5be6\u9a57\u6b21\u6578\u70ba \\(3\\)\uff0c\u6210\u529f\u6a5f\u7387\u70ba \\(0.9\\) \u7684\u4e8c\u9805\u5be6\u9a57\u3002\u82e5\u6211\u5011\u89c0\u5bdf\u9019\u500b\u4e8c\u9805\u5be6\u9a57\u4e2d\u6210\u529f\u7684\u6b21\u6578\uff0c\u5247\u7a31\u6b64\u6210\u529f\u6b21\u6578\u70ba\u4e00\u500b\u7b26\u5408\u4e8c\u9805\u5206\u914d (binomial distribution) \u7684\u96a8\u6a5f\u8b8a\u6578\uff0c\u4ee5\u201c\\(+\\)\u201d\u4ee3\u8868\u9032\u7403\uff0c\u201c\\(-\\)\u201d\u8868\u793a\u672a\u9032\u7403\uff0c\u5176\u53ef\u80fd\u7684\u7d50\u679c\u4ee5\u96a8\u6a5f\u8b8a\u6578\u53ca\u5176\u767c\u751f\u7684\u6a5f\u7387\u4ee5\u8868\u4e00\u6240\u793a\uff1a<\/p>\n

\u8868\u4e00\u3001Stephen Curry \u9023\u7f70 3 \u7403\u7684\u7d50\u679c\u3001\u96a8\u6a5f\u8b8a\u6578\u53ca\u5176\u767c\u751f\u7684\u6a5f\u7387\u3002<\/p>\n\n\n\n\n\n\n\n
\u7d50\u679c S<\/td>\n\u96a8\u6a5f\u8b8a\u6578 X\uff1a\u4e09\u6b21\u7f70\u7403\u7684\u7e3d\u9032\u7403\u6578 x<\/td>\n\u6a5f\u7387 f(x)<\/td>\n<\/tr>\n
\uff0b\uff0b\uff0b<\/td>\nx = 3<\/td>\n\\(0.9^3\\)<\/td>\n<\/tr>\n
\uff0b\uff0b\uff0d
\n\uff0b\uff0d\uff0b
\n\uff0d\uff0b\uff0b<\/td>\n		x = 2<\/td>\n\\(C^3_20.9^2(1-0.9)^1\\)<\/td>\n<\/tr>\n
\uff0b\uff0d\uff0d
\n\uff0d\uff0b\uff0d
\n\uff0d\uff0d\uff0b<\/td>\n		\u00a0x = 1<\/td>\n\\(C^3_10.9^1(1-0.9)^2\\)<\/td>\n<\/tr>\n
\uff0d\uff0d\uff0d<\/td>\nx = 0<\/td>\n\\(0.1^3\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u8868\u4e00\u4e2d\u7b2c\u4e00\u6b04\u7684\u7d50\u679c\u70ba\u96a8\u6a5f\u5be6\u9a57\u7684\u6a23\u672c\u7a7a\u9593\uff08\u5b9a\u7fa9\u70ba \\(S\\)\uff09\uff0c\u800c\u7b2c\u4e8c\u6b04\u96a8\u6a5f\u8b8a\u6578\u7684\u7d50\u679c\u5247\u662f\u900f\u904e\u4e00\u500b\u96a8\u6a5f\u8b8a\u6578\uff08\u5b9a\u7fa9\u70ba \\(X\\)\uff09\uff0c\u5c07\u6a23\u672c\u7a7a\u9593\u7684\u7d50\u679c\u8f49\u63db\u7531\u5be6\u6578 \\(\\{x: 0,~1,~2,~3\\}\\)\u3002\u7b2c\u4e09\u6b04\u8868\u8ff0\u96a8\u6a5f\u8b8a\u6578\u7684\u7d50\u679c\u4e4b\u767c\u751f\u6a5f\u7387\uff08\u8a18\u70ba \\(f(x)\\)\uff0c\\(f\\) \u70ba\u6a5f\u7387\u51fd\u6578\uff09\u3002\u89c0\u5bdf\u9032\u7403\u6578 \\(x\\) \u8207\u767c\u751f\u6a5f\u7387 \\(f(x)\\) \u7684\u95dc\u4fc2\uff0c\u6211\u5011\u53ef\u5c07\u4e0d\u540c\u9032\u7403\u6578\u7684\u6a5f\u7387\uff0c\u4ee5\u4e00\u500b\u51fd\u6578\u8868\u9054\uff1a<\/p>\n

\\(f(x)=C^3_x0.9^x(1-0.9)^{3-x},~~~~~~x=0,~1,~2,~3\\)<\/p>\n

\u6b64\u6642\u6211\u5011\u53ef\u4ee5\u8aaa\u96a8\u6a5f\u8b8a\u6578 \\(X\\) \u5c6c\u65bc\u300c\u5be6\u9a57\u6b21\u6578 \\(n = 3\\)\uff0c\u6210\u529f\u6a5f\u7387 \\(p = 0.9\\) \u7684\u4e8c\u9805\u5206\u914d\u300d\uff0c\u8a18\u70ba<\/p>\n

\\(X\\sim B(3,0.9)\\)<\/p>\n

\u82e5\u63a8\u5ee3\u81f3\u5be6\u9a57\u6b21\u6578\u70ba \\(n\\)\uff0c\u6210\u529f\u6a5f\u7387\u70ba \\(p\\)\uff0c\u5247\u6a5f\u7387\u5206\u914d\u70ba<\/p>\n

\u00a0\\(f(x)=C^n_xp^x(1-p)^{n-x},~~~~~~x=0,~1,…,~n\\)<\/p>\n

\u82e5\u6709\u4e00\u96a8\u6a5f\u8b8a\u6578 \\(X\\) \u5c6c\u65bc\u6b64\u5206\u914d\uff0c\u5247\u8a18\u70ba<\/p>\n

\\(X\\sim B(n,p)\\)<\/p>\n

\u63a5\u8457\uff0c\u6211\u5011\u8981\u8a0e\u8ad6\u4e8c\u9805\u5206\u914d\u7684\u5e73\u5747\u6578\uff08\u4ee3\u8868\u5206\u914d\u7684\u96c6\u4e2d\u8da8\u52e2\uff09\u8207\u8b8a\u7570\u6578\uff08\u4ee3\u8868\u5206\u914d\u7684\u96e2\u6563\u7a0b\u5ea6\uff09\u3002\u82e5\u6709\u4e00\u96a8\u6a5f\u8b8a\u6578 \\(X\\sim B(n,p)\\)\uff0c\u6211\u5011\u53ef\u5c07 \\(X\\) \u8996\u70ba \\(n\\) \u500b\u6210\u529f\u6a5f\u7387\u70ba \\(p\\)\uff0c\u4e14\u4e92\u76f8\u7368\u7acb\u7684\u767d\u52aa\u529b\u5206\u914d \\((X_i\\sim B(1,p))\\) \u7684\u7e3d\u548c\u3002<\/p>\n

\\(X=X_1+X_2+…+X_n\\)<\/p>\n

\u900f\u904e\u671f\u671b\u503c\u7684\u904b\u7b97\uff0c\u53ef\u4ee5\u5f97\u77e5\u5176\u4e2d\u4efb\u4f55\u4e00\u500b\u4f2f\u52aa\u529b\u5206\u914d\u7684\u5e73\u5747\u6578\u70ba<\/p>\n

\\(E(X_i)=\\sum\\limits^1_{x=0}xp^x(1-p)^{1-x}=p\\)<\/p>\n

\u8b8a\u7570\u6578\u70ba<\/p>\n

\\(Var(X_i)=E(X^2_i)-[E(X)]^2=\\sum\\limits^1_{x=0}x^2p^2(1-p)^{1-x}-p^2=p(1-p)\\)<\/p>\n

\u7531\u65bc\u539f\u96a8\u6a5f\u8b8a\u6578 \\(X\\) \u4e2d\u7684\u6240\u6709 \\(X_i\\) \u5747\u70ba\u4e92\u76f8\u7368\u7acb\u7684\u96a8\u6a5f\u8b8a\u6578\uff0c\u57fa\u65bc\u671f\u671b\u503c\u8207\u8b8a\u7570\u6578\u7684\u53ef\u52a0\u6027\uff0c\u6211\u5011\u53ef\u77e5\u539f\u96a8\u6a5f\u8b8a\u6578 \\(X\\) \u7684\u5e73\u5747\u503c\u8207\u8b8a\u7570\u6578\u70ba<\/p>\n

\\(E(X)=E(X_1+X_2+…+X_n)=E(X_1)+E(X_2)+…+E(X_n)=np\\)<\/p>\n

\\(Var(X)=Var(X_1+X_2+…+X_n)=Var(X_1)+Var(X_2)+…+Var(X_n)=np(1-p)\\)<\/p>\n

\u4e8c\u9805\u5206\u914d\u5728\u4e0d\u540c\u5be6\u9a57\u6b21\u6578\uff0c\u53ca\u4e0d\u540c\u6210\u529f\u6a5f\u7387\u7684\u5206\u914d\u5716\u5f62\u5982\u5716\u4e00\u548c\u5716\u4e8c\u3002<\/p>\n

\u53e6\u5916\uff0c\u4e8c\u9805\u5206\u914d\u4e4b\u6240\u4ee5\u88ab\u7a31\u70ba\u4e8c\u9805\u5206\u914d\uff0c\u662f\u56e0\u70ba\u5206\u914d\u7684\u6a5f\u7387\u503c\u662f\u4e8c\u9805\u5f0f\u5b9a\u7406\u7684\u4e8c\u9805\u4fc2\u6578\u3002\u4e8c\u9805\u5f0f\u5b9a\u7406\u8207\u4e8c\u9805\u5206\u914d\u4e4b\u6a5f\u7387\u7e3d\u548c\uff1a<\/p>\n

\u4e8c\u9805\u5f0f\u5b9a\u7406\uff1a\\(\\sum\\limits^n_{x=0}C^n_xa^xb^{n-x}=(a+b)^n\\)<\/p>\n

\u4e8c\u9805\u5206\u914d\u4e4b\u6a5f\u7387\u7e3d\u548c\uff1a\\(\\sum\\limits^n_{x=0}f(x)=C^n_xp^x(1-p)^{n-x}=[p+(1-p)]^n=1\\)<\/p>\n

\"73317_p1\"<\/a>

\u5716\u4e00\u3001\u4e8c\u9805\u5206\u914d\u5728\u4e0d\u540c\u6210\u529f\u6a5f\u7387\u4e0b\u7684\u5206\u914d\u5716\u5f62\u3002\uff08\u672c\u6587\u4f5c\u8005\u675c\u67cf\u6bc5\u7e6a\uff09<\/p><\/div>\n

\"73317_p2\"<\/a>

\u5716\u4e8c\u3001\u4e8c\u9805\u5206\u914d\u5728\u4e0d\u540c\u5be6\u9a57\u6b21\u6578\u4e0b\u5206\u914d\u5716\u5f62\u3002\uff08\u672c\u6587\u4f5c\u8005\u675c\u67cf\u6bc5\u7e6a\uff09<\/p><\/div>\n

\n

\u53c3\u8003\u6587\u737b<\/strong><\/p>\n

    \n
  1. Hogg, R. V., & Craig, A. T. (1970). introduction to mathematical statistics. 7th Ed. p.11<\/li>\n
  2. Hogg, R. V., Tanis, E., & Zimmerman, D. (2014).\u00a0Probability and statistical inference<\/em>. 9th Ed. p.49<\/li>\n<\/ol>\n<\/div>","protected":false},"excerpt":{"rendered":"

    \u4f2f\u52aa\u529b\u8a66\u9a57\u8207\u4e8c\u9805\u5206\u5e03 (Bernoulli Trial and Binomial Distribution) …<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[111,230,229],"tags":[10417,10415,10414,3911,10416,3508],"class_list":["post-73317","post","type-post","status-publish","format-standard","hentry","category-mathematics00","category-math06-01","category-math06","tag-probability-distribution","tag-probability-function","tag-random-variable","tag-3911","tag-10416","tag-3508","loop-entry","cat-111","cat-230","cat-229","no-thumbnail"],"views":40869,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/73317","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=73317"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/73317\/revisions"}],"predecessor-version":[{"id":86042,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/73317\/revisions\/86042"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=73317"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=73317"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=73317"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}