# Mean distance from center of multivariate Gaussian distribution

Say I have a multivariate Gaussian distribution and I want to measure the expected vector distance of points from the mean of this distribution. If I know the covariance matrix of my mvtgaussian, is there some closed form solution for this?

I'm aware there is a previous questing asking something very similar, but there were no responses to that one.

If $$X\sim\mathcal N_p(\mu,\Sigma)$$ the expectation $$\mathbb E[||X-\mu||^2]$$ (assuming the standard Euclidean distance as the chosen distance) is available as $$\mathbb E[||X-\mu||^2]= \mathbb E[\text{tr}((X-\mu)(X-\mu)^\top)] =\text{tr}\{\mathbb E[((X-\mu)(X-\mu)^\top]\}=\text{tr}\{\Sigma\}$$ The random variable $$||X-\mu||$$ is distributed as the squared root of a weighted sum of $$\chi^2_1$$ variates, the weights being the eigenvalues of $$\Sigma$$.
• This makes sense, and gives me the distribution of $|| X - \mu ||$ as the square root of a mixture of chi squares. However, how do I then get the expected value of that squared mixed chi squared distribution? Is there a solution to that?