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.