{"id":1329,"date":"2015-11-09T01:05:02","date_gmt":"2015-11-08T16:05:02","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1329"},"modified":"2021-10-21T08:26:33","modified_gmt":"2021-10-20T23:26:33","slug":"wsr201504","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr201504\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52015\uff1a\u7b2c4\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(N\\) \u3092 \\(3\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3068\u3059\u308b.\r\n\\(1\\) \u304b\u3089 \\(N\\) \u307e\u3067\u306e\u6570\u5b57\u304c\u66f8\u304b\u308c\u305f \\(N\\) \u679a\u306e\u30ab\u30fc\u30c9\u3092\u7528\u610f\u3057, A \u3068 B \u306e\u4e8c\u4eba\u3067\u6b21\u306e\u3088\u3046\u306a\u30b2\u30fc\u30e0\u3092\u884c\u3046.\r\n\u307e\u305a A \u304c , \\(1\\) \u304b\u3089 \\(N\\) \u307e\u3067\u306e\u6570\u306e\u3046\u3061\u304b\u3089 \\(1\\) \u3064\u3092\u9078\u3073\u305d\u308c\u3092 \\(K\\) \u3068\u3057, \u305d\u306e\u6570\u306f B \u306b\u77e5\u3089\u305b\u305a\u306b\u304a\u304f. \u305d\u306e\u5f8c, \u4ee5\u4e0b\u306e\u8a66\u884c\u3092\u4f55\u5ea6\u3082\u7e70\u308a\u8fd4\u3059.<br \/>\r\nB \u306f \\(N\\) \u679a\u306e\u30ab\u30fc\u30c9\u304b\u3089\u7121\u4f5c\u70ba\u306b\u4e00\u679a\u5f15\u3044\u3066 A \u306b\u305d\u306e\u6570\u3092\u4f1d\u3048, A \u306f\u5f15\u304b\u308c\u305f\u6570\u5b57\u304c \\(K\\) \u3088\u308a\u5927\u304d\u3051\u308c\u3070\u300c\u4e0a\u300d, \\(K\\) \u4ee5\u4e0b\u3067\u3042\u308c\u3070\u300c\u4ee5\u4e0b\u300d\u3068 B \u306b\u7b54\u3048, B \u306f\u305d\u306e\u7b54\u304b\u3089 \\(K\\) \u306e\u7bc4\u56f2\u3092\u7d5e\u308a\u8fbc\u3080. \u5f15\u3044\u305f\u30ab\u30fc\u30c9\u306f\u5143\u3078\u623b\u3059.<br \/>\r\n\u3053\u306e\u3068\u304d, \\(n\\) \u56de\u4ee5\u4e0b\u306e\u8a66\u884c\u3067 B \u304c \\(K\\) \u3092\u78ba\u5b9a\u3067\u304d\u308b\u78ba\u7387\u3092 \\(P _ N (n)\\) \u3067\u8868\u3059. \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(K = 1\\) \u306e\u3068\u304d, \\(P _ 3 (1) , P _ 3 (2) , P _ 3 (3)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(K = 2\\) \u306e\u3068\u304d, \\(P _ 3 (1) , P _ 3 (2) , P _ 3 (3)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\\(K = 1, 2, \\cdots , N\\) \u306b\u3064\u3044\u3066, \\(P _ N (n)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(4)<\/strong>\u3000\u81ea\u7136\u6570 \\(c\\) \u306b\u5bfe\u3057\u3066, \u6975\u9650\u5024 \\(\\displaystyle\\lim _ {N \\rightarrow \\infty}P _ N (cN)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n<hr \/>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>B \u304c \\(1\\) \u3092\u5f15\u3051\u3070, \\(K\\) \u3092\u78ba\u5b9a\u3067\u304d\u308b\u306e\u3067\r\n\\[\\begin{align}\r\nP _ 3 (1) & = \\underline{\\dfrac{1}{3}} , \\\\\r\nP _ 3 (2) & = 1 -\\left( \\dfrac{2}{3} \\right)^2 = \\underline{\\dfrac{5}{9}} , \\\\\r\nP _ 3 (3) &= 1 -\\left( \\dfrac{2}{3} \\right)^3 = \\underline{\\dfrac{19}{27}}\r\n\\end{align}\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>B \u304c \\(2\\) \u3068 \\(3\\) \u3092\u5f15\u3051\u3070, \\(K\\) \u3092\u78ba\u5b9a\u3067\u304d\u308b\u306e\u3067\r\n\\[\\begin{align}\r\nP _ 3 (1) & = \\underline{0} , \\\\\r\nP _ 3 (2) & = 1 -2 \\left( \\dfrac{2}{3} \\right)^2 +\\left( \\dfrac{1}{3} \\right)^2 \\\\\r\n& = 1 -\\dfrac{8}{9} +\\dfrac{1}{9} = \\underline{\\dfrac{2}{9}} , \\\\\r\nP _ 3 (3) & = 1 -2 \\left( \\dfrac{2}{3} \\right)^3 +\\left( \\dfrac{1}{3} \\right)^3 \\\\\r\n& = 1 -\\dfrac{16}{27} +\\dfrac{1}{27} = \\underline{\\dfrac{4}{9}}\r\n\\end{align}\\]\r\n<p><strong>(3)<\/strong><\/p>\r\n<ul>\r\n<li><p>\\(K = 1\\) \u306e\u3068\u304d, <strong>(1)<\/strong> \u3068\u540c\u69d8\u306b, B \u304c \\(1\\) \u3092\u5f15\u3051\u3070\u3088\u3044\u306e\u3067\r\n\\[\r\nP _ N (n) = 1 -\\left( 1 -\\dfrac{1}{N} \\right)^n\r\n\\]<\/li>\r\n<li><p>\\(K = 2, \\cdots N-1\\) \u306e\u3068\u304d, <strong>(2)<\/strong> \u3068\u540c\u69d8\u306b, B \u304c \\(K\\) \u3068 \\(K+1\\) \u3092\u5f15\u3051\u3070\u3088\u3044\u306e\u3067\r\n\\[\r\nP _ N (n) = 1 -2 \\left( 1 -\\dfrac{1}{N} \\right)^n +\\left( 1 -\\dfrac{2}{N} \\right)^n\r\n\\]<\/li>\r\n<li><p>\\(K = N\\) \u306e\u3068\u304d, B \u304c \\(N\\) \u3092\u5f15\u3051\u3070, \\(K\\) \u3092\u78ba\u5b9a\u3067\u304d\u308b\u306e\u3067\r\n\\[\r\nP _ N (n) = 1 -\\left( 1 -\\dfrac{1}{N} \\right)^n\r\n\\]<\/li>\r\n<\/ul>\r\n<p>\u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u78ba\u7387\u306f\r\n\\[\r\nP _ N (n) = \\underline{\\left\\{ \\begin{array}{ll} 1 -\\left( 1 -\\dfrac{1}{N} \\right)^n & ( \\ K = 1 , N \\text{\u306e\u3068\u304d} \\ ) \\\\ 1 -2 \\left( 1 -\\dfrac{1}{N} \\right)^n +\\left( 1 -\\dfrac{2}{N} \\right)^n & ( \\ K = 2 , \\cdots , N-1 \\text{\u306e\u3068\u304d} \\ ) \\end{array} \\right.}\r\n\\]\r\n<p><strong>(4)<\/strong><\/p>\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000\\(K = 1 , N\\) \u306e\u3068\u304d<br \/>\r\n\\[\\begin{align}\r\nP _ N (cN) & = 1 -\\left\\{ \\left( 1 -\\dfrac{1}{N} \\right)^{-N} \\right\\}^{-c} \\\\\r\n& \\rightarrow 1 -e^{-c} \\quad ( \\ n \\rightarrow \\infty \\text{\u306e\u3068\u304d} \\ )\r\n\\end{align}\\]<\/li>\r\n<li><p><strong>2*<\/strong>\u3000\\(K = 2 , \\cdots , N-1\\) \u306e\u3068\u304d<br \/>\r\n\\[\\begin{align}\r\nP _ N (cN) & = 1 -2 \\left\\{ \\left( 1 -\\dfrac{1}{N} \\right)^{-N} \\right\\}^{-c} +\\left\\{ \\left( 1 -\\dfrac{2}{N} \\right)^{-\\frac{N}{2}} \\right\\}^{-2c} \\\\\r\n& \\rightarrow 1 -2e^{-c} +e^{-2c} \\quad ( \\ n \\rightarrow \\infty \\text{\u306e\u3068\u304d} \\ )\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n<p>\u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u6975\u9650\u5024\u306f\r\n\\[\r\n\\displaystyle\\lim _ {N \\rightarrow \\infty}P _ N (cN) = \\underline{\\left\\{ \\begin{array}{ll} 1 -e^{-c} & ( \\ K = 1 , N \\text{\u306e\u3068\u304d} \\ ) \\\\ 1 -2e^{-c} +e^{-2c} & ( \\ K = 2 , \\cdots , N-1 \\text{\u306e\u3068\u304d} \\ ) \\end{array} \\right.}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(N\\) \u3092 \\(3\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3068\u3059\u308b. \\(1\\) \u304b\u3089 \\(N\\) \u307e\u3067\u306e\u6570\u5b57\u304c\u66f8\u304b\u308c\u305f \\(N\\) \u679a\u306e\u30ab\u30fc\u30c9\u3092\u7528\u610f\u3057, A \u3068 B \u306e\u4e8c\u4eba\u3067\u6b21\u306e\u3088\u3046\u306a\u30b2\u30fc\u30e0\u3092\u884c\u3046. \u307e\u305a A \u304c , \\(1\\) \u304b\u3089 \\ &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/wsr201504\/\">\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":[134],"tags":[147,137],"class_list":["post-1329","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2015","tag-waseda_r","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1329","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=1329"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1329\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1329"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1329"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1329"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}