{"id":725,"date":"2013-04-10T21:38:20","date_gmt":"2013-04-10T12:38:20","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=725"},"modified":"2021-11-04T16:07:24","modified_gmt":"2021-11-04T07:07:24","slug":"htb200705","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/htb200705\/","title":{"rendered":"\u4e00\u6a4b\u59272007\uff1a\u7b2c5\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(1\\) \u304c\u66f8\u304b\u308c\u305f\u30ab\u30fc\u30c9\u304c \\(1\\) \u679a, \\(2\\) \u304c\u66f8\u304b\u308c\u305f\u30ab\u30fc\u30c9\u304c \\(1\\) \u679a, \u2026, \\(n\\) \u304c\u66f8\u304b\u308c\u305f\u30ab\u30fc\u30c9\u304c \\(1\\) \u679a\u306e\u5168\u90e8\u3067 \\(n\\) \u679a\u306e\u30ab\u30fc\u30c9\u304b\u3089\u306a\u308b\u7d44\u304c\u3042\u308b.\r\n\u3053\u306e\u7d44\u304b\u3089 \\(1\\) \u679a\u3092\u629c\u304d\u51fa\u3057\u5143\u306b\u3082\u3069\u3059\u64cd\u4f5c\u3092 \\(3\\) \u56de\u884c\u3046.\r\n\u629c\u304d\u51fa\u3057\u305f\u30ab\u30fc\u30c9\u306b\u66f8\u304b\u308c\u305f\u6570\u3092 \\(a , b , c\\) \u3068\u3059\u308b\u3068\u304d, \u5f97\u70b9 \\(X\\) \u3092\u6b21\u306e\u898f\u5247 (i) , (ii) \u306b\u5f93\u3063\u3066\u5b9a\u3081\u308b.<\/p>\r\n<ol>\r\n<li><p>(i)\u3000\\(a , b , c\\) \u304c\u3059\u3079\u3066\u7570\u306a\u308b\u3068\u304d, \\(X\\) \u306f \\(a , b , c\\) \u306e\u3046\u3061\u306e\u6700\u5927\u3067\u3082\u6700\u5c0f\u3067\u3082\u306a\u3044\u5024\u3068\u3059\u308b.<\/p><\/li>\r\n<li><p>(ii)\u3000\\(a , b , c\\) \u306e\u3046\u3061\u306b\u91cd\u8907\u3057\u3066\u3044\u308b\u3082\u306e\u304c\u3042\u308b\u3068\u304d, \\(X\\) \u306f\u305d\u306e\u91cd\u8907\u3057\u305f\u5024\u3068\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n<p>\\(1 \\leqq k \\leqq n\\) \u3092\u307f\u305f\u3059 \\(k\\) \u306b\u5bfe\u3057\u3066, \\(X = k\\) \u3068\u306a\u308b\u78ba\u7387\u3092 \\(p _ k\\) \u3068\u3059\u308b.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(p _ k\\) \u3092 \\(n\\) \u3068 \\(k\\) \u3067\u8868\u305b.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(p _ k\\) \u304c\u6700\u5927\u3068\u306a\u308b \\(k\\) \u3092 \\(n\\) \u3067\u8868\u305b.<\/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<ul>\r\n<li><p>\u898f\u5247 (i) \u306b\u3088\u3063\u3066, \\(X=k\\) \u3068\u306a\u308b\u78ba\u7387\u306f\r\n\\[\r\n\\dfrac{3! (k-1)(n-k)}{n^3} = \\dfrac{6(k-1)(n-k)}{n^3}\r\n\\]<\/li>\r\n<li><p>\u898f\u5247 (ii) \u306b\u3088\u3063\u3066, \\(X=k\\) \u3068\u306a\u308b\u78ba\u7387\u306f\r\n\\[\r\n\\dfrac{{} _ {3} \\text{C} {} _ 2 (n-1)}{n^3} +\\dfrac{1}{n^3} = \\dfrac{3n-2}{n^3}\r\n\\]<\/li>\r\n<\/ul>\r\n<p>\u3088\u3063\u3066\r\n\\[\\begin{align}\r\np _ k & = \\dfrac{6(k-1)(n-k)}{n^3} +\\dfrac{3n-2}{n^3} \\\\\r\n& = \\underline{\\dfrac{-6k^2 +6(n+1)k -3n-2}{n^3}}\r\n\\end{align}\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p><strong>(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\np _ {k+1} -p _ k & = \\dfrac{6}{n^3} \\left\\{ (k+1)^2 -k^2 +(n-1) \\right\\} \\\\\r\n& = \\dfrac{6(2k-n)}{n^3}\r\n\\end{align}\\]\r\n\\(p _ {k+1} -p _ k = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nk = \\dfrac{n}{2}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(n\\) \u306e\u5947\u5076\u3067\u5834\u5408\u5206\u3051\u3057\u3066<\/p>\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000\\(n\\) \u304c\u5076\u6570\u306e\u3068\u304d\r\n\\[\r\np _ 1 \\lt \\cdots \\lt p _ {\\frac{n}{2}} = p _ {\\frac{n}{2}+1} \\gt \\cdots p _ n\r\n\\]\r\n\u306a\u306e\u3067, \\(p _ k\\) \u304c\u6700\u5927\u3068\u306a\u308b\u306e\u306f\r\n\\[\r\nk = \\dfrac{n}{2} , \\ \\dfrac{n}{2} +1\r\n\\]<\/li>\r\n<li><p><strong>2*<\/strong>\u3000\\(n\\) \u304c\u5947\u6570\u306e\u3068\u304d\r\n\\[\r\np _ 1 \\lt \\cdots \\lt p _ {\\frac{n-1}{2}} \\lt p _ {\\frac{n+1}{2}} \\gt p _ {\\frac{n+3}{2}} \\gt \\cdots >p _ n\r\n\\]\r\n\u306a\u306e\u3067, \\(p _ k\\) \u304c\u6700\u5927\u3068\u306a\u308b\u306e\u306f\r\n\\[\r\nk = \\dfrac{n+1}{2}\r\n\\]<\/li>\r\n<\/ol>\r\n<p>\u3088\u3063\u3066, \u6c42\u3081\u308b \\(k\\) \u306e\u5024\u306f\r\n\\[\r\nk = \\underline{\\left\\{ \\begin{array}{ll} \\dfrac{n}{2} , \\dfrac{n}{2} +1 & ( \\ n \\text{\u304c\u5076\u6570\u306e\u3068\u304d} ) \\\\ \\dfrac{n+1}{2} & ( \\ n \\text{\u304c\u5947\u6570\u306e\u3068\u304d} ) \\end{array} \\right.}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(1\\) \u304c\u66f8\u304b\u308c\u305f\u30ab\u30fc\u30c9\u304c \\(1\\) \u679a, \\(2\\) \u304c\u66f8\u304b\u308c\u305f\u30ab\u30fc\u30c9\u304c \\(1\\) \u679a, \u2026, \\(n\\) \u304c\u66f8\u304b\u308c\u305f\u30ab\u30fc\u30c9\u304c \\(1\\) \u679a\u306e\u5168\u90e8\u3067 \\(n\\) \u679a\u306e\u30ab\u30fc\u30c9\u304b\u3089\u306a\u308b\u7d44\u304c\u3042\u308b. \u3053\u306e\u7d44\u304b\u3089 \\( &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/htb200705\/\">\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":[107],"tags":[146,109],"class_list":["post-725","post","type-post","status-publish","format-standard","hentry","category-hitotsubashi_2007","tag-hitotsubashi","tag-109"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/725","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=725"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/725\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=725"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=725"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=725"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}