cp's OEIS Frontend

A246665 Decimal expansion of the asymptotic probability of success in the full-information secretary problem with uniform distribution when the number of applicants is also uniformly distributed.

%I A246665 #27 Feb 16 2025 08:33:23
%S A246665 4,3,5,1,7,0,8,0,5,5,8,0,1,2,7,6,5,8,0,5,9,1,8,9,9,1,2,8,4,7,8,5,8,4,
%T A246665 1,0,4,2,7,9,6,2,5,9,4,7,5,3,4,7,0,2,4,7,0,2,9,7,9,1,2,3,0,4,4,3,9,0,
%U A246665 6,6,5,8,7,5,4,4,3,0,3,3,5,7,8,4,9,9,7,6,6,2,8,6,8,5,0,2,6,5,9
%N A246665 Decimal expansion of the asymptotic probability of success in the full-information secretary problem with uniform distribution when the number of applicants is also uniformly distributed.
%C A246665 In this variant of the secretary problem, the applicants' values are independently distributed on a known interval, like in A242674; and the number of applicants is itself a random variable with uniform distribution on 1..n (and then the limit n -> infinity is taken), like in A325905. So we have more information than in the variant considered in A325905 but less information than in the variant considered in A242674. Hence A325905 < this constant < A242674. - _Andrey Zabolotskiy_, Sep 14 2019
%D A246665 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.15 Optimal stopping constants, p. 361.
%H A246665 Steven R. Finch, <a href="https://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, Sec. 5.15 Optimal Stopping Constants, pp. 53-55, arXiv:2001.00578 [math.HO], 2020.
%H A246665 Zdzisław Porosiński, <a href="https://doi.org/10.1016/S0167-7152(01)00201-2">On best choice problems having similar solutions</a>, Statistics & Probability Letters, 56 (2002), 321-327.
%H A246665 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/SultansDowryProblem.html">Sultan's Dowry Problem</a>
%H A246665 Wikipedia, <a href="http://en.wikipedia.org/wiki/Secretary_problem">Secretary problem</a>
%F A246665 (1 - e^a)*Ei(-a) - (e^(-a) + a*Ei(-a))*(gamma + log(a) - Ei(a)), where a is A246664, gamma is Euler's constant and Ei is the exponential integral function.
%e A246665 0.43517080558012765805918991284785841042796259475347024702979123...
%t A246665 a = x /. FindRoot[E^x*(1 - EulerGamma - Log[x] + ExpIntegralEi[-x]) - (EulerGamma + Log[x] - ExpIntegralEi[x]) == 1, {x, 2}, WorkingPrecision -> 102]; (1 - E^a)*ExpIntegralEi[-a] - (E^-a + a*ExpIntegralEi[-a])*(EulerGamma + Log[a] - ExpIntegralEi[a]) // RealDigits // First
%Y A246665 Cf. A246664, A068985, A325905, A242674.
%K A246665 nonn,cons,easy
%O A246665 0,1
%A A246665 _Jean-François Alcover_, Sep 01 2014