Covariance of normally distributed random variables

Question

If $ X \sim N(0,1) $ and given $ X = x $ then $ Y \sim N(x,1) $

I want to find the $ Cov(X,Y) $ using the relationship stated above.

My attempt:

$ Cov(X,Y) = E[XY] - E[X]E[Y] \\ E[X] = 0\\ Cov(X,Y) = E[XY] \\ E[XY] = E[E[XY|X=x]]$

I am not sure how to proceed from there.. Do I integrate the joint distribution?

Answer 1

$$E[XY] = E[XE[Y|X]] $$ Now as $E[Y|X] = X$: $$ = E[X^2] = 1 $$ and $$ E[X]E[Y] = 0 $$

you get $cov(X,Y) = 1$.

Answer 2

You should be careful with your conditioning. What is true that $E[XY] = E[E[XY|X]]$, and not $E[XY]=E[E[XY|X=x]]$.

So with $E[XY] = E[E[XY|X]]$, you can integrate with respect to the distribution of $X$.

$E[E[XY|X]] = \int E[XY|X=x]f_X(x)dx.$