Let $f: \R ^d \to \R $ be a normal density with mean $\mu \in \R ^d$ and covariance $\Sigma \in \mathbfsf{S} ^d$.

\[ \mu = \bmat{\mu _x \\ \mu _y} \quad \text{ and } \quad \Sigma = \bmat{ \Sigma _{xx} & \Sigma _{xy} \\ \Sigma _{yx} & \Sigma _{yy} } \]

The conditional density $f_{x \mid y}(\xi , \gamma )$ is $\normal{\bar{\mu }(\gamma )}{\bar{\Sigma }}$ where

\[ \bar{\mu }(\gamma ) = \mu _1 + \Sigma _{xy}\Sigma _{yy}^{-1}(\gamma - \mu _y) \text{ and } \bar{\Sigma } = \Sigma _{xx} - \Sigma _{xy}\Sigma _{yy}^{-1}\Sigma _{yx}. \]