Suppose that \(X\) and \(Y\) have a continuous joint distribution for which the joint p.d.f. is as follows:
\(f_{X,Y}(x,y) = \begin{cases} 2x - \frac{1}{2}y & \text{for } 0 \le x \le 1 \text{ and } 0 \le y \le 2, \\ 0 & \text{otherwise} \end{cases}\)
Find \(E(Y|X)\) and \(Var(Y|X)\). And now suppose that \(X = 0.5\). What is the expected value of \(Y\); also, how much does \(Y\) vary by, on average?