{"id":47134,"date":"2013-10-25T08:57:59","date_gmt":"2013-10-25T00:57:59","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=47134"},"modified":"2021-10-06T16:19:45","modified_gmt":"2021-10-06T08:19:45","slug":"%e7%8f%ad%e4%bd%9b%e5%ae%9a%e5%be%8b-benfords-law","status":"publish","type":"post","link":"http:\/\/localhost\/%e7%8f%ad%e4%bd%9b%e5%ae%9a%e5%be%8b-benfords-law\/","title":{"rendered":"\u73ed\u4f5b\u5b9a\u5f8b"},"content":{"rendered":"

\u73ed\u4f5b\u5b9a\u5f8b (Benford’s Law)<\/span>
\n\u570b\u7acb\u5c4f\u6771\u9ad8\u7d1a\u4e2d\u5b78\u6578\u5b78\u79d1\u694a\u74ca\u8339\u8001\u5e2b<\/span><\/strong><\/strong><\/p>\n

\u73fe\u884c\u7684\u9ad8\u4e2d\u6559\u79d1\u66f8\u4e2d\uff0c\u6709\u500b\u975e\u5e38\u6709\u610f\u601d\u7684\u984c\u76ee\uff1a<\/p>\n

\u5be9\u8a08\u5de5\u4f5c\u8005\u6703\u4f7f\u7528\u73ed\u4f5b<\/span>\u6cd5\u5247\u4f86\u67e5\u5e33\uff0c\u73ed\u4f5b<\/span>\u6cd5\u5247\u662f\uff1a\u300c\u9280\u884c\u5b58\u6b3e\u7684\u6700\u9ad8\u4f4d\u6578\u5b57\u662f $$a$$\u00a0<\/i>\u8005\u7684\u6bd4\u4f8b\u7d04\u70ba $$\\log(1+\\frac{1}{a})$$\u300d\ufe52\u6839\u64da\u73ed\u4f5b<\/span>\u6cd5\u5247\ufe50\u9280\u884c\u5b58\u6b3e\u7684\u6700\u9ad8\u4f4d\u6578\u5b57\u662f $$4,5,6$$ \u6216 $$7$$ \u8005\u7684\u6bd4\u4f8b\u7d04\u6709
\n$$(1)~20\\%$$ \u00a0 $$(2)~30\\%$$ \u00a0 $$(3)~40\\%$$ \u00a0 $$(4)~50\\%$$ \u00a0 $$(5)~60\\%$$ .<\/p>\n

\u9019\u500b\u984c\u76ee\u7531\u7c21\u55ae\u7684\u5c0d\u6578\u904b\u7b97\u6027\u8cea\uff0c\u5982\u4e0b\u5217\u7b97\u5f0f\u5f97\u5230\u7b54\u6848 $$(2)~30\\%$$\u3002<\/p>\n

$$\\log \\left( {1 + \\frac{1}{4}} \\right) + \\log \\left( {1 + \\frac{1}{5}} \\right) + \\log \\left( {1 + \\frac{1}{6}} \\right) + \\log \\left( {1 + \\frac{1}{7}} \\right) \\\\= \\log \\left( {\\frac{5}{4} \\times \\frac{6}{5} \\times \\frac{7}{6} \\times \\frac{8}{7}} \\right)=\\log 2\\approx 0.301$$<\/p>\n

\u300c\u73ed\u4f5b\u5b9a\u5f8b\u300d\u53c8\u7a31\u70ba\u9996\u4f4d\u6578\u5b57\u6cd5\u5247 ( First-Digit Law )\u3002\u7136\u800c\uff0c\u70ba\u4ec0\u9ebc\u8aaa\u5b83\u6709\u8da3\u5462\uff1f\u9996\u5148\uff0c\u6211\u5011\u7531 $$1$$ \u5230 $$9$$ \u6309\u7167\u73ed\u4f5b<\/span>\u6cd5\u5247\u9010\u4e00\u7b97\u51fa\u6bcf\u500b\u6700\u9ad8\u4f4d\u6578\u5b57\u7684\u6bd4\u4f8b\uff1a<\/p>\n

$$\\begin{array}{lll}\\log \\left( {1 + \\frac{1}{1}} \\right) \\approx 0.301030&\\log \\left( {1 + \\frac{1}{4}} \\right) \\approx 0.096910& \\log \\left( {1 + \\frac{1}{7}} \\right) \\approx 0.057992\\\\ \\log \\left( {1 + \\frac{1}{2}} \\right) \\approx 0.176091&\\log \\left( {1 + \\frac{1}{5}} \\right) \\approx 0.079181&\\log \\left( {1 + \\frac{1}{8}} \\right) \\approx 0.051153\\\\\\log \\left( {1 + \\frac{1}{3}} \\right) \\approx 0.124939&\\log \\left( {1 + \\frac{1}{6}} \\right) \\approx 0.066947&\\log \\left( {1 + \\frac{1}{{\\rm{9}}}} \\right) \\approx 0.045757\\end{array}$$<\/p>\n

\u76f4\u89ba\u4e0a\u6211\u5011\u4ee5\u70ba\u6bcf\u500b\u6700\u9ad8\u4f4d\u6578\u5b57\u7684\u6bd4\u5217\u662f\u76f8\u540c\u7684\uff0c\u4f46\u4e8b\u5be6\u4e0a\uff0c\u6700\u9ad8\u4f4d\u6578\u5b57\u7684\u6bd4\u4f8b\u5206\u5e03\u4e26\u4e0d\u5747\u52fb\uff0c\u6700\u9ad8\u4f4d\u6578\u5b57 $$1$$ \u51fa\u73fe\u7684\u6bd4\u4f8b\u5927\u7d04\u662f $$30\\%$$\uff0c\u6578\u5b57 $$2$$ \u6216 $$3$$ \u7684\u6bd4\u4f8b\u7e3d\u5171\u7d04 $$30\\%$$\uff0c\u6578\u5b57 $$4, 5, 6, 7, 8$$ \u6216 $$9$$ \u7684\u6bd4\u4f8b\u7e3d\u5171\u7d04\u662f $$40\\%$$ \uff0c\u9019\u500b\u73fe\u8c61\u548c\u76f4\u89ba\u4e0d\u76f8\u7b26\u5408\uff0c\u6bcf\u500b\u6578\u5b57\u51fa\u73fe\u7684\u6a5f\u7387\u4e26\u4e0d\u4e00\u6a23\u3002\u6700\u9ad8\u4f4d\u6578\u5b57\u6108\u5c0f\uff0c\u51fa\u73fe\u7684\u6a5f\u7387\u6108\u5927\uff01\u5e95\u4e0b\uff0c\u6211\u5011\u7528\u9577\u689d\u5716\u8868\u793a\uff0c\u4e00\u76ee\u77ad\u7136\u52dd\u904e\u5343\u8a00\u842c\u8a9e\u3002<\/p>\n

\"47134_p01\"<\/a><\/p>\n

\u5728\u771f\u5be6\u4e16\u754c\u4e2d\uff0c\u5f88\u591a\u985e\u578b\u7684\u6578\u64da\u800c\u4e14\u662f\u81ea\u7136\u7522\u751f\u7684\u6578\u64da\u901a\u5e38\u90fd\u9075\u5b88\u300c\u73ed\u4f5b\u5b9a\u5f8b\u300d\uff0c\u5c24\u5176\u662f\u8ca1\u52d9\u6578\u64da\u3002\u56e0\u6b64\uff0c\u53ef\u4f7f\u7528\u6b64\u4e00\u81ea\u7136\u6cd5\u5247\u6aa2\u9a57\u8ca1\u52d9\u8cc7\u6599\u7684\u53ef\u4fe1\u5ea6\uff0c\u751a\u81f3\u7cfe\u8209\u507d\u9020\u7684\u5047\u5e33\u3001\u63ed\u767c\u8a50\u6b3a\u884c\u70ba\u3002<\/p>\n

\u300c\u73ed\u4f5b\u5b9a\u5f8b\u300d\u6709\u500b\u8457\u540d\u7684\u61c9\u7528\u5be6\u4f8b\uff0c\u6b64\u6848\u582a\u7a31\u7f8e\u570b\u83ef\u76db\u9813\u5dde\u53f2\u4e0a\u6700\u5927\u6295\u8cc7\u8a50\u6b3a\u6848\uff0c\u5176\u638f\u7a7a\u91d1\u984d\u9ad8\u9054\u4e00\u5104\u7f8e\u5143\uff0c\u53d7\u6ce2\u53ca\u7684\u6295\u8cc7\u4eba\u591a\u9054 $$5000$$ \u4eba\u3002\u8a50\u9a19\u4e3b\u8b00\u51f1\u6587\u2027\u52de\u502b\u65af( Kevin Lawrence )\u548c\u5176\u540c\u5925\u5011\u5411\u6295\u8cc7\u4eba\u7c4c\u8cc7\uff0c\u5275\u8fa6\u4e00\u5bb6\u5065\u8eab\u4ff1\u6a02\u90e8\uff0c\u4e0d\u904e\u4ed6\u5011\u537b\u5c07\u8cc7\u91d1\u82b1\u7528\u5728\u79c1\u4eba\u4eab\u6a02\u4e0a\uff0c\u70ba\u4e86\u63a9\u98fe\u632a\u7528\u516c\u6b3e\u7684\u4e0d\u6cd5\u884c\u70ba\uff0c\u628a\u8cc7\u91d1\u5728\u6d77\u5916\u7a7a\u6bbc\u7a7a\u53f8\u548c\u9280\u884c\u4e4b\u9593\u8f49\u4f86\u8f49\u53bb\u505a\u5047\u5e33\uff0c\u88fd\u9020\u51fa\u751f\u610f\u8208\u9686\u7684\u5047\u8c61\u3002\u5e78\u597d\uff0c\u9451\u8b58\u6703\u8a08\u5bb6\u9054\u6d1b\u2027\u9438\u745e\u723e( Darrell Dorrell )\u8b58\u7834\u6b64\u6295\u8cc7\u9a19\u5c40\uff0c\u4ed6\u5c07\u4e03\u842c\u591a\u7b46\u652f\u7968\u53ca\u532f\u6b3e\u7684\u76f8\u95dc\u6578\u64da\uff0c\u8207\u300c\u73ed\u4f5b\u5b9a\u5f8b\u300d\u7684\u6700\u9ad8\u4f4d\u6578\u5b57\u5206\u5e03\u6bd4\u5c0d\uff0c\u767c\u73fe\u9019\u4e9b\u6578\u64da\u7121\u6cd5\u901a\u904e\u6aa2\u9a57\u3002\u6700\u5f8c\u8017\u8cbb\u4e09\u5e74\u53f8\u6cd5\u8abf\u67e5\uff0c\u51f1\u6587\u2027\u52de\u502b\u65af\u88ab\u5224\u5211 $$20$$ \u5e74\u5165\u76e3\u3002<\/p>\n

\u53c8\u4f8b\u5982\uff0c\u7f8e\u570b\u570b\u7a05\u5c40\u7814\u7a76\u300c\u73ed\u4f5b\u5b9a\u5f8b\u300d\u4ee5\u63ea\u51fa\u9003\u6f0f\u7a05\u884c\u70ba\uff0c\u751a\u81f3\u4e5f\u6709\u7814\u7a76\u54e1\u7528\u9019\u500b\u5b9a\u5f8b\u6aa2\u9a57\u7f8e\u570b\u7e3d\u7d71\u67ef\u6797\u9813\u7684\u5831\u7a05\u8cc7\u6599\u3002\u56e0\u6b64\uff0c\u300c\u73ed\u4f5b\u5b9a\u5f8b\u300d\u5728\u8ca1\u52d9\u5be9\u8a08\u7684\u9632\u5f0a\u5999\u7528\uff0c\u6709\u500b\u8cbc\u5207\u50b3\u795e\u7684\u5c01\u865f\uff1a\u300c\u5047\u5e33\u524b\u661f\u300d\u3002<\/p>\n

\u300c\u73ed\u4f5b\u5b9a\u5f8b\u300d\u662f\u5982\u4f55\u88ab\u767c\u73fe\u7684\u5462\uff1f 1881 \u5e74\uff0c\u7f8e\u570b\u5929\u6587\u5b78\u5bb6\u7d10\u5eb7( Simon Newcomb )\u89c0\u5bdf\u5230\u5c0d\u6578\u8868\u66f8\u7c4d\u524d\u9762\u7684\u66f8\u9801\u6bd4\u5f8c\u9762\u7684\u7834\u820a\uff0c\u540c\u5115\u79d1\u5b78\u5bb6\u5011\u6240\u8655\u7406\u7684\u6578\u64da\uff0c\u6700\u9ad8\u4f4d\u6578\u5b57 $$1$$ \u7684\u6b21\u6578\u8981\u6bd4\u6578\u5b57 $$2$$ \u4f86\u7684\u591a\uff0c\u4f9d\u6b64\u985e\u63a8\uff0c\u6578\u5b57 $$9$$ \u7684\u66f8\u9801\u662f\u6700\u4e7e\u6de8\u7684\uff0c\u9019\u4e9b\u6578\u64da\u7684\u5448\u73fe\u504f\u5411\u8f03\u5c0f\u6578\u5b57\u7684\u5206\u5e03\u3002<\/p>\n

1938 \u5e74\uff0c\u7f8e\u570b\u7269\u7406\u5b78\u5bb6\u73ed\u4f5b ( Frank Benford )\u4e5f\u6ce8\u610f\u5230\u540c\u6a23\u7684\u4e8b\u60c5\uff0c\u4e26\u4e14\u63d0\u51fa\u767c\u8868\u3002\u76f4\u5230 1995 \u5e74\uff0c\u55ac\u6cbb\u4e9e\u7406\u5de5\u5b78\u9662\u7684\u6578\u5b78\u5bb6\u5e0c\u723e( Ted Hill )\u624d\u7d66\u51fa\u73ed\u4f5b\u5b9a\u5f8b\u7684\u8b49\u660e\u30022012 \u5e74\uff0c\u7d10\u6fa4\u897f\u5927\u5b78\u6703\u8a08\u5b78\u6559\u6388\u5c3c\u683c\u91cc\u5c3c( Mark Nigrini )\u8457\u4f5c\u7684\u5c08\u66f8\u300a\u73ed\u4f5b\u5b9a\u5f8b\u300b\u51fa\u7248\uff0c\u66f8\u4e2d\u8a0e\u8ad6\u73ed\u4f5b\u5b9a\u5f8b\u5728\u6cd5\u52d9\u6703\u8a08\u3001\u5be9\u8a08\u67e5\u5e33\u548c\u8a50\u6b3a\u5075\u67e5\u7684\u61c9\u7528\uff0c\u53e6\u5916\u4ed6\u4e5f\u7814\u767c\u6578\u5b57\u5206\u6790\u7cfb\u7d71\uff0c\u53ef\u5feb\u901f\u6aa2\u67e5\u8cc7\u6599\u5eab\u6578\u64da\u662f\u5426\u9020\u5047\u3002<\/p>\n

\u5047\u7684\u771f\u4e0d\u4e86\uff0c\u771f\u7684\u5047\u4e0d\u4e86\uff01\u8fa8\u8b58\u771f\u5047\u6191\u85c9\u7684\u4e0d\u662f\u5e78\u904b\u4e4b\u795e\u7684\u7737\u9867\uff0c\u800c\u662f\u904b\u7528\u77e5\u8b58\uff0c\u770b\u900f\u8868\u8c61\uff0c\u76f4\u6307\u771f\u76f8\u3002<\/p>\n

\n

\u53c3\u8003\u8cc7\u6599<\/strong><\/p>\n

    \n
  1. \u66fc\u7f85\u8fea\u8afe\u8457\u3001\u80e1\u5b88\u4ec1\u8b6f(2012)\uff0c\u300a\u9189\u6f22\u8d70\u8def-\u6a5f\u7387\u5982\u4f55\u5de6\u53f3\u4f60\u6211\u7684\u547d\u904b\u548c\u6a5f\u6703\u300b\uff0c\u53f0\u5317\uff1a\u5929\u4e0b\u9060\u898b\u51fa\u7248\u793e\u3002<\/li>\n
  2. \u8a31\u5fd7\u8fb2\u4e3b\u7de8(2013) \uff0c\u300a\u666e\u901a\u9ad8\u7d1a\u4e2d\u5b78\u6578\u5b78\u300b\u7b2c\u4e00\u518a\uff0c \u53f0\u5317\uff1a\u9f8d\u9a30\u6587\u5316\u3002<\/li>\n
  3. http:\/\/www.dfi.wa.gov\/sd\/kevin_laurence_sentenced.htm<\/a><\/li>\n
  4. http:\/\/en.wikipedia.org\/wiki\/Benford%27s_law<\/a><\/li>\n<\/ol>\n<\/div>","protected":false},"excerpt":{"rendered":"

    \u73ed\u4f5b\u5b9a\u5f8b (Benford’s Law) \u570b\u7acb\u5c4f\u6771\u9ad8\u7d1a\u4e2d\u5b78\u6578\u5b78\u79d1\u694a\u74ca\u8339\u8001\u5e2b \u73fe\u884c\u7684\u9ad8\u4e2d\u6559\u79d1\u66f8\u4e2d\uff0c\u6709…<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[111],"tags":[7940,426,2384,5672],"class_list":["post-47134","post","type-post","status-publish","format-standard","hentry","category-mathematics00","tag-benford","tag-426","tag-2384","tag-5672","loop-entry","cat-111","no-thumbnail"],"views":15090,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/47134","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=47134"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/47134\/revisions"}],"predecessor-version":[{"id":87350,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/47134\/revisions\/87350"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=47134"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=47134"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=47134"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}