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Decimal expansion of the expectation of the maximum of a size 8 sample from a normal (0,1) distribution.
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%I #5 Jun 16 2014 11:32:19

%S 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,

%T 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,

%U 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

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

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

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

%F 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.

%e 1.423600306045277753078324649306257253...

%t 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

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

%K nonn,cons

%O 1,2

%A _Jean-François Alcover_, Jun 16 2014