Given the constants $\{a,b,c,d,e,f$}, I want to compute the conditional mean $\text{E}[Z|S_1,S_2]$ and the conditional variance $\text{Var}[Z|S_1,S_2]$, with:
$Z=a+bX_1+cX_2+dY_1+eY_2+fY_3$
Is the following true?
$\text{E}[Z|S_1,S_2]=a+b\text{E}[X_1|S_1,S_2]+c\text{E}[X_2|S_1,S_2]$
and
$\text{Var}[Z|S_1,S_2]=b^2\text{Var}[X_1|S_1,S_2]+c^2\text{Var}[X_2|S_1,S_2]+d^2\sigma_{Y_1}^2+e^2\sigma_{Y_2}^2+f^2\sigma_{Y_3}^2+bc\text{Cov}[X_1,X_2|S_1,S_2]+2de\text{Cov}[Y_1,Y_2]+2df\text{Cov}[Y_1,Y_3]+2ef\text{Cov}[Y_2,Y_3]$
where $\text{Cov}[X_1,X_2|S_1,S_2]=\text{E}[X_1X_2|S_1,S_2]-\text{E}[X_1|S_1,S_2]\text{E}[X_2|S_1,S_2]$
Assume $S_1=X_1+\epsilon_{X_1}, S_2=X_2+\epsilon_{X_2}$ (where $\epsilon_{X_1}\sim \mathcal N(0,\sigma_{\epsilon_{X_1}}^2)$ and $\epsilon_{X_2}\sim \mathcal N(0,\sigma_{\epsilon_{X_2}}^2)$) and the following joint distributions:
$\begin{pmatrix} X_1 \\ X_2 \end{pmatrix}$ $\sim \mathcal N$ $\bigg(\begin{pmatrix} \mu_{X_1} \\ \mu_{X_2} \end{pmatrix}, \begin{pmatrix} \sigma_{X_1}^2 & \rho_{X_1X_2}\sigma_{X_1}\sigma_{X_2}\\ * & \sigma_{X_2}^2 \end{pmatrix}\bigg)$
$\begin{pmatrix} Y_1 \\ Y_2 \\ Y_3 \end{pmatrix}$ $\sim \mathcal N$ $\Bigg(\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} \sigma_{Y_1}^2 & \rho_{Y_1Y_2}\sigma_{Y_1}\sigma_{Y_2} & \rho_{Y_1Y_3}\sigma_{Y_1}\sigma_{Y_3}\\ * & \sigma_{Y_2}^2 & \rho_{Y_2Y_3}\sigma_{Y_2}\sigma_{Y_3}\\ * & * & \sigma_{Y_3}^2 \end{pmatrix}\Bigg)$
Assume also that $(X_1,X_2)$ and $(S_1,S_2)$ are independent from $(Y_1,Y_2,Y_3)$.
