Mean of squared $ L_2 $ norm of Gaussian random vector

Question

In my studies of probability, I have recently came across the following task:

Let us assume we have an $N$ dimensional Gaussian random vector $ X $ with zero mean and known (not necessarily diagonal) covariance matrix $ \Sigma $. I am interested in the mean of the following random variable $ \lvert \lvert X \rvert \rvert _2 ^2 $. In simple words, is there an expression for the mean of the squared $ L_2 $ norm of an $ N $ dimensional Gaussian random vector with general covariance matrix?

I certainly appreciate all help on this.

Answer 1

The expectation of sum equals to the sum of expectations whenever they are exist: $$E\lvert \lvert X \rvert \rvert _2 ^2=E[X_1^2+\ldots+X_N^2]=E[X_1^2]+\ldots+E[X_N^2] = \text{Var}[X_1]+\ldots+\text{Var}[X_N]=\text{tr}(\Sigma).$$

Here $\text{tr}(\Sigma)$ is the trace of covariance matrix, i.e. the sum of the elements on the main diagonal of $\Sigma$. The answer is not affected by any covariance.