If $(X,Y,Z)$ is jointly normal, what is the distribution of $(X,Y)\mid Z$?

Question

Let $(X,Y,Z)$ be a jointly normal random vector. I am interested in the distribution of $(X,Y)\mid Z$. It is normal obviously, but what is its mean and variance matrix in terms of the mean and variance matrix of $(X,Y,Z)$ ?

Answer 1

Suppose \begin{align} & \left[ \begin{array}{c} U \\ V \end{array} \right] \sim \operatorname N_{p+q} \left( \left[ \begin{array}{c} \mu \\ \nu \end{array} \right], \left[ \begin{array}{lc} A & B \\ B^\top & C \end{array} \right] \right) \\[8pt] \text{where } & U,\mu\in\mathbb R^{p\times1}, \quad V,\nu\in\mathbb R^{q\times1}, \\ & A\in\mathbb R^{p\times p}, \quad B\in\mathbb R^{p\times q}, \quad C\in\mathbb R^{q\times q}. \end{align} Then $$ \operatorname E(U\mid V) = \mu + B C^{-1}(V-\nu) $$ $$ \operatorname{var}(U\mid V) = A - BC^{-1} B^\top. $$ The way to prove this is somewhat interesting but more work than I'm going to do at this hour.

As applied to your three jointly normal random variables, it says this: $$ \operatorname E \left( \left[ \begin{array}{c} X \\ Y \end{array} \right] \,\Big|\, V \right) = \left[ \begin{array}{c} \operatorname EX \\ \operatorname EY \end{array} \right] + \left[ \begin{array}{c} \frac{\operatorname{cov}(X,Z)}{\operatorname{var} Z} \\[4pt] \frac{\operatorname{cov}(Y,Z)}{\operatorname{var} Z} \end{array} \right] (Z-\operatorname EZ) $$

\begin{align} & \operatorname{var} \left( \left[ \begin{array}{c} X \\ Y \end{array} \right] \,\Big|\, Z \right) \\[10pt] = {} & \left[ \begin{array}{cc} \operatorname{var} X & \operatorname{cov}(X,Y) \\ \operatorname{cov}(X,Y) & \operatorname{var}(Y) \end{array} \right] - \frac 1 {\operatorname{var}Z} \left[ \begin{array}{cc} \big(\operatorname{cov}(X,Z)\big)^2 & \operatorname{cov}(X,Z)\operatorname{cov}(Y,Z) \\ \operatorname{cov}(X,Z)\operatorname{cov}(Y,Z) & \big( \operatorname{cov}(Y,Z) \big)^2 \end{array} \right]. \end{align}