{"id":6467,"date":"2022-05-23T13:36:44","date_gmt":"2022-05-23T04:36:44","guid":{"rendered":"https:\/\/hatsudy.com\/jp\/?p=6467"},"modified":"2022-05-26T04:34:17","modified_gmt":"2022-05-25T19:34:17","slug":"negative-binomial-distribution","status":"publish","type":"post","link":"https:\/\/hatsudy.com\/jp\/negative-binomial-distribution.html","title":{"rendered":"\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\uff1a\u78ba\u7387\u95a2\u6570\u306e\u516c\u5f0f\u306e\u610f\u5473\u3068\u671f\u5f85\u5024\uff08\u5e73\u5747\uff09\u30fb\u5206\u6563\u306e\u8a08\u7b97\u65b9\u6cd5"},"content":{"rendered":"\n<p>\u4e8c\u8005\u629e\u4e00\u306e\u7d50\u679c\u306b\u306a\u308b\u30b1\u30fc\u30b9\u306f\u591a\u304f\u3001\u305d\u3046\u3057\u305f\u5834\u5408\u306b\u5229\u7528\u3055\u308c\u308b\u306e\u304c\u4e8c\u9805\u5206\u5e03\u3067\u3059\u3002\u4e00\u65b9\u3001\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u3068\u3044\u3046\u78ba\u7387\u5206\u5e03\u304c\u5b58\u5728\u3057\u307e\u3059\u3002<\/p>\n<p>\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u3067\u306f\u3001\u5931\u6557\u56de\u6570\\(x\\)\u3092\u5229\u7528\u3057\u307e\u3059\u3002\u4e8c\u9805\u5206\u5e03\u3067\u306f\u6210\u529f\u56de\u6570\u306e\u307f\u3092\u5229\u7528\u3057\u3066\u78ba\u7387\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002\u305d\u308c\u306b\u5bfe\u3057\u3066\u3001\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u3067\u306f\u6210\u529f\u56de\u6570\u3060\u3051\u3067\u306a\u304f\u3001\u5931\u6557\u56de\u6570\u3092\u8003\u616e\u3057\u3066\u78ba\u7387\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u305d\u308c\u3067\u306f\u3001\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u306e\u78ba\u7387\u95a2\u6570\u306f\u3069\u306e\u3088\u3046\u306b\u306a\u3063\u3066\u3044\u308b\u306e\u3067\u3057\u3087\u3046\u304b\u3002\u307e\u305f\u3001\u671f\u5f85\u5024\uff08\u5e73\u5747\uff09\u3084\u5206\u6563\u306f\u3069\u306e\u3088\u3046\u306b\u8a08\u7b97\u3059\u308c\u3070\u3044\u3044\u306e\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n<p>\u4e8c\u9805\u5206\u5e03\u3068\u975e\u5e38\u306b\u4f3c\u3066\u3044\u308b\u3082\u306e\u306e\u3001\u65b0\u305f\u306a\u6982\u5ff5\u3068\u306a\u308b\u306e\u304c\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u3067\u3059\u3002\u305d\u3053\u3067\u3001\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u306e\u7279\u5fb4\u3084\u516c\u5f0f\u306e\u610f\u5473\u3001\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u8a08\u7b97\u65b9\u6cd5\u3092\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n<h2>\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u3068\u306f\u4f55\u304b\uff1f\u96e2\u6563\u578b\u78ba\u7387\u5206\u5e03\u3067\u306e\u78ba\u7387\u5206\u5e03<\/h2>\n<p>\u7d50\u679c\u304c\u4e8c\u3064\u3060\u3051\u306e\u8a66\u884c\u3092\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3068\u3044\u3044\u307e\u3059\u3002\u30b3\u30a4\u30f3\u3092\u6295\u3052\u308b\u3068\u304d\u3001\u7d50\u679c\u306f\u8868\u307e\u305f\u306f\u88cf\u3067\u3059\u3002\u307e\u305f\u30b5\u30a4\u30b3\u30ed\u3092\u6295\u3052\u308b\u3068\u304d\u3001\u7d50\u679c\u306f1\u306e\u76ee\u307e\u305f\u306f\u305d\u308c\u4ee5\u5916\u3067\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u65bd\u884c\u304c\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3067\u3059\u3002<\/p>\n<p>\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u4f55\u5ea6\u3082\u884c\u3046\u3053\u3068\u306b\u3088\u3063\u3066\u4e8c\u9805\u5206\u5e03\u3092\u5f97\u3089\u308c\u307e\u3059\u3002\u540c\u69d8\u306b\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u4f55\u5ea6\u3082\u884c\u3046\u3053\u3068\u306b\u3088\u308a\u3001\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u3092\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n<p>\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u3067\u306f\u3001\u6210\u529f\u78ba\u7387\\(p\\)\u306e\u30a4\u30d9\u30f3\u30c8\u306b\u3064\u3044\u3066\u3001\\(k\\)\u56de\u6210\u529f\u3059\u308b\u307e\u3067\u306b\\(x\\)\u56de\u306e\u5931\u6557\u3092\u3059\u308b\u3068\u304d\u306e\u78ba\u7387\u5206\u5e03\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u884c\u3046\u305f\u3081\u3001\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u306f\u96e2\u6563\u578b\u78ba\u7387\u5206\u5e03\u306b\u5206\u985e\u3055\u308c\u307e\u3059\u3002<span style=\"color: #0000ff;\">\u4e8c\u9805\u5206\u5e03\u3068\u307b\u3068\u3093\u3069\u540c\u3058\u3067\u3059\u304c\u3001\u5931\u6557\u56de\u6570\u304c\u542b\u307e\u308c\u308b\u306e\u304c\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u306e\u7279\u5fb4\u3067\u3059\u3002<\/span><\/p>\n<h3>\u4e00\u822c\u7684\u306a\u4e8c\u9805\u5206\u5e03\u3068\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u306e\u9055\u3044<\/h3>\n<p>\u305d\u308c\u3067\u306f\u3001\u5177\u4f53\u7684\u306b\u4e8c\u9805\u5206\u5e03\u3068\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u3067\u3069\u306e\u3088\u3046\u306a\u9055\u3044\u304c\u3042\u308b\u306e\u304b\u306b\u3064\u3044\u3066\u3001\u3088\u308a\u8a73\u7d30\u306b\u78ba\u8a8d\u3057\u3066\u3044\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\u30b5\u30a4\u30b3\u30ed\u3092\u6295\u3052\u308b\u3068\u304d\u30011\u306e\u76ee\u304c\u51fa\u305f\u3089\u6210\u529f\u30011\u306e\u76ee\u4ee5\u5916\u304c\u51fa\u305f\u3089\u5931\u6557\u3068\u3057\u307e\u3059\u30021\u306e\u76ee\u304c\u51fa\u308b\u78ba\u7387\u306f\\(\\displaystyle\\frac{1}{6}\\)\u3067\u3059\u3002\u307e\u305f\u30011\u306e\u76ee\u4ee5\u5916\u304c\u51fa\u308b\u78ba\u7387\u306f\\(\\displaystyle\\frac{5}{6}\\)\u3067\u3059\u3002\u305d\u308c\u3067\u306f\u3001\u30b5\u30a4\u30b3\u30ed\u30925\u56de\u6295\u3052\u30661\u306e\u76ee\u304c2\u56de\u51fa\u308b\u78ba\u7387\u306f\u3044\u304f\u3089\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n<p>\u4e8c\u9805\u5206\u5e03\u3092\u7406\u89e3\u3057\u3066\u3044\u308b\u5834\u5408\u3001\u78ba\u7387\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\\(_5C_2\\left(\\displaystyle\\frac{1}{6}\\right)^2\\left(\\displaystyle\\frac{5}{6}\\right)^3\u22520.161\\)<\/p>\n<p>\u4e00\u65b9\u3067\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u3067\u306f\u3069\u306e\u3088\u3046\u306a\u8a08\u7b97\u306b\u306a\u308b\u306e\u3067\u3057\u3087\u3046\u304b\u3002\u5148\u307b\u3069\u3068\u540c\u3058\u3088\u3046\u306b\u3001\u30b5\u30a4\u30b3\u30ed\u30925\u56de\u6295\u3052\u308b\u5834\u9762\u3092\u8003\u3048\u307e\u3057\u3087\u3046\u3002\u30b5\u30a4\u30b3\u30ed\u3092\u6295\u3052\u30661\u306e\u76ee\u304c3\u56de\u51fa\u308b\u307e\u3067\u306b\u30011\u306e\u76ee\u4ee5\u5916\u304c2\u56de\u51fa\u308b\u78ba\u7387\u306f\u3044\u304f\u3089\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n<p>\u9ad8\u6821\u6570\u5b66\u3092\u5b66\u3093\u3067\u3044\u308b\u5834\u5408\u3001\u5168\u54e1\u304c\u3053\u306e\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u601d\u3044\u307e\u3059\u3002\u30b5\u30a4\u30b3\u30ed\u3092\u6295\u3052\u30663\u56de\u76ee\u306b1\u306e\u76ee\u304c\u51fa\u308b\u305f\u3081\u306b\u306f\u3001\u4ee5\u4e0b\u306e\u6761\u4ef6\u3067\u306a\u3051\u308c\u3070\u3044\u3051\u307e\u305b\u3093\u3002<\/p>\n<ul>\n<li>\u305d\u308c\u307e\u3067\u306b1\u306e\u76ee\u304c2\u56de\u51fa\u3066\u304a\u308a\u3001\u304b\u30641\u306e\u76ee\u4ee5\u5916\u304c2\u56de\u51fa\u3066\u304a\u308a\u30015\u56de\u76ee\u306b1\u306e\u76ee\u304c\u51fa\u308b\u3002<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/hatsudy.com\/jp\/wp-content\/uploads\/2022\/05\/kgje5-1.jpg\" alt=\"\" width=\"500\" height=\"130\" class=\"aligncenter size-full wp-image-6518\" \/><\/p>\n<p>\u305d\u306e\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u8a08\u7b97\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\\(_4C_2\\left(\\displaystyle\\frac{1}{6}\\right)^2\\left(\\displaystyle\\frac{5}{6}\\right)^2\u00d7\\displaystyle\\frac{1}{6}\u22520.0193\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\\(k\\)\u56de\u6210\u529f\uff081\u306e\u76ee\u304c\u51fa\u308b\uff09\u307e\u3067\u306b\u3001\\(x\\)\u56de\u5931\u6557\u3059\u308b\uff081\u306e\u76ee\u4ee5\u5916\u304c\u51fa\u308b\uff09\u78ba\u7387\u3092\u51fa\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n<h2>\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u306e\u78ba\u7387\u95a2\u6570<\/h2>\n<p>\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u3068\u3044\u3046\u306e\u306f\u3001\u6982\u5ff5\u306f\u96e3\u3057\u304f\u3042\u308a\u307e\u305b\u3093\u3002\\(k\\)\u56de\u306e\u6210\u529f\u3059\u308b\u305f\u3081\u306b\u306f\u3001\u305d\u308c\u307e\u3067\u306b\\(k-1\\)\u56de\u306e\u6210\u529f\u3092\u3057\u3066\u3044\u306a\u3051\u308c\u3070\u3044\u3051\u307e\u305b\u3093\u3002\u307e\u305f\\(k\\)\u56de\u306e\u6210\u529f\u3092\u3059\u308b\u307e\u3067\u306b\\(x\\)\u56de\u306e\u5931\u6557\u3092\u3059\u308b\u5834\u5408\u3001\\(k\\)\u56de\u76ee\u306e\u6210\u529f\u306e\u4e00\u3064\u524d\u306b\u5408\u8a08\u3067\\(k+x-1\\)\u306e\u8a66\u884c\u3092\u3059\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002\u305d\u3057\u3066\u3001\u6700\u5f8c\uff08\\(k\\)\u56de\u76ee\uff09\u3067\u6210\u529f\u3057\u307e\u3059\u3002<\/p>\n<p>\u3064\u307e\u308a<span style=\"color: #ff0000;\">\\(k+x-1\\)\u56de\u306e\u8a66\u884c\u306e\u4e2d\u304b\u3089\u3001\\(x\\)\u56de\u306e\u5931\u6557\u3092\u9078\u3076\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/span>\u30b5\u30a4\u30b3\u30ed\u306e\u4f8b\u3067\u3042\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/hatsudy.com\/jp\/wp-content\/uploads\/2022\/05\/kgje6.jpg\" alt=\"\" width=\"550\" height=\"134\" class=\"aligncenter size-full wp-image-6520\" \/><\/p>\n<p>\u305d\u306e\u305f\u3081\u6210\u529f\u78ba\u7387\u304c\\(p\\)\u306e\u5834\u5408\u3001\u4ee5\u4e0b\u306e\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\\(_{k+x-1}C_xp^{k-1}(1-p)^x\u00d7p\\)<\/p>\n<p>\\(=_{k+x-1}C_xp^k(1-p)^x\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u306e\u78ba\u7387\u95a2\u6570\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002\u9ad8\u6821\u6570\u5b66\u3067\u5b66\u3076\u78ba\u7387\u306e\u521d\u6b69\u3092\u7406\u89e3\u3057\u3066\u3044\u308c\u3070\u3001\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u306e\u78ba\u7387\u95a2\u6570\u3092\u51fa\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\u7d71\u8a08\u5b66\u306e\u6559\u79d1\u66f8\u3067\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u3092\u5b66\u3076\u5834\u5408\u3001\u6700\u521d\u306b\u516c\u5f0f\u304c\u51fa\u3055\u308c\u307e\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u307b\u3068\u3093\u3069\u306e\u4eba\u306f\u7406\u89e3\u3067\u304d\u307e\u305b\u3093\u3002\u305d\u3053\u3067\u3001\u516c\u5f0f\u304c\u3082\u3064\u610f\u5473\u3092\u5b66\u3073\u307e\u3057\u3087\u3046\u3002\\(k\\)\u56de\u6210\u529f\u3059\u308b\u307e\u3067\u306b\\(k-1\\)\u56de\u3068\\(x\\)\u56de\u306e\u8a66\u884c\u3092\u3057\u3066\u3044\u308b\u3053\u3068\u3092\u7406\u89e3\u3059\u308c\u3070\u3001\u306a\u305c\u5148\u307b\u3069\u63d0\u793a\u3057\u305f\u516c\u5f0f\u3092\u5f97\u3089\u308c\u308b\u306e\u304b\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<h3>\u671f\u5f85\u5024\uff08\u5e73\u5747\uff09\u3092\u5f97\u308b\u516c\u5f0f<\/h3>\n<p>\u6b21\u306b\u3001\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u3067\u671f\u5f85\u5024\uff08\u5e73\u5747\uff09\u3092\u5f97\u308b\u516c\u5f0f\u3092\u5b66\u3073\u307e\u3057\u3087\u3046\u3002\u3069\u306e\u3088\u3046\u306b\u3057\u3066\u671f\u5f85\u5024\u306e\u8a08\u7b97\u3092\u3059\u308c\u3070\u3044\u3044\u306e\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n<p>\u6210\u529f\u78ba\u7387\u304c\\(p\\)\u3067\u3042\u308a\u3001\u6210\u529f\u56de\u6570\u304c\u5408\u8a08\u3067\\(k\\)\u56de\u306e\u5834\u5408\u3001\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u306e\u671f\u5f85\u5024\\(E(X)\\)\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(E(X)=\\displaystyle\\frac{k(1-p)}{p}\\)<\/li>\n<\/ul>\n<p>\u4f8b\u3048\u3070\u30b5\u30a4\u30b3\u30ed\u3092\u6295\u3052\u3066\u6210\u529f\u78ba\u7387\u304c\\(\\displaystyle\\frac{1}{6}\\)\u3067\u3042\u308a\u3001\u5408\u8a08\u30673\u56de\u6210\u529f\u3059\u308b\u5834\u5408\u3001\u671f\u5f85\u5024\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\\(\\displaystyle\\frac{3\\left(1-\\displaystyle\\frac{1}{6}\\right)}{\\displaystyle\\frac{1}{6}}=15\\)<\/p>\n<p>\u3053\u306e\u3088\u3046\u306b\u3001<span style=\"color: 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