$X \sim N(0,\sigma_x^2)$ and $Y \sim N(0, \sigma_y^2)$, $\rho(X,Y) = \rho$. Find E[X|Y].
Attempt to the solution: Firstly, I tried to use that $cov(X,Y) = \rho_{XY}\sigma_x\sigma_y = cov(Y, E[X|Y])$, but I didnt manage to find something meaninful. Then I tried to use the definition $E[X|Y]:=\int{ {{f_{XY}(x,y)}\over{_{Y}(y)}}}$, but I am struggling with finding the joint distribution of X and Y. What can I do?
If $X$ and $Y$ are jointly normal then $cov (X-cY,Y)=0$ when $c=\frac {EXY} {EY^{2}}=\frac {\rho} {\sigma_y^{2}}$. This implies that $X-cY$ and $Y$ are independent (since they have a joint normal distribution). Hence $E(X|Y)=E(X-cY|Y)+cE(Y|Y)=0+cY=cY$.