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.


1 Answer 1


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 $\Sigma$.

  • $\begingroup$ 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? $\endgroup$
    – RTbecard
    Nov 7, 2022 at 11:57
  • $\begingroup$ Not that I know, otherwise, obviously, I would have given it! $\endgroup$
    – Xi'an
    Nov 7, 2022 at 12:14

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