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

1,2

Comments

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

References

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

Links

Table of n, a(n) for n=1..100.

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.

Example

1.423600306045277753078324649306257253...

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

Crossrefs

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

Sequence in context: A123403 A276957 A275847 * A098317 A371325 A095185

Adjacent sequences: A243958 A243959 A243960 * A243962 A243963 A243964

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 16 2014

Status

approved