= Radial moments of an isotropic Gaussian vector
{title2=$\mathbb E|\mathbf X|^{2n}=2^n\Gamma(n+d/2)v^n/\Gamma(d/2)$}
For $d$ <independent> centered <normal random variables> of common <variance> $v>0$, $|\mathbf X|^2/v$ has a <chi-squared distribution> with $d$ <statistical degrees of freedom>[degrees of freedom]. Its <probability density function>[density] gives the displayed <moments> by a <gamma integral>. In three dimensions, $\mathbb E|\mathbf X|^{2n}=(2n+1)!!v^n$. Radial odd <moments> need not vanish, although any odd <Cartesian product> does.
Back to article page
