A243961 Decimal expansion of the expectation of the maximum of a size 8 sample from a normal (0,1) distribution.

1, 4, 2, 3, 6, 0, 0, 3, 0, 6, 0, 4, 5, 2, 7, 7, 7, 5, 3, 0, 7, 8, 3, 2, 4, 6, 4, 9, 3, 0, 6, 2, 5, 7, 2, 5, 3, 0, 8, 6, 7, 2, 5, 2, 7, 0, 6, 9, 4, 8, 3, 1, 4, 3, 2, 2, 2, 5, 9, 1, 7, 5, 5, 4, 7, 8, 3, 5, 5, 5, 1, 2, 6, 8, 5, 2, 8, 1, 4, 2, 1, 6, 4, 2, 8, 9, 8, 8, 6, 5, 9, 7, 6, 9, 2, 7, 5, 5, 3, 7

Offset: 1

Author

Jean-François Alcover, Jun 16 2014

Keywords

• nonn
• cons

Comments

According to Steven Finch, no exact expression of this moment mu(8) is known, unlike the moments mu(n) for n<8.

Examples

			1.423600306045277753078324649306257253...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Crossrefs

Cf. A087197 mu(2), A243446 mu(3), A243448 mu(4), A243453 mu(5), A243523 mu(6), A243524 mu(7).

Programs

Mathematica

digits = 100; m0 = 5; dm = 5; f[x_] := 1/ Sqrt[2*Pi]*Exp[-x^2/2]; F[x_] := 1/2*Erf[x/Sqrt[2]] + 1/2; Clear[mu8]; mu8[m_] := mu8[m] = 8*NIntegrate[x*F[x]^7*f[x], {x , -m , m}, WorkingPrecision -> digits+5, MaxRecursion -> 20]; mu8[m0]; mu8[m = m0 + dm]; While[RealDigits[mu8[m]] != RealDigits[mu8[m - dm]], Print["m = ", m]; m = m + dm]; RealDigits[mu8[m], 10, digits] // First

Formula

integral_(-infinity..infinity) 8*x*F(x)^7*f(x) dx, where f(x) is the normal (0,1) density and F(x) its cumulative distribution.