40.
Suppose that (X, Y) is a discrete random vector with PMF $p_{X,Y}$,
where
$P((X, Y) = (x, y)) = egin{cases} 0.5 & \text{if } (x, y) = (1, 2); \ 0.4 & \text{if } (x, y) = (0, -3); \ 0.1 & \text{if } (x, y) = (1, -3); \ 0 & \text{otherwise.} \end{cases}$
(a) Find $E(X|Y = y)$ and $Var(X|Y = y)$ for $y \in \{2, -3\}$.
(b) Find $Var(E(X|Y))$, $E(Var(X|Y))$ and $Var(X)$. Verify the
equality
$Var(X) = Var(E(X|Y)) + E(Var(X|Y))$
based on your answers.
