Conditional expectation based on two bivariate normal RVs?

Question

Suppose $X$ and $Y$ are bivariate normal, both with mean 0 and sd 1. Considering the correlation $\rho$, what should $E[X|X,Y]$ be? I think the answer should be just the RV $X$ as we conditioned $X$ itself. But is it possible that, since $Y$ and $X$ are correlated, it has some effect on the conditional expectation?

Thanks!

Answer 1

HINT: It doesn't care what is the relation between $X$ and $Y$. By definition you have that

$$ E[X|Z]:=E[X|\sigma(Z)] $$

In your case $Z=(X,Y)$. Thus it is enough to prove that $\sigma (X)\subset \sigma (X,Y)$

Answer 2

$E(X|X,Y)=X$ by definition of conditional expectation.