A376922 Variance of n-th power of a standard normal random variable.

1, 2, 15, 96, 945, 10170, 135135, 2016000, 34459425, 653836050, 13749310575, 316126087200, 7905853580625, 213439785208650, 6190283353629375, 191894675132160000, 6332659870762850625, 221641908024728441250, 8200794532637891559375, 319830558102716120460000

Offset

1,2

Comments

The variance of the n-th sample moment is exactly a(n) / k for sample size k.

For a non-standard normal r.v. X ~ N(0, sigma^2), Var(X^n) = a(n) sigma^(2n).

References

M. G. Kendall and A. Stuart, The Advanced Theory of Statistics Volume 1, Charles Griffin & Company, 1963, page 229.

Links

Table of n, a(n) for n=1..20.

Formula

a(n) = M(2n) - M(n)^2, where M(n) = A123023(n) are the moments of N(0, 1).

Example

If Z ~ N(0, 1), then Var(Z) = 1, Var(Z^2) = 2, Var(Z^3) = 15, etc.

Crossrefs

Cf. A001147, A103639, A123023.

Sequence in context: A356578 A362768 A381284 * A258390 A378001 A027080

Adjacent sequences: A376919 A376920 A376921 * A376923 A376924 A376925

Keyword

nonn

Author

Adam M. Scherlis, Oct 10 2024

Status

approved