{"id":1011,"date":"2015-04-04T23:26:13","date_gmt":"2015-04-04T14:26:13","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1011"},"modified":"2021-03-09T16:33:10","modified_gmt":"2021-03-09T07:33:10","slug":"tkr201404","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201404\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2014\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\\(p ,q\\) \u306f\u5b9f\u6570\u306e\u5b9a\u6570\u3067, \\(0 \\lt p \\lt 1\\) , \\(q \\gt 0\\) \u3092\u307f\u305f\u3059\u3068\u3059\u308b. \u95a2\u6570\r\n\\[\r\nf(x) = (1-p) x +(1-x)( 1-e^{-qx} )\r\n\\]\r\n\u3092\u8003\u3048\u308b. \r\n\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. \u5fc5\u8981\u3067\u3042\u308c\u3070, \u4e0d\u7b49\u5f0f \\(1+x \\leqq e^x\\) \u304c\u3059\u3079\u3066\u306e\u5b9f\u6570 \\(x\\) \u306b\u5bfe\u3057\u3066\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u306a\u3057\u306b\u7528\u3044\u3066\u3088\u3044.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(0 \\lt x \\lt 1\\) \u306e\u3068\u304d, \\(0 \\lt f(x) \\lt 1\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(x _ 0\\) \u306f \\(0 \\lt x _ 0 \\lt 1\\) \u3092\u307f\u305f\u3059\u5b9f\u6570\u3068\u3059\u308b. \u6570\u5217 \\(\\left\\{ x _ n \\right\\}\\) \u306e\u5404\u9805 \\(x _ n \\ ( n = 1 , 2 , 3 , \\cdots )\\) \u3092, \r\n\\[\r\nx _ n = f( x _ {n-1} )\r\n\\]\r\n\u306b\u3088\u3063\u3066\u9806\u6b21\u5b9a\u3081\u308b. \\(p \\gt q\\) \u3067\u3042\u308b\u3068\u304d, \r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} x _ n = 0\r\n\\]\r\n\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(p \\lt q\\) \u3067\u3042\u308b\u3068\u304d, \r\n\\[\r\nc = f(c) , \\quad 0 \\lt c \\lt 1\r\n\\]\r\n\u3092\u307f\u305f\u3059\u5b9f\u6570 \\(c\\) \u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

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

    \\(0 \\lt x \\lt 1\\) \u306a\u306e\u3067, \\(f(x)\\) \u306f, \\(1-p\\) \u3068 \\(1 -e^{-qx}\\) \u306e\u52a0\u91cd\u5e73\u5747\u3067\u3042\u308b.
    \r\n\\(0 \\lt p \\lt 1\\) \u3088\u308a\r\n\\[\r\n0 \\lt 1-p \\lt 1\r\n\\]\r\n\\(q \\gt 0\\) \u3088\u308a, \\(0 \\lt e^{-qx} \\lt 1\\) \u306a\u306e\u3067\r\n\\[\r\n0 \\lt 1 -e^{-qx} \\lt 1\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n0 \\lt f(x) \\lt 1\r\n\\]\r\n

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

    \\(1+x \\leqq e^x\\) ...[A] \u304c, \u3059\u3079\u3066\u306e\u5b9f\u6570\u306b\u3064\u3044\u3066\u6210\u7acb\u3059\u308b\u306e\u3067, \\(x\\) \u3092 \\(-qx\\) \u306b\u7f6e\u304d\u63db\u3048\u308c\u3070\r\n\\[\\begin{align}\r\n1 -qx & \\leqq e^{-qx} \\\\\r\n\\text{\u2234} \\quad 1 -e^{-qx} & \\leqq qx \\quad ... [1] .\r\n\\end{align}\\]\r\n[1] \u3092\u7528\u3044\u308c\u3070, \\(0 \\lt x \\lt 1\\) \u306b\u5bfe\u3057\u3066\r\n\\[\\begin{align}\r\nf(x) & \\leqq (1-p) x +( 1 -x ) qx \\\\\r\n& = ( 1-p+q ) x -qx^2 \\\\\r\n& \\lt ( 1-p+q ) x\r\n\\end{align}\\]\r\n\u6761\u4ef6 \\(p \\gt q\\) \u3088\u308a, \\(0 \\lt 1-p+q \\lt 1\\) ...[2] \u3067\u3042\u308b\u3053\u3068\u3068, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n0 \\lt f(x) \\lt ( 1-p+q ) x \\quad ... [3]\r\n\\]\r\n\u3053\u3053\u3067, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066\r\n\\[\r\n0 \\lt x _ n \\lt ( 1-p+q )^n x _ 0 \\quad ... [ \\text{\uff0a} ]\r\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092, \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(n=1\\) \u306e\u3068\u304d
      \r\n\u6761\u4ef6\u3088\u308a, \\(0 \\lt x _ 0 \\lt 1\\) \u306a\u306e\u3067, [3] \u3092\u7528\u3044\u308c\u3070\r\n\\[\r\n0 \\lt x _ 1 \\lt ( 1-p+q ) x _ 0\r\n\\]\r\n\u3086\u3048\u306b, [\uff0a]\u304c\u6210\u7acb\u3057\u3066\u3044\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(n=k \\ ( k = 1, 2, \\cdots )\\) \u306e\u3068\u304d
      \r\n[\uff0a]\u304c\u6210\u7acb\u3059\u308b, \u3059\u306a\u308f\u3061\r\n\\[\r\n0 \\lt x _ k \\lt ( 1-p+q )^k x _ 0\r\n\\]\r\n\u3068\u4eee\u5b9a\u3059\u308b\u3068, [3] \u3092\u7528\u3044\u3066\r\n\\[\r\n0 \\lt f( x _ k ) \\lt ( 1-p+q ) x _ k \\\\\r\n\\text{\u2234} \\quad 0 \\lt x _ {k+1} \\lt ( 1-p+q )^{k+1} x _ 0\r\n\\]\r\n\u3064\u307e\u308a, \\(n=k+1\\) \u306e\u3068\u304d\u3082, [\uff0a]\u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      1*<\/strong>, 2*<\/strong>\u3088\u308a, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, [\uff0a] \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u305f.
      \r\n[2] \u306b\u6ce8\u610f\u3059\u308c\u3070\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} ( 1-p+q )^n x _ 0 = 0\r\n\\]\r\n\u306a\u306e\u3067, \u306f\u3055\u307f\u3046\u3061\u306e\u539f\u7406\u3088\u308a\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} x _ n = 0\r\n\\]\r\n

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

      \\(g(x) = f(x) -x\\) \u3068\u304a\u3044\u3066, \\(g(c) = 0 \\ ( 0 \\lt c \\lt 1 )\\) \u3068\u306a\u308b\u5b9f\u6570 \\(c\\) \u306e\u5b58\u5728\u3092\u793a\u305b\u3070\u3088\u3044. \r\n\\[\r\ng(0) = 0 , \\quad g(1) = -p \\lt 0 \\quad ... [4]\r\n\\]\r\n\\(p \\lt q\\) \u3088\u308a, \\(0 \\lt \\dfrac{p}{q} \\lt 1\\) \u3067\u3042\u308a\r\n\\[\\begin{align}\r\ng \\left( \\dfrac{p}{q} \\right) & = (1-p) \\dfrac{p}{q} +\\left( 1-\\dfrac{p}{q} \\right) \\left( 1-e^{-p} \\right) \\\\\r\n& = \\dfrac{1}{qe^p} \\left\\{ p(1-p) +(q-p)(e^p-1) \\right\\} \\\\\r\n& \\gt 0 \\quad ... [5]\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, [4] [5] \u3088\u308a, \\(\\dfrac{p}{q} \\lt x \\lt 1\\) \u306b\u304a\u3044\u3066 \\(g(x) = 0\\) \u3068\u306a\u308b\u5b9f\u6570 \\(x\\) \u304c\u5c11\u306a\u304f\u3068\u3082 \\(1\\) \u3064\u5b58\u5728\u3059\u308b.
      \r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(p ,q\\) \u306f\u5b9f\u6570\u306e\u5b9a\u6570\u3067, \\(0 \\lt p \\lt 1\\) , \\(q \\gt 0\\) \u3092\u307f\u305f\u3059\u3068\u3059\u308b. \u95a2\u6570 \\[ f(x) = (1-p) x +(1-x)( 1-e^{-qx} ) \\] \u3092\u8003\u3048\u308b. \u4ee5 … \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":[113],"tags":[139,112],"class_list":["post-1011","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2014","tag-tokyo_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1011","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=1011"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1011\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1011"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1011"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1011"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}