A243964 Decimal expansion of the variance of the maximum of a size 8 sample from a normal (0,1) distribution.
3, 7, 2, 8, 9, 7, 1, 4, 3, 2, 8, 6, 7, 2, 8, 9, 9, 4, 2, 2, 0, 2, 1, 1, 2, 2, 8, 7, 6, 2, 1, 1, 4, 6, 0, 2, 1, 7, 6, 3, 5, 9, 2, 9, 2, 0, 0, 0, 4, 6, 7, 3, 7, 5, 7, 9, 5, 7, 8, 4, 9, 1, 7, 6, 7, 2, 4, 8, 9, 4, 6, 2, 1, 5, 3, 8, 5, 0, 7, 7, 7, 9, 6, 3, 0, 6, 7, 5, 7, 3, 9, 8, 0, 1, 0, 4, 5, 7, 6, 2, 9
Offset: 0
Examples
0.3728971432867289942202112287621146...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.
Crossrefs
Programs
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Mathematica
digits = 101; 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["m1 = ", m]; m = m+dm]; m8 = mu8[m]; Clear[v, m]; v[m_] := v[m] = 8*NIntegrate[x^2*F[x]^7*f[x], {x, -m , m}, WorkingPrecision -> digits+5, MaxRecursion -> 20]; v[m0]; v[m = m0+dm]; While[RealDigits[v[m]] != RealDigits[v[m-dm]], Print["m2 = ", m]; m = m+dm]; v8 = v[m]-m8^2; RealDigits[v8, 10, digits] // First
Formula
integral_(-infinity..infinity) 8*x^2*F(x)^7*f(x) dx - mu(8)^2, where f(x) is the normal (0,1) density and F(x) its cumulative distribution, mu(8) being the moment A243961.
Comments