{"id":1713,"date":"2020-07-08T11:00:00","date_gmt":"2020-07-08T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1713"},"modified":"2020-07-19T20:41:49","modified_gmt":"2020-07-19T11:41:49","slug":"binomial-distribution","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/binomial-distribution","title":{"rendered":"\uff12\u9805\u5206\u5e03"},"content":{"rendered":"\n\n\n\n\n<p>\uff12\u9805\u5206\u5e03\uff08binomial distribution\uff09\u306f\u96e2\u6563\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3001\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u8907\u6570\u56de\u884c\u3063\u305f\u969b\u306b\u5f97\u3089\u308c\u308b\u30c7\u30fc\u30bf\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u6210\u529f\u78ba\u7387\\(p\\)\u306e\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\\(n\\)\u56de\u884c\u3063\u305f\u3082\u306e\u3092\uff12\u9805\u5206\u5e03\u3068\u3044\u3044\u3001\u8a18\u53f7\\(X\\sim Bin(n,\\ p)\\)\u3067\u3088\u304f\u8868\u3055\u308c\u307e\u3059\u3002\u3053\u306e\u8a18\u4e8b\u3067\u306f\u3001\uff12\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\u3068\u4e3b\u306a\u6027\u8cea\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<div id=\"toc_container\" class=\"toc_light_blue no_bullets\"><p class=\"toc_title\">\u76ee\u6b21<\/p><ul class=\"toc_list\"><li><a href=\"#i\"><span class=\"toc_number toc_depth_1\">1<\/span> \uff12\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831<\/a><ul><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_2\">1.1<\/span> \u78ba\u7387\u95a2\u6570\u3068\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570<\/a><\/li><\/ul><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_1\">2<\/span> \uff12\u9805\u5206\u5e03\u306e\u30dd\u30a2\u30bd\u30f3\u8fd1\u4f3c<\/a><\/li><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_1\">3<\/span> \uff12\u9805\u5206\u5e03\u3068\u30d9\u30fc\u30bf\u5206\u5e03<\/a><\/li><li><a href=\"#F\"><span class=\"toc_number toc_depth_1\">4<\/span> \uff12\u9805\u5206\u5e03\u3068\\(F\\)\u5206\u5e03<\/a><\/li><li><a href=\"#i-5\"><span class=\"toc_number toc_depth_1\">5<\/span> \uff12\u9805\u5206\u5e03\u306e\u518d\u751f\u6027<\/a><\/li><li><a href=\"#i-6\"><span class=\"toc_number toc_depth_1\">6<\/span> \uff12\u9805\u5206\u5e03\u3068\u8d85\u5e7e\u4f55\u5206\u5e03<\/a><\/li><li><a href=\"#i-7\"><span class=\"toc_number toc_depth_1\">7<\/span> \u30d1\u30e9\u30e1\u30fc\u30bf\u63a8\u5b9a<\/a><\/li><li><a href=\"#i-8\"><span class=\"toc_number toc_depth_1\">8<\/span> \uff12\u9805\u5206\u5e03\u306e\u4e71\u6570\u767a\u751f\u65b9\u6cd5<\/a><\/li><li><a href=\"#i-9\"><span class=\"toc_number toc_depth_1\">9<\/span> \uff12\u9805\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u5206\u5e03<\/a><\/li><\/ul><\/div>\n<h2><span id=\"i\">\uff12\u9805\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831<\/span><\/h2>\n\n\n\n<p>\u203b\u3000\u8868\u306f\u6a2a\u306b\u30b9\u30af\u30ed\u30fc\u30eb\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>\u30d1\u30e9\u30e1\u30fc\u30bf<\/td><td>\\(n,\\ p\\)\uff08\\(p\\)\u306f\u6210\u529f\u78ba\u7387\u3001\\(n\\)\u306f\u8a66\u884c\u56de\u6570\u3092\u8868\u3057\u307e\u3059\u3002\uff09<\/td><\/tr><tr><td>\u78ba\u7387\u5909\u6570\u306e\u7bc4\u56f2<\/td><td>\\(0\\leq x \\leq n\\)\u3000\u3000\\(x\\)\u306f\u6574\u6570\uff08\u6210\u529f\u56de\u6570\u3092\u8868\u3057\u307e\u3059\u3002\uff09<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cumulative-distribution-function\">\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570<\/a><\/td><td>\\(\\displaystyle\\sum_{k=0}^{x}\\  _{n}C_{k}p^{k}(1-p)^{n-k}\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/probability-function\">\u78ba\u7387\u95a2\u6570<\/a><\/td><td>\\(_{n}C_{x}p^{x}(1-p)^{n-x}\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mgf\">\u7a4d\u7387\u6bcd\u95a2\u6570<\/a><\/td><td>\\(\\{ pe^{t}+(1-p) \\}^{n}\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/pgf\">\u78ba\u7387\u6bcd\u95a2\u6570<\/a><\/td><td>\\(\\left\\{pt+ (1-p) \\right\\}^{n}\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cf\">\u7279\u6027\u95a2\u6570<\/a><\/td><td>\\(\\{ pe^{it}+(1-p) \\}^{n}\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cgf\">\u30ad\u30e5\u30e0\u30e9\u30f3\u30c8\u6bcd\u95a2\u6570<\/a><\/td><td>\\(n\\log\\{ pe^{t}+(1-p) \\}\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cf\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/moment\">\u30e2\u30fc\u30e1\u30f3\u30c8\uff08\u539f\u70b9\u307e\u308f\u308a\uff09<\/a><\/a><\/td><td><div style=\"overflow-x: auto;\">\\begin{align}<br>\\ \\ \\ \\ \\ 1\u6b21\uff08\u671f\u5f85\u5024\uff09&amp; np \\\\<br>\\ \\ \\ \\ \\ 2\u6b21\u3000\u3000\u3000\u3000\u3000&amp; np\\{np+(1-p)\\} \\\\<br>\\ \\ \\ \\ \\ 3\u6b21\u3000\u3000\u3000\u3000\u3000&amp; np\\{(n-1)(n-2)p^{2}+3p(n-1)+1\\}<br>\\end{align}<\/div><\/td><\/tr><tr><td>\u6f38\u5316\u5f0f\uff08\u539f\u70b9\u307e\u308f\u308a\u306e<br>\\(k\\)\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8\uff09<\/td><td>\\(\\mu_{k}^{\\prime}\\)\u3092\\(k\\)\u6b21\u306e\u539f\u70b9\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u3068\u3057\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br> \\mu_{1}^{\\prime} &amp;= np \\\\<br> \\mu_{k+1}^{\\prime} &amp;= \\left\\{ n\\mu_{k}^{\\prime}+(1-p)\\cdot\\displaystyle\\frac{d}{dp}\\mu_{k}^{\\prime} \\right\\},\\ \\ \\ k=1,2,\\cdots<br>\\end{align}<\/div><\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/moment\">\u30e2\u30fc\u30e1\u30f3\u30c8\uff08\u5e73\u5747\u307e\u308f\u308a\uff09<\/a><\/td><td><div style=\"overflow-x: auto;\">\\begin{align}<br>2\u6b21\uff08\u5206\u6563\uff09 &amp; np(1-p) \\\\<br>3\u6b21\u3000\u3000\u3000 \u3000&amp; np(1-p)\\{(1-p)-p\\} \\\\<br>4\u6b21\u3000\u3000\u3000 \u3000&amp; np(1-p)\\{1+3p(1-p)(n-2)\\}<br>\\end{align}<\/div><\/td><\/tr><tr><td>\u6f38\u5316\u5f0f\uff08\u5e73\u5747\u307e\u308f\u308a\u306e<br>\\(k\\)\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8\uff09<\/td><td>\\(\\mu_{k}\\)\u3092\\(k\\)\u6b21\u306e\u5e73\u5747\u307e\u308f\u308a\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u3068\u3057\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mu_{0} &amp;= 1 \\\\<br>\\mu_{1} &amp;= 0 \\\\<br>\\mu_{k+1} &amp;= p(1-p)\\left( nk\\mu_{k-1}+\\displaystyle\\frac{d}{dp}\\mu_{k} \\right),\\ \\ \\ k=1,2,\\cdots<br>\\end{align}<\/div><\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/moment\">\u968e\u4e57\u30e2\u30fc\u30e1\u30f3\u30c8<\/a><\/td><td>\\(<br>\\mu_{[k]} = \\left\\{<br>\\begin{array}{c}<br>_{n}P_{k}\\ p^{k},\\ \\ \\ k\\leq n \\\\<br>0,\\ \\ \\ k>0<br>\\end{array}\\right.<br>\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mean-definition\">\u671f\u5f85\u5024<\/a><\/td><td>\\( \\displaystyle np \\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/variance\">\u5206\u6563<\/a><\/td><td>\\(\\displaystyle np(1-p) \\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/skewness\">\u6b6a\u5ea6<\/a><\/td><td>\\(\\displaystyle \\frac{(1-p)-p}{\\sqrt{np(1-p)}}\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/kurtosis\">\u5c16\u5ea6<\/a><\/td><td>\\( 3+\\displaystyle\\frac{1-6p(1-p)}{np(1-p)} \\)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h3>\u8a3c\u660e\u4e00\u89a7<\/h3>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/dgeod-mv\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u5e7e\u4f55\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570\uff09-100x100.png\" alt=\"\u5e7e\u4f55\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u5e7e\u4f55\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u6c42\u3081\u65b9\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011<\/div>\n    <div class=\"blogcard_excerpt\">\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u3053\u306e\u8a18\u4e8b\u3067\u306f\u5e7e\u4f55\u5206\u5e03\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u3092\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u671f\u5f85\u5024\u30fb\u5206\u6563\u306e\u6c42\u3081\u65b9\u304c\u5206\u304b\u3089\u306a\u3044\u65b9\u306f\u662f\u975e\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n\n  <div class=\"blogcard\">\n  <a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/dgeod-mc\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u5e7e\u4f55\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09-100x100.png\" alt=\"\u5e7e\u4f55\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u5e7e\u4f55\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9<\/div>\n    <div class=\"blogcard_excerpt\">\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606 \u3053\u306e\u8a18\u4e8b\u3067\u306f\u5e7e\u4f55\u5206\u5e03\u306e\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u3092\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u7a4d\u7387\u6bcd\u95a2\u6570\u30fb\u7279\u6027\u95a2\u6570\u306e\u6c42\u3081\u65b9\u304c\u5206\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h3><span id=\"i-2\">\u78ba\u7387\u95a2\u6570\u3068\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570<\/span><\/h3>\n\n\n\n<p>\\(n=10\\)\u306e\u3068\u304d\u3001\\(p=0.2,\\ 0.5,\\ 0.8\\)\u306b\u5bfe\u5fdc\u3059\u308b\uff12\u9805\u5206\u5e03\u306e\u78ba\u7387\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"756\" height=\"447\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\uff12\u9805\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570-n10\uff09.png\" alt=\"\" class=\"wp-image-1746\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\uff12\u9805\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570-n10\uff09.png 756w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\uff12\u9805\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570-n10\uff09-300x177.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\uff12\u9805\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570-n10\uff09-320x189.png 320w\" sizes=\"(max-width: 756px) 100vw, 756px\" \/><\/figure>\n\n\n\n<p>\\(p=0.4\\)\u306e\u3068\u304d\u3001\\(n=10,\\ 20,\\ 30\\)\u306b\u5bfe\u5fdc\u3059\u308b\uff12\u9805\u5206\u5e03\u306e\u78ba\u7387\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"980\" height=\"618\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\uff12\u9805\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570-p0.4\uff09-1.png\" alt=\"\" class=\"wp-image-1749\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\uff12\u9805\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570-p0.4\uff09-1.png 980w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\uff12\u9805\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570-p0.4\uff09-1-300x189.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\uff12\u9805\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570-p0.4\uff09-1-768x484.png 768w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\uff12\u9805\u5206\u5e03\uff08\u78ba\u7387\u95a2\u6570-p0.4\uff09-1-320x202.png 320w\" sizes=\"(max-width: 980px) 100vw, 980px\" \/><\/figure>\n\n\n\n<h2><span id=\"i-3\">\uff12\u9805\u5206\u5e03\u306e\u30dd\u30a2\u30bd\u30f3\u8fd1\u4f3c<\/span><\/h2>\n\n\n\n<p>\u78ba\u7387\u5909\u6570\\(X\\)\u306f\uff12\u9805\u5206\u5e03\\(X\\sim Bin(n,\\, p)\\)\u306b\u5f93\u3044\u3001\u78ba\u7387\u5909\u6570\\(Y\\)\u306f\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\\(Y\\sim Po(\\lambda_{B})\\)\u306b\u5f93\u3063\u3066\u3044\u308b\u3068\u3057\u307e\u3059\u3002\u305f\u3060\u3057<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\lambda_{B} &amp;= \\frac{(2n-x)p}{2-p}<br>\\end{align}<\/div><br>\u3067\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u4e8c\u9805\u5206\u5e03\u306e\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\\(\\mathrm{P}(X\\leq x)\\)\u306f\u3001\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u3067\u8fd1\u4f3c\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3064\u307e\u308a\u3001<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mathrm{P}(X\\leq x) \\approx \\mathrm{P}(Y\\leq \\lambda_{B})<br>\\end{align}<\/div><br>\u304c\u6210\u7acb\u3057\u307e\u3059\u3002\u3053\u306e\u8fd1\u4f3c\u3092\u30dc\u30eb\u30b7\u30a7\u30d5\u306e\u8fd1\u4f3c\u5f0f\u3068\u3044\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u203b\u3000\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306b\u3064\u3044\u3066\u306f\uff1c\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\uff1e\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2><span id=\"i-4\">\uff12\u9805\u5206\u5e03\u3068\u30d9\u30fc\u30bf\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u3000\\(X\\sim Bin(n,\\ p)\\)\u3001\\(Y\\sim Beta(x,\\ n-x+1)\\)\u3068\u3057\u307e\u3059\u3002\u305f\u3060\u3057\u3001\\(Y\\)\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\\(x,\\ n-x+1\\)\u306e\u30d9\u30fc\u30bf\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u3067\u3059\u3002\u3053\u306e\u3068\u304d\u3001\\(X\\)\u306e\u4e0a\u5074\u78ba\u7387\\(P(X\\geq x)\\)\u3092\u3001\u30d9\u30fc\u30bf\u5206\u5e03\u3092\u7528\u3044\u3066\u6b21\u306e\u3088\u3046\u306b\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mathrm{P}(X\\geq x) = \\mathrm{P}(Y\\leq p)<br>\\end{align}<\/div><\/p>\n\n\n\n<p>\u3059\u306a\u308f\u3061\u3001<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\sum_{k=x}^{n}\\ _{n}C_{k}p^{k}(1-p)^{n-k} = \\frac{1}{B(x,\\ n-x+1)}\\int_{0}^{p}y^{x-1}(1-y)^{n-x}dy<br>\\end{align}<\/div><br>\u304c\u6210\u7acb\u3057\u307e\u3059\u3002\u305f\u3060\u3057\\(B(\\alpha,\\ \\beta)\\)\u306f\u30d9\u30fc\u30bf\u95a2\u6570\u3067\u3042\u308a<br><div style=\"overflow-x: auto;\">\\begin{align}<br>B(\\alpha,\\ \\beta) = \\int_{0}^{1}y^{\\alpha-1}(1-y)^{\\beta-1}dy<br>\\end{align}<\/div><br>\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u203b\u3000\u30d9\u30fc\u30bf\u5206\u5e03\u306b\u3064\u3044\u3066\u306f\uff1c\u30d9\u30fc\u30bf\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\uff1e\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2><span id=\"F\">\uff12\u9805\u5206\u5e03\u3068\\(F\\)\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u3000\\(X\\sim Bin(n,\\ p)\\)\u3001\\(Y\\sim F(m_{1},\\ m_{2})\\)\u3068\u3057\u307e\u3059\u3002\u305f\u3060\u3057\u3001\\(Y\\)\u306f\u81ea\u7531\u5ea6\\(m_{1},\\ m_{2}\\)\u306e\\(F\\)\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u3067\u3059\u3002\u3053\u306e\u3068\u304d\u3001\\(X\\)\u306e\u4e0a\u5074\u78ba\u7387\\(P(X\\geq x)\\)\u3092\u3001\\(F\\)\u5206\u5e03\u3092\u7528\u3044\u3066\u6b21\u306e\u3088\u3046\u306b\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mathrm{P}(X\\geq x) = \\mathrm{P}(Y\\geq \\frac{m_{2}(1-p)}{m_{1}p})<br>\\end{align}<\/div><br>\u305f\u3060\u3057\u3001<br><div style=\"overflow-x: auto;\">\\begin{align}<br>m_{1} = 2(n-x+1),\\ \\ m_{2}=2x<br>\\end{align}<\/div><br>\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u203b\u3000\\(F\\)\u5206\u5e03\u306b\u3064\u3044\u3066\u306f\uff1c\\(F\\)\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\uff1e\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2><span id=\"i-5\">\uff12\u9805\u5206\u5e03\u306e\u518d\u751f\u6027<\/span><\/h2>\n\n\n\n<p>\\(X_{i}\\sim Bin(n_{i},\\ p)\\)\uff08\\(i=1,\\cdots,n\\)\uff09\u306b\u4e92\u3044\u306b\u72ec\u7acb\u306b\u5f93\u3063\u3066\u3044\u308b\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u3053\u306e\u78ba\u7387\u5909\u6570\u306e\u548c\u3082\uff12\u9805\u5206\u5e03\u306b\u5f93\u3044\u3001<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\sum_{i=1}^{n}X_{i} = X_{1}+X_{2}+\\cdots+X_{n}\\sim Bin\\left(\\sum_{i=1}^{n}n_{i},\\ p\\right)<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2><span id=\"i-6\">\uff12\u9805\u5206\u5e03\u3068\u8d85\u5e7e\u4f55\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u3000\u30d1\u30e9\u30e1\u30fc\u30bf\\(N,\\ M,\\ n\\)\u306e\u8d85\u5e7e\u4f55\u5206\u5e03<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\frac{_{M}C_{x}\\ _{N-M}C_{n-x}}{_{N}C_{n}}<br>\\end{align}<\/div><br>\u306f\\(p=\\displaystyle\\frac{M}{N}\\)\u3092\u4e00\u5b9a\u306b\u4fdd\u3061\u3064\u3064\u3001\\(N\\)\u304c\u5341\u5206\u306b\u5927\u304d\u3051\u308c\u3070\u3001\uff12\u9805\u5206\u5e03<br><div style=\"overflow-x: auto;\">\\begin{align}<br>_{n}C_{x}p^{x}(1-p)^{n-x}<br>\\end{align}<\/div><br>\u306b\u53ce\u675f\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u203b\u3000\u8d85\u5e7e\u4f55\u5206\u5e03\u306b\u3064\u3044\u3066\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/hypergeometric-distribution(\u65b0\u3057\u3044\u30bf\u30d6\u3067\u958b\u304f)\">\uff1c\u8d85\u5e7e\u4f55\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\uff1e<\/a>\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<h2><span id=\"i-7\">\u30d1\u30e9\u30e1\u30fc\u30bf\u63a8\u5b9a<\/span><\/h2>\n\n\n\n<p>\u30d1\u30e9\u30e1\u30fc\u30bf\\(p\\)\u306e\u63a8\u5b9a\u91cf\\(\\widehat{p}\\)\u306f\u30e2\u30fc\u30e1\u30f3\u30c8\u6cd5\u3001\u6700\u5c24\u6cd5\u3069\u3061\u3089\u306e\u5834\u5408\u3082<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\widehat{p} = \\frac{x}{n}<br>\\end{align}<\/div><br>\u3067\u4e0e\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u306e\u63a8\u5b9a\u91cf\u306f\\(p\\)\u306e\u6700\u5c0f\u5206\u6563\u4e0d\u504f\u63a8\u5b9a\u91cf\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h2><span id=\"i-8\">\uff12\u9805\u5206\u5e03\u306e\u4e71\u6570\u767a\u751f\u65b9\u6cd5<\/span><\/h2>\n\n\n\n<p>\u4e00\u69d8\u5206\u5e03\u3092\u7528\u3044\u3066\u4e71\u6570\u3092\u767a\u751f\u3055\u305b\u307e\u3059\u3002<\/p>\n\n\n\n<ol><li>\u533a\u9593\\((0,\\ 1)\\)\u306e\u72ec\u7acb\u306b\u5f93\u3046\u4e00\u69d8\u4e71\u6570\\(U_{i}\\)\uff08\\(i=1,\\cdots,n\\)\uff09\u3092\u767a\u751f\u3055\u305b\u307e\u3059\u3002<\/li><li>\\(U_{i}&lt;p\\)\u306e\u3068\u304d\u3001\u305d\u306e\\(i\\)\u756a\u76ee\u306e\u8a66\u884c\u306f\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306b\u304a\u3044\u3066\u6210\u529f\u3057\u305f\u3082\u306e\u3068\u3057\u307e\u3059\u3002<\/li><\/ol>\n\n\n\n<p>\u203b\u3000\u4e00\u69d8\u5206\u5e03\u306b\u3064\u3044\u3066\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/continuous-uniform-distribution\">\uff1c\u4e00\u69d8\u5206\u5e03\u306e\u57fa\u672c\u60c5\u5831\uff1e<\/a>\u3092\u53c2\u7167<br>\u203b\u3000\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306b\u3064\u3044\u3066\u306f<a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/bernoulli-distribution\">\uff1c\u30d9\u30eb\u30cc\u30fc\u30a4\u5206\u5e03\uff1e<\/a>\u3092\u53c2\u7167<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2><span id=\"i-9\">\uff12\u9805\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\uff12\u9805\u5206\u5e03\u3068\u95a2\u9023\u6df1\u3044\u5206\u5e03\u3092\u56f3\u30fb\u8868\u3067\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002\u5404\u5206\u5e03\u306e\u8a73\u3057\u3044\u60c5\u5831\u306f\u8868\u306e\u4e2d\u306e\u30ea\u30f3\u30af\u304b\u3089\u304a\u9858\u3044\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"738\" height=\"514\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7.png\" alt=\"\" class=\"wp-image-2055\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7.png 738w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7-300x209.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/07\/\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u306e\u95a2\u4fc2\u30de\u30c3\u30d7-320x223.png 320w\" sizes=\"(max-width: 738px) 100vw, 738px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/bernoulli-distribution\">\u30d9\u30eb\u30cc\u30fc\u30a4\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\uff11\u56de\u884c\u3046\u3068\u304d\u306e\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/categorical-distribution\">\u30ab\u30c6\u30b4\u30ea\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">1\u56de\u306e\u8a66\u884c\u3067\\(k\\)\u901a\u308a\u306e\u30d1\u30bf\u30fc\u30f3\u306e\u4e2d\u304b\u3089\u3069\u308c\u304b\u304c\u5f97\u3089\u308c\u308b\u53ef\u80fd\u6027\u304c\u3042\u308b\u8a66\u884c\u3092\u8868\u3059\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/bind\">\uff12\u9805\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\\(n\\)\u56de\u306e\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3067\\(x\\)\u56de\u6210\u529f\u3059\u308b\u3068\u304d\u306e\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/dgeod\">\u5e7e\u4f55\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u8907\u6570\u56de\u884c\u3046\u4e0a\u3067\u3001\u521d\u3081\u3066\u6210\u529f\u3059\u308b\u307e\u3067\u306e<br>\u8a66\u884c\u56de\u6570\u3092\u8868\u3059\u5206\u5e03<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/dhgeod\">\u8d85\u5e7e\u4f55\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u8907\u6570\u56de\u884c\u3063\u3066\u3044\u304f\u4e0a\u3067\u3001\u305d\u306e\u90fd\u5ea6\u6210\u529f\u78ba\u7387\u304c<br>\u5909\u5316\u3059\u308b\u5206\u5e03\uff08\u6709\u9650\u500b\u306e\u30a2\u30bf\u30ea\u304f\u3058\u306a\u3069\uff09<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/nbiob\">\u8ca0\u306e\uff12\u9805\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u8907\u6570\u56de\u884c\u3046\u4e0a\u3067\u3001\\(k\\)\u56de\u6210\u529f\u3059\u308b\u307e\u3067\u306e<br>\u5931\u6557\u3059\u308b\u56de\u6570\u3092\u8868\u3059\u5206\u5e03\uff08\\(k=1\\)\u306e\u5834\u5408\u3001\u5e7e\u4f55\u5206\u5e03\u306b\u306a\u308a\u307e\u3059\uff09<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/dmuld\">\u591a\u9805\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u6210\u529f\u30fb\u5931\u6557\u306e\uff12\u7a2e\u985e\u3060\u3051\u3067\u306a\u304f\u3001\u8a66\u884c\u306e\u7d50\u679c\u304c<br>\u8907\u6570\u500b\u3042\u308b\u5834\u5408\u306e\u5206\u5e03\u3092\u8868\u3057\u307e\u3059<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/distribution\/dpoid\">\u30dd\u30a2\u30bd\u30f3\u5206\u5e03<\/a><\/td><td class=\"has-text-align-left\" data-align=\"left\">\u6210\u529f\u78ba\u7387\u304c\u6975\u7aef\u306b\u5c0f\u3055\u304f\uff08\u3064\u307e\u308a\u6ec5\u591a\u306b\u8d77\u3053\u3089\u306a\u3044\uff09\u3001<br>\u8a66\u884c\u56de\u6570\u304c\u6975\u7aef\u306b\u5927\u304d\u3044\u5834\u5408\u306e\u5206\u5e03<\/td><\/tr><\/tbody><\/table><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>\uff12\u9805\u5206\u5e03\uff08binomial distribution\uff09\u306f\u96e2\u6563\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3001\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u8907\u6570\u56de\u884c\u3063\u305f\u969b\u306b\u5f97\u3089\u308c\u308b\u30c7\u30fc\u30bf\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u6210 &#8230; 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