How can I get $\mathbf{Cov}(X,Y)$ from those conditions?

Question

\begin{align*} \mathbf{E}(Y)&=1, \\ \mathbf{Var}(X)&=1, \\ \mathbf{E}(YㅣX)&= 1+X, \\ \mathbf{E}(\mathbf{Var}(YㅣX))&=2. \end{align*}

From these conditions, I found that $\mathbf{E}(Y)=1$, $\mathbf{E}(X)=0$, $\mathbf{E}(Y^2)=4$ , $\mathbf{E}(X^2)=1$ and I know that $\mathbf{Cov}(X,Y)=\mathbf{E}(XY)-\mathbf{E}(X)\mathbf{E}(Y)$.

How can I get $\mathbf{E}(XY)$ to find $\mathbf{Cov}(X,Y)$?

Thanks for reading.

Answer 1

You may use the Law of Total Expectation: $$\mathsf E(XY)~{=\mathsf E(\mathsf E(XY\mid X))\\~\vdots}$$

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Alternatively, you may use the Law of Total Covariance:$$\mathsf{Cov}(X,Y)=\mathsf{Cov}\big(\mathsf E(X\mid X),\mathsf E(Y\mid X)\big)+\mathsf E\big(\mathsf{Cov}(X,Y\mid X)\big)$$