Conditional expectation of three-dimensional Gaussian random variable

Question

Suppose $X_1$ $X_2$ and $X_3$ are multivariate-Gaussian distributed: $$(X_1,X_2,X_3)\sim\mathcal N (0,\Sigma)$$ Is there a simple expression for the following conditional expectation in terms of the matrix entries $\Sigma_{i,j}$? $$\mathbb{E}[X_1X_2|X_3=x],\quad x\in\mathbb{R}$$ Maybe something in similar fashion to this simpler result: $$\mathbb{E}[X_1|X_3=x]=\frac{\Sigma_{1,3}}{\Sigma_{3,3}}x.$$

Answer 1

You can easily extend from the simple result by making the variables into matrix and vectors:$$ \mu_{\mathbf v}=\Sigma_{\mathbf v,3}\Sigma_{3,3}^{-1}\mathbf v $$ where $\mathbf v =[X_1,X_2]^\top$

This is a specific case for conditioning on Multivariate Gaussians which general formula is found on Wikipedia