I am trying to understand and master simple tricks with conditional expectation computations.
Consider a random variable U with $E(U)=C_1$ and $\text{Var}(U)=\sigma^2$. Let $V$ be another random variable with a linear conditional mean and constant conditional variance: $$ E(V\mid U)=A+BU \\ \text{Var}(V\mid U)=\sigma_{V\mid U}^2 $$ Where $A,B,\sigma \text{ and } \sigma_{V\mid U}$ are constants.
I managed to find $E(V), \text{Var}(V)$ etc. But I am unable to compute the Covariance of $U$ and $V$: When applying the definition: $Cov(U,V)=E(UV)-E(U)E(V)$ I am not able to solve $E(UV)$ or to see how to approach the solution. I realise there exist countless similar type of exercices on conditional expectations with their solutions but since $U$ and $V$ are not independent I cannot apply the usual tricks.
PS: I try to apply $E(UV)=E(E(UV\mid V))$ but what really is an obstacle for me, is to understand how to go from $E(V\mid U)$ to $E(UV\mid U)$. Once I will understand that step I will have a better grasp on the topic.
Any idea or advice toward the solution would be much appreciated, thank you!
Apply:$$\mathbb E[UV]=\mathbb E[\mathbb E(UV|U)]=\mathbb E[U\mathbb E(V|U)]$$