Let $X \sim N(0,1)$ and let $Y$ be a RV such that $E[Y|X=x] = ax+b$ and $Var [Y|X=x]=1.$
Find $E[XY]$.
I found $E[Y]$ using the law of total expectation, and I found $Var[Y]$ using the law of total variance.
Now I am not sure how to proceed since I do not know the joint density function of $X$ and $Y$.
I was thinking that maybe they were independent and then this becomes simple, however, I am not sure that that they are independent.
Any guidance would be appreciated.