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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
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 (list; constant; graph; refs; listen; history; text; internal format)
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 -> is taken), like in A325905. So we have more information than in the variant considered in A325905 but less information then 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

Table of n, a(n) for n=0..98.

Steven R. Finch, Errata and Addenda to Mathematical Constants. p. 45.

Steven R. Finch, Errata and Addenda to Mathematical Constants, January 22, 2016. [Cached copy, with permission of the author]

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

Cf. A246664, A068985, A325905, A242674.

Sequence in context: A292612 A316254 A029934 * A274260 A011397 A081665

Adjacent sequences:  A246662 A246663 A246664 * A246666 A246667 A246668

KEYWORD

nonn,cons,easy

AUTHOR

Jean-François Alcover, Sep 01 2014

STATUS

approved

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Last modified October 21 17:26 EDT 2019. Contains 328303 sequences. (Running on oeis4.)