High dimensional spherical Gaussian

Question

I would like to know how to find the expected squared distance to the origin of a $d$-dimensional spherical Gaussian centered at the origin with variance $\sigma^2$. Thanks for the help

Answer 1

By using spherical coordinates it is a one dimensional problem, where the density function is $f(r)=Kr^{d-1}e^{-\frac{r^2}{2\sigma^2}}$ for $r\ge0$ The expected square distance is then $\frac{\int\limits_0^\infty r^2f(r)dr}{\int\limits_0^\infty f(r)dr}=d\sigma^2$.