Conditional mean formula

Question

Let's consider three random variables $X,Y,Z \sim N(0,1)$ and the correlation matrix is $$\Sigma =\begin{pmatrix} 1 & e^{-1} & e^{-2} \\ e^{-1} & 1 & e^{-1} \\ e^{-2} & e^{-1} & 1\end{pmatrix}.$$ I would like to find $E(X|Y=y,Z=z)$ and the formula is $$E(X|Y=y,Z=z)=EX+ \Sigma\ \Sigma_{YZ}^{-1}((y,z)-(EY,EZ)).$$ Perhaps I don't quite understand the formula but the data correlation matrix $\Sigma_{YX}$ is a $2\times 2$ matrix, while $\Sigma$ is a $3\times 3$ matrix. How is the formula applied in this case, where we are conditioning on two random variables?

Answer 1

There are two typos in your formula, it should read $$E(X|Y=y,Z=z)=EX+ \Sigma_{\color{red}{X,YZ}}\ \Sigma_{YZ}^{-1}\left(\color{red}{\left(\matrix{y\\ z}\right)-\left(\matrix{EY\\ EZ}\right)}\right).$$ where in your case $\Sigma_{{X,YZ}}$ stands for the $1\times 2$ matrix $\left(\matrix{e^{-1} & e^{-2}}\right)$.

It is also important to note that, as highlighted, the vectors on the right are column vectors.

See Conditional distributions from Multivariate Normal Distributions for more information.