{"id":687,"date":"2013-03-13T20:55:36","date_gmt":"2013-03-13T11:55:36","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=687"},"modified":"2021-09-16T07:12:11","modified_gmt":"2021-09-15T22:12:11","slug":"ngr200704b","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr200704b\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2007\uff1a\u7b2c4\u554f(b)"},"content":{"rendered":"<hr \/>\n<p>\u888b\u306e\u4e2d\u306b\u8d64\u3068\u9ec4\u3068\u9752\u306e\u7389\u304c \\(1\\) \u500b\u305a\u3064\u5165\u3063\u3066\u3044\u308b.\r\n\u300c\u3053\u306e\u888b\u304b\u3089\u7389\u3092 \\(1\\) \u500b\u53d6\u308a\u51fa\u3057, \u51fa\u305f\u7389\u3068\u540c\u3058\u8272\u306e\u7389\u3092\u888b\u306e\u4e2d\u306b \\(1\\) \u500b\u8ffd\u52a0\u3059\u308b\u300d\u3068\u3044\u3046\u64cd\u4f5c\u3092 \\(N\\) \u56de\u7e70\u308a\u8fd4\u3057\u305f\u5f8c, \u8d64\u306e\u7389\u304c\u888b\u306e\u4e2d\u306b \\(m\\) \u500b\u3042\u308b\u78ba\u7387\u3092 \\(p _ N (m)\\) \u3068\u3059\u308b.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\u9023\u6bd4 \\(p _ 3 (1) : p _ 3 (2) : p _ 3 (3) : p _ 3 (4)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\u4e00\u822c\u306e \\(N\\) \u306b\u5bfe\u3057 \\(p _ N (m) \\ ( 1 \\leqq m \\leqq N+1 )\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n<hr \/>\r\n<!--more-->\r\n<h2>\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\\(N\\) \u56de\u64cd\u4f5c\u3092\u3059\u308b\u3068, \u888b\u306e\u4e2d\u306e\u7389\u306e\u6570\u306f \\(N+3\\) \u500b\u306b\u306a\u308b.<br \/>\r\n\u307e\u305f, \u8d64\u3044\u7389\u306e\u500b\u6570 \\(m\\) \u306e\u7bc4\u56f2\u306f, \\(1 \\leqq m \\leqq N+1 \\quad ... [1]\\) .\r\n\\(N\\) \u56de\u306e\u64cd\u4f5c\u306e\u5f8c\u306b, \u8d64\u3044\u7389\u304c \\(m\\) \u500b\u3042\u308b\u306e\u306f<\/p>\r\n<ul>\r\n<li><p>\\(N-1\\) \u56de\u306e\u64cd\u4f5c\u306e\u5f8c\u306b, \u8d64\u3044\u7389\u304c \\(m-1\\) \u500b\u3042\u308a, \\(N\\) \u56de\u76ee\u3067\u8d64\u3044\u7389\u304c\u51fa\u305f\u3068\u304d<\/p><\/li>\r\n<li><p>\\(N-1\\) \u56de\u306e\u64cd\u4f5c\u306e\u5f8c\u306b, \u8d64\u3044\u7389\u304c \\(m\\) \u500b\u3042\u308a, \\(N\\) \u56de\u76ee\u3067\u8d64\u4ee5\u5916\u306e\u8272\u306e\u7389\u304c\u51fa\u305f\u3068\u304d<\/p><\/li>\r\n<\/ul>\r\n<p>\u306e \\(2\\) \u901a\u308a\u304c\u3042\u308b\u306e\u3067\r\n\\[\r\np _ N (m) = \\dfrac{m-1}{N+2} p _ {N-1} (m-1) +\\dfrac{N+2-m}{N+2} p _ {N-1} (m) \\quad ... [2] \\ .\r\n\\]\r\n\u305f\u3060\u3057, [1] \u3088\u308a\r\n\\[\r\np _ N (0) = p _ N (N+1) = 0 \\quad ... [3] \\ .\r\n\\]\r\n\u3044\u307e\r\n\\[\r\np _ 1 (1) = \\dfrac{2}{3} , \\ p _ 1 (2) = \\dfrac{1}{3} \\ .\r\n\\]\r\n\u306a\u306e\u3067, [2] [3] \u306b\u3057\u305f\u304c\u3063\u3066\u9806\u6b21\u6c42\u3081\u308b\u3068\r\n\\[\\begin{align}\r\np _ 2 (1) & = \\dfrac{3}{4} p _ 1 (1) = \\dfrac{1}{2} , \\\\\r\np _ 2 (2) & = \\dfrac{1}{4} p _ 1 (1) +\\dfrac{1}{2} p _ 1 (2) = \\dfrac{1}{3} , \\\\\r\np _ 2 (3) & = \\dfrac{1}{2} p _ 1 (2) = \\dfrac{1}{6} , \\\\\r\np _ 3 (1) & = \\dfrac{4}{5} p _ 2 (1) = \\dfrac{2}{5} , \\\\\r\np _ 3 (2) & = \\dfrac{1}{5} p _ 2 (1) +\\dfrac{3}{5} p _ 2 (2) = \\dfrac{3}{10} , \\\\\r\np _ 3 (3) & = \\dfrac{2}{5} p _ 2 (2) +\\dfrac{2}{5} p _ 2 (3) = \\dfrac{1}{5} , \\\\\r\np _ 3 (4) & = \\dfrac{3}{5} p _ 2 (3) = \\dfrac{1}{10} \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u9023\u6bd4\u306f\r\n\\[\r\np _ 3 (1) : p _ 3 (2) : p _ 3 (3) : p _ 3 (4) = \\underline{4 : 3: 2 : 1} \\ .\r\n\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p><strong>(1)<\/strong> \u306e\u904e\u7a0b\u3088\u308a\r\n\\[\\begin{align}\r\np _ 1 (1) : p _ 1 (2) & = 2 : 1 \\\\\r\np _ 2 (1) : p _ 2 (2) : p _ 2 (3) & = 3 : 2 : 1 \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u4e00\u822c\u306e \\(N\\) \u306b\u3064\u3044\u3066\r\n\\[\r\nP _ N (1) : p _ N (2) : \\cdots : p _ N (N+1) = (N+1) : N : \\cdots : 1 \\ .\r\n\\]\r\n\u3059\u306a\u308f\u3061, \\(1 \\leqq m \\leqq N+1\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\np _ N (m) & = \\dfrac{N+2-m}{1+2+ \\cdots +(N+1)} \\\\\r\n& = \\dfrac{2 (N+2-m)}{(N+1)(N+2)} \\quad ... [ \\text{A} ] \\ .\r\n\\end{align}\\]\r\n\u3068\u63a8\u6e2c\u3055\u308c\u308b.<br \/>\r\n[A] \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092, \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000\\(N = 1 , \\ m = 1 , 2\\) \u306e\u3068\u304d<br \/>\r\n<strong>(1)<\/strong> \u306e\u904e\u7a0b\u3088\u308a\r\n\\[\r\np _ 1 (1) = \\dfrac{2}{3} , \\ p _ 1 (2) = \\dfrac{1}{3} \\ .\r\n\\]\r\n\u306a\u306e\u3067, [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<li><p><strong>2*<\/strong>\u3000\\(N = k , \\ 1 \\leqq m \\leqq k+1 \\ ( k \\geqq 1 )\\) \u306e\u3068\u304d<br \/>\r\n[A]\u304c\u6210\u7acb\u3059\u308b, \u3059\u306a\u308f\u3061\r\n\\[\r\np _ k (m) = \\dfrac{2 (k+2-m)}{(k+1)(k+2)} \\ .\r\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068, [2] [3] \u3092\u7528\u3044\u3066\r\n<ul>\r\n<li><p>\\(m = 1\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\np _ {k+1} (1) & = \\dfrac{k+2}{k+3} p _ {k} (1) \\\\\r\n& = \\dfrac{k+2}{k+3} \\cdot \\dfrac{1}{k+2} \\\\\r\n& = \\dfrac{1}{k+3} \\ .\r\n\\end{align}\\]<\/li>\r\n<li><p>\\(m = k+2\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\np _ {k+1} (k+2) & = \\dfrac{k+1}{k+3} p _ {k} (k+1) \\\\\r\n& = \\dfrac{k+1}{k+3} \\cdot \\dfrac{1}{(k+1)(k+2)} \\\\\r\n& = \\dfrac{1}{(k+2)(k+3)} \\ .\r\n\\end{align}\\]<\/li>\r\n<li><p>\\(2 \\leqq m \\leqq k+1\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\np _ {k+1} (m) & = \\dfrac{m-1}{k+3} p _ {k} (m-1) +\\dfrac{k+3-m}{k+3} p _ {k} (m) \\\\\r\n& = \\dfrac{m-1}{k+3} \\cdot \\dfrac{2 (k+3-m)}{(k+1)(k+2)} +\\dfrac{k+3-m}{k+3} \\cdot \\dfrac{2 (k+2-m)}{(k+1)(k+2)} \\\\\r\n& = \\dfrac{2 (k+3-m) \\left\\{ (m-1) + (k+2-m) \\right\\}}{(k+1)(k+2)(k+3)} \\\\\r\n& = \\dfrac{2 (k+3-m)}{(k+2)(k+3)} \\ .\r\n\\end{align}\\]<\/li>\r\n<\/ul>\r\n\u4ee5\u4e0a\u3088\u308a, \\(N = k+1 , \\ 1 \\leqq m \\leqq k+2\\) \u306e\u3068\u304d\u3082, [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n<p>\u4ee5\u4e0a\u3088\u308a, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(N\\) \u306b\u3064\u3044\u3066, [A] \u304c\u6210\u7acb\u3057\r\n\\[\r\np _ N (m) = \\underline{\\dfrac{2 (N+2-m)}{(N+1)(N+2)}} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u888b\u306e\u4e2d\u306b\u8d64\u3068\u9ec4\u3068\u9752\u306e\u7389\u304c \\(1\\) \u500b\u305a\u3064\u5165\u3063\u3066\u3044\u308b. \u300c\u3053\u306e\u888b\u304b\u3089\u7389\u3092 \\(1\\) \u500b\u53d6\u308a\u51fa\u3057, \u51fa\u305f\u7389\u3068\u540c\u3058\u8272\u306e\u7389\u3092\u888b\u306e\u4e2d\u306b \\(1\\) \u500b\u8ffd\u52a0\u3059\u308b\u300d\u3068\u3044\u3046\u64cd\u4f5c\u3092 \\(N\\) \u56de\u7e70\u308a\u8fd4\u3057\u305f\u5f8c, \u8d64\u306e\u7389\u304c\u888b\u306e\u4e2d\u306b \\ &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/ngr200704b\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[97],"tags":[143,109],"class_list":["post-687","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2007","tag-nagoya_r","tag-109"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/687","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=687"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/687\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=687"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=687"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=687"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}