{"id":157,"date":"2011-12-01T17:58:18","date_gmt":"2011-12-01T08:58:18","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=157"},"modified":"2021-10-29T10:18:26","modified_gmt":"2021-10-29T01:18:26","slug":"wsr201005","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr201005\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52010\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\u8868\u306e\u51fa\u308b\u78ba\u7387\u304c \\(p \\ ( 0 \\lt p \\lt 1 )\\) , \u88cf\u304c\u51fa\u308b\u78ba\u7387\u304c \\(1-p\\) \u306e\u786c\u8ca8\u304c\u3042\u308b.\r\n\\(n\\) \u3092\u81ea\u7136\u6570\u3068\u3059\u308b. \u3053\u306e\u786c\u8ca8\u3092 \\(2n\\) \u56de\u6295\u3052\u305f\u3068\u304d, \u8868\u304c \\(n+1\\) \u56de\u4ee5\u4e0a\u51fa\u308b\u78ba\u7387\u3092 \\(P _ n\\) \u3068\u3059\u308b. \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(P _ 2 , P _ 3\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(P _ 3 \\gt P _ 2\\) \u3068\u306a\u308b \\(p\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(P _ {n+1} - P _ n = p^{n+1} ( 1-p )^n ( ap+b )\\) \u3068\u306a\u308b \\(a , b\\) \u3092 \\(n\\) \u3092\u7528\u3044\u3066\u8868\u305b. \u305f\u3060\u3057 \\(a , b\\) \u306f \\(p\\) \u3092\u542b\u307e\u306a\u3044\u3068\u3059\u308b.<\/p><\/li>\r\n

  4. (4)<\/strong>\u3000\\(p = \\dfrac{7}{16}\\) \u306e\u3068\u304d, \\(P _ n\\) \u3092\u6700\u5927\u306b\u3059\u308b \\(n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \\(n = 2\\) \u306e\u3068\u304d, \\(4\\) \u56de\u6295\u3052\u3066 \\(3\\) \u56de\u4ee5\u4e0a\u8868\u304c\u51fa\u308c\u3070\u3088\u3044\u306e\u3067\r\n\\[\r\nP _ 2 = {} _ 4 \\text{C} {} _ 3 \\, p^3 (1-p) +p^4 = \\underline{p^3 (4-3p)}\r\n\\]\r\n\\(n = 3\\) \u306e\u3068\u304d, \\(6\\) \u56de\u6295\u3052\u3066 \\(4\\) \u56de\u4ee5\u4e0a\u8868\u304c\u51fa\u308c\u3070\u3088\u3044\u306e\u3067\r\n\\[\\begin{align}\r\nP _ 3 & = {} _ 6 \\text{C} {} _ 4 \\, p^4 (1-p)^2 +{} _ 6 \\text{C} {} _ 5 \\, p^5 (1-p) +p^6 \\\\\r\n& = p^4 \\left\\{ 15 (1-p)^2 +6p (1-p) +p^2 \\right\\} \\\\\r\n& = \\underline{p^4 ( 15 -24p +10p^2 )}\r\n\\end{align}\\]\r\n

    (2)<\/strong><\/p>\r\n

    \\(0 \\lt p \\lt 1\\) \u306a\u306e\u3067, \\(P _ 3 \\gt P _ 2\\) \u3088\u308a\r\n\\[\\begin{align}\r\np ( 15 -24p +10p^2 ) & \\gt 4-3p \\\\\r\n5p^3 -12p^2 +9p -2 & \\gt 0 \\\\\r\n( 1-p )^2 ( 5p-2 ) & \\gt 0 \\\\\r\n\\text{\u2234} \\quad p \\gt \\dfrac{2}{5} &\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\underline{\\dfrac{2}{5} \\lt p \\lt 1}\r\n\\]\r\n

    (3)<\/strong><\/p>\r\n

    \u78ba\u7387 \\(P _ {n+1}\\) \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5834\u5408\u5206\u3051\u3057\u3066\u6c42\u3081\u308b.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(2n\\) \u56de\u307e\u3067\u306b\u8868\u304c \\(n+1\\) \u56de\u51fa\u3066\u3044\u308b\u78ba\u7387\u3092 \\(Q _ n\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nQ _ n = {} _ {2n} \\text{C} {} _ {n+1} \\, p^{n+1} (1-p)^{n-1}\r\n\\]\r\n\u3053\u306e\u5f8c \\(2\\) \u56de\u306e\u3046\u3061 \\(1\\) \u56de\u3067\u3082\u8868\u304c\u51fa\u308c\u3070, \u8868\u304c\u5408\u8a08 \\(n+2\\) \u56de\u4ee5\u4e0a\u51fa\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(2n\\) \u56de\u307e\u3067\u306b\u8868\u304c \\(n\\) \u56de\u51fa\u3066\u3044\u308b\u78ba\u7387\u3092 \\(R _ n\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nR _ n = {} _ {2n} \\text{C} {} _ {n} \\, p^{n} (1-p)^{n}\r\n\\]\r\n\u3053\u306e\u5f8c \\(2\\) \u56de\u3068\u3082\u8868\u304c\u51fa\u308c\u3070, \u8868\u304c\u5408\u8a08 \\(n+2\\) \u56de\u4ee5\u4e0a\u51fa\u308b.<\/p><\/li>\r\n

    3. 3*<\/strong>\u3000\\(2n\\) \u56de\u307e\u3067\u306b\u8868\u304c \\(n+2\\) \u56de\u4ee5\u4e0a\u51fa\u3066\u3044\u308b\u78ba\u7387\u306f \\(P _ n -Q _ n\\) \u3067\u3042\u308a, \u3053\u306e\u5f8c \\(2\\) \u56de\u306e\u7d50\u679c\u306b\u304b\u304b\u308f\u3089\u305a, \u8868\u304c\u5408\u8a08 \\(n+2\\) \u56de\u4ee5\u4e0a\u51fa\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      1*<\/strong> \uff5e 3*<\/strong> \u3088\u308a,\r\n\\[\\begin{align}\r\nP _ {n+1} & = \\left\\{ 1 -(1-p)^2 \\right\\} Q _ n + p^2 R _ n + \\left( P _ n -Q _ n \\right) \\\\\r\n& = P _ n + p^2 R _ n - (1-p)^2 Q _ n\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nP _ {n+1} -P _ n & = {} _ {2n} \\text{C} {} _ {n} \\, p^{n+2} (1-p)^{n} -{} _ {2n} \\text{C} {} _ {n+1} \\, p^{n+1} (1-p)^{n+1} \\\\\r\n& = p^{n+1} (1-p)^n \\left\\{ {} _ {2n} \\text{C} {} _ {n} \\, p -{} _ {2n} \\text{C} {} _ {n+1} \\, (1-p) \\right\\} \\\\\r\n& = p^{n+1} (1-p)^n \\left( {} _ {2n+1} \\text{C} {} _ {n+1} \\, p -{} _ {2n} \\text{C} {} _ {n+1} \\right)\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\underline{a = {} _ {2n+1} \\text{C} {} _ {n+1} , \\ b = -{} _ {2n} \\text{C} {} _ {n+1}}\r\n\\]\r\n

      (4)<\/strong><\/p>\r\n

      (3)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070, \\(P _ {n+1} -P _ n \\gt 0\\) \u3088\u308a\r\n\\[\\begin{align}\r\n{} _ {2n+1} \\text{C} {} _ {n+1} \\, p -{} _ {2n} \\text{C} {} _ {n+1} & \\gt 0 \\\\\r\n\\dfrac{7}{16} \\cdot \\dfrac{(2n+1)!}{(n+1)! n!} -\\dfrac{(2n)!}{(n+1)! (n-1)!} & \\gt 0 \\\\\r\n7(2n+1) -16n & \\gt 0 \\\\\r\n\\text{\u2234} \\quad n \\lt \\dfrac{7}{2} &\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066 \\(P _ n\\) \u306e\u5927\u5c0f\u306f\u4ee5\u4e0b\u306e\u901a\u308a.\r\n\\[\r\nP _ 1 \\lt P _ 2 \\lt P _ 3 \\lt P _ 4 \\gt P _ 5 \\gt \\cdots\r\n\\]\r\n\u3086\u3048\u306b\u6c42\u3081\u308b\u5024\u306f\r\n\\[\r\n\\underline{n = 4}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u8868\u306e\u51fa\u308b\u78ba\u7387\u304c \\(p \\ ( 0 \\lt p \\lt 1 )\\) , \u88cf\u304c\u51fa\u308b\u78ba\u7387\u304c \\(1-p\\) \u306e\u786c\u8ca8\u304c\u3042\u308b. \\(n\\) \u3092\u81ea\u7136\u6570\u3068\u3059\u308b. \u3053\u306e\u786c\u8ca8\u3092 \\(2n\\) \u56de\u6295\u3052\u305f\u3068\u304d, \u8868\u304c \\(n+1\\) \u56de\u4ee5\u4e0a … \u7d9a\u304d\u3092\u8aad\u3080 →<\/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":[37],"tags":[147,14],"class_list":["post-157","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2010","tag-waseda_r","tag-14"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/157","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=157"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/157\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=157"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=157"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=157"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}