Bivariate-Normal Conditional Expectation

Question

$X$ and $Y$ are iid standard normal random variables. Assume $a, b, c, d$ and $u$ are constants. Calculate $E( cX + dY | aX + bY = u)$

Answer 1

Let \begin{align*} \beta = \frac{ac+bd}{a^2+b^2}. \end{align*} Then $cX+dY - \beta(aX+bY)$ and $aX+bY$ are independent. Therefore, \begin{align*} E( cX + dY \mid aX + bY = u)= \beta u. \end{align*}