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.

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, 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, 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

Offset: 0

Author

Jean-François Alcover, Sep 01 2014

Keywords

• nonn
• cons
• easy

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

Examples

			0.43517080558012765805918991284785841042796259475347024702979123...
		

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

Crossrefs

Cf. A246664, A068985, A325905, A242674.

Programs

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

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.