$X$ and $Y$ are two random variables drawn independently from standard normal distribution $\mathcal{N}(0,1)$. Set $Z:=3X+4Y$. What is the conditional expectation of $X$ given $Z$, i.e., $E[X|Z]?$
By symmetry, I guess $E[X|Z] = \frac{1}{7}Z$. Not sure if it is correct..
More generally, if $X \sim\mathcal{N}(0,\sigma_1^2)$ and $Y \sim\mathcal{N}(0,\sigma_2^2)$ are independent, any ideas how to calculate $E[X|X+Y]$?