Let P be a random variable having pdf
\begin{equation*}
f(p) = \begin{cases} 3p^2 & 0 < p < 1\\ 0 & \text{otherwise} \end{cases}
\end{equation*}
Suppose that given $P = p$, X and Y are conditionally independent discrete random
variables with
$P(X = k|P = p) = P(Y = k|P = p) = p(1-p)^k$,
$k = 0, 1, 2, ...$
(a) (10 points) Obtain the pmf of X, i.e., $P(X = k)$, $k = 0, 1, 2, ...$
(b) (10 points) Obtain the pmf of $Z = \text{max}\{X, Y\}$.
(c) (10 points) Obtain the conditional pdf of P given $X = x$ and $Y = y$, $x, y = 0, 1, 2, ...$
