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A243964
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Decimal expansion of the variance of the maximum of a size 8 sample from a normal (0,1) distribution.
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0
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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
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OFFSET
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0,1
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COMMENTS
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According to Steven Finch, no exact expression of this moment is known.
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.
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LINKS
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FORMULA
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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.
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EXAMPLE
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0.3728971432867289942202112287621146...
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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