Example 4.1.10. Let $X_1$ and $X_2$ be discrete random variables. Suppose the conditional pmf of $X_1$ given $X_2$ and the marginal distribution of $X_2$ are given by
$p(x_1|x_2) = \binom{x_2}{x_1} (\frac{1}{2})^{x_1}$, $x_1 = 0, 1, \dots, x_2$
$p(x_2) = \frac{2}{3}(\frac{1}{3})^{x_2 - 1}$, $x_2 = 1, 2, 3, \dots$
Find the mgf of $X_1$.