[Math] Expected value of norm of multivariate normal distribution random vector

Question

Let $X$ is a random vector size $p$ from multivariate normal distribution $\mathcal{N}$($0$, $\sigma$ $I$), $I$ is identity matrix.
I want to find the expected value of reciprocal of norm like this $$E\left(\lVert x \rVert^{-1}\right)$$
I have tried to search for this in many statistics literature but didn't find anything.
I hope someone can help me to solve this problem.

Answer 1

Let $Y = \lVert X \rVert^2$. Then $Y/\sigma^2 \sim \chi^2_p$. And you wish to find $EY^{-1/2}$. This is $$ \frac{1}{\sigma}\cdot\dfrac{1}{2^{p/2}\Gamma(p/2)}\int_0^\infty y^{-1/2} y^{p/2 - 1} e^{-y/2}dy$$ $$ = \frac{1}{\sigma} \dfrac{2^{(p-1)/2}\Gamma((p-1)/2)}{2^{p/2}\Gamma(p/2)} = \frac{1}{\sigma\sqrt{2}}\dfrac{\Gamma((p-1)/2)}{\Gamma(p/2)}$$