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選取兩個實數 x 和 y，在 (0,1) 內均勻隨機選取。均勻分佈。 機率 P_(even) 是多少？其中 [x/y]，[r] 表示 最接近整數函式，為 偶數？

答案如下。

P(a<x/y<b)={P(ay<x<by) for 0<=a<b<1; P(x/b<y<x/a) for 1<a<b
(1)
={int_0^1int_(ay)^(by)dxdy=1/2(b-a) for 0<=a<b<1; int_0^1int_(x/b)^(x/a)dydx=1/(2a)-1/(2b) for 1<a<b
(2)

所以

P_(even)=P(0<x/y<1/2)+sum_(n=1)^(infty)P(2n-1/2<x/y<2n+1/2)
(3)
=1/2(1/2-0)+sum_(n=1)^(infty)[1/(2(2n-1/2))-1/(2(2n+1/2))]
(4)
=1/4+sum_(n=1)^(infty)(1/(4n-1)-1/(4n+1))
(5)
=1/4+(1/3-1/5+1/7-1/9+...)
(6)
=1/4+(1-tan^(-1)1)
(7)
=5/4-pi/4
(8)
=1/4(5-pi)
(9)
=0.46460...
(10)

(OEIS A091651).

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參考文獻

American Mathematics Competitions. Problem B3 in "William Lowell Putnam Competition Archive, 1993" http://www.unl.edu/amc/a-activities/a7-problems/putnam/.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, p. 139, 2003.Losinski, L. F.; Alexanderson, G. L.; and Larson, L. C. "The Fifty-Fourth William Lowell Putnam Mathematical Competition." Amer. Math. Monthly 101, 725-734, 1994.Scholes, J. "54th Putnam 1993." http://www.kalva.demon.co.uk/putnam/putn93.html.Sloane, N. J. A. Sequence A091651 in "The On-Line Encyclopedia of Integer Sequences."

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實數選取

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Weisstein, Eric W. "實數選取"。來自 —— 資源。 https://mathworld.tw/RealNumberPicking.html

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