# Conditional Expectation of a normal distribution [duplicate]

say we have a multivariate normal distribution with $${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$$

The conditional expection is $$\overline{\boldsymbol\mu}=\boldsymbol\mu_1+\Sigma_{12}{\Sigma_{22}}^{-1}({\boldsymbol a}-\boldsymbol\mu_2)$$

Is this saying that for each value of $$a$$ in a dataset you minus the mean and multiply by the covariance matrices to obtain the conditional expectation or is it a double summation of $$a - \boldsymbol\mu_2$$? I am slightly confused as to what the terms actually are and how to calculate them.

• @Xi'an Hi, thanks for link. I think I understand. So it is the sub matrices multiplying these values? Apr 17 '21 at 12:37