Expectation of the product of two jointly Gaussian random variables

Question

This is an extension to the previous popular question. Say I have two joint Gaussian random variables $$ (Z_1,Z_2)\sim N(0,\Sigma), $$ where $\Sigma\in\mathbb{R}^{2\times 2}$. What the precise formula for $\mathbb{E}[Z_1Z_2]$? Thanks.

Answer 1

$\mathbb E\left[Z_1Z_2\right] = \text{Cov}\left[Z_1, Z_2\right] + \mathbb E\left[Z_1\right]\mathbb E\left[Z_2\right] = \Sigma_{1,2} + 0 = \Sigma_{1,2}$