$E(Cx+Dy | Ax+By=u)$

Question

It is given that $x$ and $y$ are standard normal, iid random variables. It is also given that $Ax+By=u$. Find $E(Cx+Dy | Ax+By=u)$.

Answer 1

Let $z_1 = Cx+Dy$ and $z_2=Ax+By$, such that $(z_1, z_2) \sim \mathcal{N}_2(\mu,\Sigma)$ with $\mu$ a zero vector and $$\Sigma =\left( \begin{array}{cc}C^2+D^2 & AC+DB \\ AC+DB & A^2+B^2 \end{array}\right).$$ Now note that $E(z_1|z_2=u)=\frac{AC+DB}{A^2+B^2}u$. See this page for some explanations on multivariave normality.