Conditional Expectaction 3 variables Suppose $X,Y$ and $Z$ are multivariate normal with means and full covariance matrix. The conditional expectation $E(X | Y)$ is well know. What is the conditional expectation of $E(X | Y,Z)$ if $Y$ and $Z$ (and $X$) are correlated? Standard textbooks only seem to cover the case when $Y$ and $Z$ are uncorrelated. 
 A: If $\mathbf{x} \in \mathbb{R}^n, \mathbf{y} \in \mathbb{R}^m$ are jointly Gaussian,
\begin{align}
\begin{pmatrix}\mathbf{x} \\ \mathbf{y}\end{pmatrix}
\sim 
\mathcal{N}\left(
\begin{pmatrix} \mathbf{a} \\ \mathbf{b} \end{pmatrix},
\begin{pmatrix} \mathbf{A} & \mathbf{C} \\ \mathbf{C}^\top & \mathbf{B} \end{pmatrix}
 \right),
\end{align}
then (Rasmussen & Williams, 2006, Chapter A.2)
\begin{align}
\mathbf{x} \mid \mathbf{y} \sim \mathcal{N}\left(\mathbf{a} + \mathbf{CB}^{-1}(\mathbf{y} - \mathbf{b}), \mathbf{A} - \mathbf{CB}^{-1}\mathbf{C}^\top \right).
\end{align}
In your case,
\begin{align}
\mathbf{x} &= x, \\
\mathbf{y} &= \begin{pmatrix} y \\ z \end{pmatrix}, \\
\mathbf{a} &= \mu_x, \\
\mathbf{b} &= \begin{pmatrix} \mu_y \\ \mu_z\end{pmatrix}, \\
\mathbf{A} &= \sigma_{xx}^2, \\
\mathbf{B} &= \begin{pmatrix} \sigma_{yy}^2 & \sigma_{yz}^2 \\ \sigma_{zy}^2 & \sigma_{zz}^2 \end{pmatrix},\\
\mathbf{C} &= \begin{pmatrix} \sigma_{xy}^2 & \sigma_{xz}^2 \end{pmatrix},
\end{align}
where $\sigma_{xz}^2$ is the covariance between $x$ and $z$. Hence,
$$E[x \mid y, z] = \mu_x + \begin{pmatrix} \sigma_{xy}^2 & \sigma_{xz}^2 \end{pmatrix}
\begin{pmatrix} \sigma_{yy}^2 & \sigma_{yz}^2 \\ \sigma_{zy}^2 & \sigma_{zz}^2 \end{pmatrix}^{-1} \begin{pmatrix} y - \mu_y \\ z - \mu_z \end{pmatrix}.$$
