OFFSET
0,1
COMMENTS
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
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.15 Optimal stopping constants, p. 361.
LINKS
Steven R. Finch, Errata and Addenda to Mathematical Constants, Sec. 5.15 Optimal Stopping Constants, pp. 53-55, arXiv:2001.00578 [math.HO], 2020.
Zdzisław Porosiński, On best choice problems having similar solutions, Statistics & Probability Letters, 56 (2002), 321-327.
Eric Weisstein's MathWorld, Sultan's Dowry Problem
Wikipedia, Secretary problem
FORMULA
(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.
EXAMPLE
0.43517080558012765805918991284785841042796259475347024702979123...
MATHEMATICA
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
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Sep 01 2014
STATUS
approved