{"id":1569,"date":"2020-07-01T11:00:00","date_gmt":"2020-07-01T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1569"},"modified":"2020-07-11T12:44:02","modified_gmt":"2020-07-11T03:44:02","slug":"continuous-uniform-distribution","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/continuous-uniform-distribution","title":{"rendered":"\u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\uff09"},"content":{"rendered":"\n\n\n\n\n<p>\u4e00\u69d8\u5206\u5e03\uff08uniform distribution\uff09\u306f\u6700\u3082\u30b7\u30f3\u30d7\u30eb\u306a\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u4e00\u69d8\u5206\u5e03\u306f\u5225\u540d\u3001\u77ed\u5f62\u5206\u5e03\uff08rectangular distribution\uff09\u3068\u3082\u547c\u3070\u308c\u307e\u3059\u3002\u4e00\u69d8\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\)\u3092\u30d1\u30e9\u30e1\u30fc\u30bf\\(\\alpha,\\ \\beta\\)\u3092\u7528\u3044\u3066\u3001\u8a18\u53f7\\(X\\sim U(\\alpha,\\ \\beta)\\)\u3067\u3088\u304f\u8868\u3055\u308c\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> \u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\uff09\u306e\u57fa\u672c\u60c5\u5831<\/a><ul><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_2\">1.1<\/span> \u78ba\u7387\u5bc6\u5ea6\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> \u6a19\u6e96\u4e00\u69d8\u5206\u5e03<\/a><\/li><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_1\">3<\/span> \u4e00\u69d8\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\u548c<\/a><\/li><li><a href=\"#i-5\"><span class=\"toc_number toc_depth_1\">4<\/span> \u30d1\u30e9\u30e1\u30fc\u30bf\u306e\u63a8\u5b9a\u65b9\u6cd5\uff08\u6700\u5c24\u6cd5\uff09<\/a><ul><li><a href=\"#i-6\"><span class=\"toc_number toc_depth_2\">4.1<\/span> \u6700\u5c24\u63a8\u5b9a\u91cf<\/a><\/li><li><a href=\"#i-7\"><span class=\"toc_number toc_depth_2\">4.2<\/span> \u4e0d\u504f\u63a8\u5b9a\u91cf<\/a><\/li><\/ul><\/li><li><a href=\"#Ualpha_beta\"><span class=\"toc_number toc_depth_1\">5<\/span> \u4e00\u69d8\u5206\u5e03\\(U(\\alpha,\\ \\beta)\\)\u306e\u4e71\u6570\u306e\u767a\u751f\u306e\u4ed5\u65b9<\/a><\/li><\/ul><\/div>\n<h2><span id=\"i\">\u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\uff09\u306e\u57fa\u672c\u60c5\u5831<\/span><\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>\u30d1\u30e9\u30e1\u30fc\u30bf<\/td><td>\\(\\alpha,\\ \\beta\\)<\/td><\/tr><tr><td>\u78ba\u7387\u5909\u6570\u306e\u7bc4\u56f2<\/td><td>\\(\\alpha\\leq x \\leq \\beta\\)<\/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\\frac{x-\\alpha}{\\beta-\\alpha}\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/probability-function\">\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570<\/a><\/td><td>\\(\\displaystyle\\frac{1}{\\beta-\\alpha}\\)<\/td><\/tr><tr><td>\u9006\u5206\u5e03\u95a2\u6570\uff08\u78ba\u7387\\(p\\)\uff09<\/td><td>\\(\\alpha+p(\\beta-\\alpha)\\)<\/td><\/tr><tr><td>\u9006\u751f\u5b58\u95a2\u6570\uff08\u78ba\u7387\\(p\\)\uff09<\/td><td>\\(\\beta-p(\\beta-\\alpha)\\)<\/td><\/tr><tr><td>\u5371\u967a\u5ea6\u95a2\u6570<\/td><td>\\(\\displaystyle\\frac{1}{\\beta-x}\\)<\/td><\/tr><tr><td>\u7d2f\u7a4d\u5371\u967a\u5ea6\u95a2\u6570<\/td><td>\\(-\\log\\displaystyle\\frac{\\beta-x}{\\beta-\\alpha}\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mgf\">\u7a4d\u7387\u6bcd\u95a2\u6570<\/a><\/td><td>\\(\\displaystyle\\frac{\\exp(\\beta t)-\\exp(\\alpha t)}{t(\\beta-\\alpha)}\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/cf\">\u7279\u6027\u95a2\u6570<\/a><\/td><td>\\(\\displaystyle\\frac{\\exp(i\\beta t)-\\exp(i\\alpha t)}{it(\\beta-\\alpha)}\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/moment\">\\(k\\)\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8\uff08\u539f\u70b9\u307e\u308f\u308a\uff09<\/a><\/td><td>\\(\\displaystyle\\frac{\\beta^{k+1}-\\alpha^{k+1}}{(\\beta-\\alpha)(k+1)}\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/moment\">\\(k\\)\u6b21\u30e2\u30fc\u30e1\u30f3\u30c8\uff08\u5e73\u5747\u307e\u308f\u308a\uff09<\/a><\/td><td><div style=\"overflow-x: auto;\">\\begin{align}<br> \\left\\{<br>\\begin{array}{ccc}<br>&amp; 0 &amp;\uff08k\u304c\u5947\u6570\uff09\\\\<br>&amp; \\displaystyle\\frac{ \\displaystyle\\left(\\frac{\\beta-\\alpha}{2}\\right)^{k} }{ k+1 }&amp;\uff08k\u304c\u5076\u6570\uff09<br>\\end{array}\\right. <br>\\end{align}<\/div><\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mean-definition\">\u671f\u5f85\u5024<\/a><\/td><td>\\(\\displaystyle\\frac{\\alpha+\\beta}{2}\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/variance\">\u5206\u6563<\/a><\/td><td>\\(\\displaystyle\\frac{(\\beta-\\alpha)^{2}}{12} \\)<\/td><\/tr><tr><td>\u5e73\u5747\u504f\u5dee<\/td><td>\\(\\displaystyle\\frac{\\beta-\\alpha}{4}\\)<\/td><\/tr><tr><td>\u4e2d\u4f4d\u6570<\/td><td>\\(\\displaystyle\\frac{\\alpha+\\beta}{2}\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/skewness\">\u6b6a\u5ea6<\/a><\/td><td>\\(0\\)<\/td><\/tr><tr><td><a href=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/kurtosis\">\u5c16\u5ea6<\/a><\/td><td>\\(\\displaystyle\\frac{9}{5}\\)<\/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\/cud-mv\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\u4e00\u69d8\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09-100x100.png\" alt=\"\u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\uff09\u306e\u671f\u5f85\u5024\u3068\u5206\u6563\u3010\u8a3c\u660e\u4ed8\u304d\u3067\u89e3\u8aac\u3011\" width=\"90\" height=\"90\" \/><\/div>\n   <div class=\"blogcard_content\">\n    <div class=\"blogcard_title\">\u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\uff09\u306e\u671f\u5f85\u5024\u3068\u5206\u6563\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\u2606\u2606\u2606\u2606 \u3053\u306e\u8a18\u4e8b\u3067\u306f\u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\uff09\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\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\/cud-mc\">\n   <div class=\"blogcard_thumbnail\"><img src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\u4e00\u69d8\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09-100x100.png\" alt=\"\u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\uff09\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\">\u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\uff09\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\u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\uff09\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\u2026<\/div>\n   <\/div>\n   <div class=\"clear\"><\/div>\n  <\/a>\n  <\/div>\n\n\n\n<h3><span id=\"i-2\">\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3068\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570<\/span><\/h3>\n\n\n\n<p>\\(\\alpha=0,\\ \\beta=5\\)\u306e\u3068\u304d\u306e\u4e00\u69d8\u5206\u5e03\uff08\u9023\u7d9a\u578b\uff09\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3068\u7d2f\u7a4d\u5206\u5e03\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=\"977\" height=\"705\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\u4e00\u69d8\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09.png\" alt=\"\" class=\"wp-image-1604\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\u4e00\u69d8\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09.png 977w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\u4e00\u69d8\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09-300x216.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\u4e00\u69d8\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09-768x554.png 768w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\u4e00\u69d8\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09-320x231.png 320w\" sizes=\"(max-width: 977px) 100vw, 977px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"977\" height=\"705\" src=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\u4e00\u69d8\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09.png\" alt=\"\" class=\"wp-image-1605\" srcset=\"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\u4e00\u69d8\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09.png 977w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\u4e00\u69d8\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09-300x216.png 300w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\u4e00\u69d8\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09-768x554.png 768w, https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\u4e00\u69d8\u5206\u5e03\uff08\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570\uff09-320x231.png 320w\" sizes=\"(max-width: 977px) 100vw, 977px\" \/><\/figure>\n\n\n\n<p> <\/p>\n\n\n\n<h2><span id=\"i-3\">\u6a19\u6e96\u4e00\u69d8\u5206\u5e03<\/span><\/h2>\n\n\n\n<p>\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304c<br><div style=\"overflow-x: auto;\">\\begin{align}<br>f(u)=u\\ \\ \\ \\ (0\\leq u\\leq 1)<br>\\end{align}<\/div><br>\u3068\u306a\u308b\u4e00\u69d8\u5206\u5e03\u3092\u7279\u306b\u6a19\u6e96\u4e00\u69d8\u5206\u5e03\u3068\u8a00\u3044\u307e\u3059\uff08\\(\\alpha=0,\\ \\beta=1\u306e\u3068\u304d\u3067\u3059\\)\uff09\u3002\u6a19\u6e96\u4e00\u69d8\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\sim U(0,\\ 1)\\)\u306b\u3064\u3044\u3066\u671f\u5f85\u5024\u3001\u5206\u6563\u306f\u305d\u308c\u305e\u308c<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mathrm{E}[X]=\\frac{1}{2}\\ \\ \\ \\mathrm{Var}[X]=\\frac{1}{12}<br>\\end{align}<\/div><br>\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3001\\(\\displaystyle2\\sqrt{3}\\left( X-\\frac{1}{2} \\right)\\)\u306f\u671f\u5f85\u5024\\(0\\)\u3001\u5206\u6563\\(1\\)\u3068\u306a\u308b\u6a19\u6e96\u5316\u3055\u308c\u305f\u4e00\u69d8\u5206\u5e03\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p> <\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2><span id=\"i-4\">\u4e00\u69d8\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\u306e\u548c<\/span><\/h2>\n\n\n\n<p>\\(X_{1},X_{2},\\cdots,X_{n}\\)\u306f\u6a19\u6e96\u4e00\u69d8\u5206\u5e03\\(U(0,\\ 1)\\)\u306b\u72ec\u7acb\u306b\u5f93\u3063\u3066\u3044\u308b\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u78ba\u7387\u5909\u6570\u3092\u548c\\(S=\\displaystyle\\sum_{i=0}^{n}X_{i}\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\left\\{<br>\\begin{array}{ccc}<br>&amp; \\displaystyle\\frac{1}{(n-1)!}\\sum_{j=0}^{k}(-1)^{j}\\ _{n}C_{j}(s-j)^{n-j} \\ \\ \\ \\ \\ \\ &amp;k\\leq s \\leq k+1,\\ \\ 0\\leq k \\leq n-1\\\\<br>&amp; 0&amp;\uff08\u305d\u306e\u4ed6\uff09<br>\\end{array}\\right.<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u5206\u5e03\u306f\u30a2\u30fc\u30a6\u30a3\u30f3\u30fb\u30db\u30fc\u30eb\u5206\u5e03\uff08Irwin-Hall distribution\uff09\u3068\u547c\u3070\u308c\u308b\u3082\u306e\u3067\u3059\u3002<br>\u3000\u3082\u3063\u3068\u4e00\u822c\u7684\u306b\\(X_{1},X_{2},\\cdots,X_{n}\\)\u304c\u4e00\u69d8\u5206\u5e03\\(U(0,\\ \\beta)\\)\u306b\u72ec\u7acb\u306b\u5f93\u3063\u3066\u3044\u308b\u5834\u5408\u3001\\(S\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\left\\{<br>\\begin{array}{ccc}<br>&amp; \\displaystyle\\frac{1}{(n-1)!\\beta^{n}}\\sum_{j=0}^{k}(-1)^{j}\\ _{n}C_{j}(s-\\beta j)^{n-1} \\ \\ \\ \\ \\ \\ &amp;k\\leq s\\beta^{-1} \\leq k+1,\\ \\ 0\\leq k \\leq n-1\\\\<br>&amp; 0&amp;\uff08\u305d\u306e\u4ed6\uff09<br>\\end{array}\\right.<br>\\end{align}<\/div><\/p>\n\n\n\n<p>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u3000<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2><span id=\"i-5\">\u30d1\u30e9\u30e1\u30fc\u30bf\u306e\u63a8\u5b9a\u65b9\u6cd5\uff08\u6700\u5c24\u6cd5\uff09<\/span><\/h2>\n\n\n\n<h3><span id=\"i-6\">\u6700\u5c24\u63a8\u5b9a\u91cf<\/span><\/h3>\n\n\n\n<p>\u63a8\u5b9a\u3057\u305f\u3044\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092<br><div style=\"overflow-x: auto;\">\\begin{align}<br>f(x) =\\frac{1}{\\theta}\\ \\ \\ 0\\leq \\theta\\leq \\theta<br>\\end{align}<\/div>\u3000<br>\u3068\u3059\u308b\u3068\u304d\u3001\\(\\theta\\)\u306e\u6700\u5c24\u63a8\u5b9a\u91cf\\(\\widehat{\\theta}\\)\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\widehat{\\theta} = \\max\\left( X_{1},\\ \\cdots\\ ,X_{n} \\right)<br>\\end{align}<\/div><br>\u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002\u3053\u306e\u63a8\u5b9a\u91cf\u306e\u671f\u5f85\u5024\u304a\u3088\u3073\u5206\u6563\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mathrm{E}\\left[ \\widehat{\\theta} \\right] &amp;= \\frac{n\\theta}{n+1} \\\\<br>\\mathrm{Var}\\left[ \\widehat{\\theta} \\right] &amp;= \\frac{n\\theta^{2}}{(n+1)^{2}(n+2)}<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h3><span id=\"i-7\">\u4e0d\u504f\u63a8\u5b9a\u91cf<\/span><\/h3>\n\n\n\n<p>\u3000\\(\\widehat{\\theta}\\)\u3092\\(\\theta\\)\u306e\u6700\u5c24\u63a8\u5b9a\u91cf\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\\(\\theta\\)\u306e\u4e0d\u504f\u63a8\u5b9a\u91cf\\(\\tilde{\\theta}\\)\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\tilde{\\theta} &amp;= \\frac{n+1}{n}\\widehat{\\theta}<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u4e0d\u504f\u63a8\u5b9a\u91cf\u306e\u671f\u5f85\u5024\u30fb\u5206\u6563\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>\\mathrm{E}\\left[ \\tilde{\\theta} \\right] &amp;= \\theta \\\\<br>\\mathrm{Var}\\left[ \\tilde{\\theta} \\right] &amp;= \\frac{\\theta^{2}}{n(n+2)}<br>\\end{align}<\/div><br>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<p>\u3000<\/p>\n\n\n\n<h2><span id=\"Ualpha_beta\">\u4e00\u69d8\u5206\u5e03\\(U(\\alpha,\\ \\beta)\\)\u306e\u4e71\u6570\u306e\u767a\u751f\u306e\u4ed5\u65b9<\/span><\/h2>\n\n\n\n<p>\u3000\u4e00\u69d8\u4e71\u6570\\(X\\sim U(0,\\ 1)\\)\u3092\u767a\u751f\u3055\u305b\u308b\u3053\u3068\u304c\u3067\u304d\u308c\u3070\u3001\\(U(\\alpha,\\ \\beta)\\)\u306b\u5f93\u3046\u4e71\u6570\u306f<br><div style=\"overflow-x: auto;\">\\begin{align}<br>Y = \\alpha+(\\beta-\\alpha)X<br>\\end{align}<\/div><br>\u3067\u751f\u6210\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4e00\u69d8\u5206\u5e03\uff08uniform distribution\uff09\u306f\u6700\u3082\u30b7\u30f3\u30d7\u30eb\u306a\u9023\u7d9a\u578b\u306e\u78ba\u7387\u5206\u5e03\u3067\u3059\u3002\u4e00\u69d8\u5206\u5e03\u306f\u5225\u540d\u3001\u77ed\u5f62\u5206\u5e03\uff08rectangular  &#8230; <\/p>\n","protected":false},"author":1,"featured_media":1604,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[84],"tags":[96],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/06\/\u4e00\u69d8\u5206\u5e03\uff08\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff09.png","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1569"}],"collection":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/comments?post=1569"}],"version-history":[{"count":51,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1569\/revisions"}],"predecessor-version":[{"id":1940,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1569\/revisions\/1940"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/1604"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=1569"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=1569"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=1569"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}