<s>[INST] Calculating conditional expectation and variance of multivariate normal 
Suppose $\mathbf{Y} = \begin{bmatrix} Y_1 \\ Y_2 \\ Y_3 \end{bmatrix}
 \sim N(\mu, \Sigma)$ where $\mu = \begin{bmatrix} 1 \\ 2 \\ 3
 \end{bmatrix}$ and $\Sigma = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 5 & 1
 \\ 1 & 1 & 3 \end{bmatrix}$. Calculate $\mathbb{E}[Y_3 | Y_1 = y_1,
 Y_2 = y_2]$ and $\mathbb{V}[Y_3 | Y_1 = y_1, Y_2 = y_2]$.

I know the general formula in the case when: $\mathbf{Y} = \begin{bmatrix} \mathbf{Y}_1 \\ \mathbf{Y}_2 \end{bmatrix} \sim N\left(\begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}, \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{bmatrix} \right)$ then $\mathbf{Y}_2 | \mathbf{Y}_1 = \mathbf{y}_1 \sim N(\mu_2 + \Sigma_{21} \Sigma_{11}^{-1} ( \mathbf{y}_1 - \mu_1), \Sigma_{22} - \Sigma_{21} \Sigma_{11}^{-1} \Sigma_{12})$. However, I am unsure of how to apply it here, could someone please show me?
 [/INST] Partition the multivariate normal random vector 
$\mathbf{Y}$
consisting of two subvectors as 
$\mathbf{Y}=\left(\begin{array}{c}
\mathbf{Y^{(1)}}\\
\mathbf{Y^{(2)}}\\
\end{array} \right)$
where 
$\mathbf{Y^{(1)}}=\left(\begin{array}{c}
Y_{1}\\
Y_{2}\\
\end{array} \right) $
and 
$\mathbf{Y^{(2)}}=\left(\begin{array}{c}
Y_{3}
\end{array} \right)$. 
Accordingly partition the mean vector  as
 $\mathbf{\mu}=\left(\begin{array}{c}
\mathbf{\mu^{(1)}}\\
\mathbf{\mu^{(2)}}\\
\end{array} \right)$
where 
$\mathbf{\mu^{(1)}}=\left(\begin{array}{c}
\mu_{1}\\
\mu_{2}\\
\end{array} \right) =\left(\begin{array}{c}
1\\
2\\
\end{array} \right)$
and 
$\mathbf{\mu^{(2)}}=\left(\begin{array}{c}
\mu_{3}
\end{array} \right) = 3$.
and variance-covariance matrix as
$\Sigma = \left(\begin{array}{ccc}
2 & -1 & 1\\
-1 & 5 & 1\\
1 &  1 & 3\\
\end{array} \right) = \left(\begin{array}{cc|c}
2 & -1 & 1\\
-1 & 5 & 1\\
\hline
1 &  1 & 3\\
\end{array} \right) = \left(\begin{array}{cc}
\Sigma_{11}&\Sigma_{12}\\
\Sigma_{21}&\Sigma_{22}\\
\end{array} \right)$
Plug-in the values in the expressions for mean and variance you have stated above to get the solution.
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