Suppose that a random variable $\mathbf{X}$ follows a multivariate Normal distribution, i.e. $\mathbf{X}\sim N(\mathbf{\mu},\mathbf{\Sigma})$, where $\mathbf{\mu}=(\mu_1,\ldots,\mu_N)^T$ and $\mathbf{\Sigma}$ is the $(N\times N)$ covariance matrix.
From what i know, we have the following expectations,
- $\mathbb{E}[\mathbf{X}] = \mathbf{\mu}$;
- $\mathbb{E}[\mathbf{X}^T\mathbf{X}] = \mathbf{\mu}^T\mathbf{\mu} + \text{Tr}(\mathbf{\Sigma})$;
- $\mathbb{E}[\mathbf{X}\mathbf{X}^T] = \mathbf{\mu}\mathbf{\mu}^T + \mathbf{\Sigma}$.
However, I cannot figure out how to calculate the expectation, $\mathbb{E}[\mathbf{X}^2_i]$ for $i=1,\ldots,N$. How to compute this?