Negative Binomial as a Poisson–Gamma Mixture

Let $\Lambda \sim \text{Gamma}(r, \beta)$ with density $f_\Lambda(\lambda) = \frac{\beta^r}{\Gamma(r)} \lambda^{r-1} e^{-\beta \lambda}$ for $\lambda > 0$, and let $X \mid \Lambda = \lambda \sim \text{Poisson}(\lambda)$.

(a) Write down $P(X = k \mid \Lambda = \lambda)$ and compute the marginal PMF $P(X = k)$ by integrating over $\Lambda$.

(b) Show that $P(X = k) = \binom{k + r - 1}{k} p^k (1-p)^r$ where $p = \frac{1}{1+\beta}$, and identify the distribution.

(c) Use the mixture representation to find $E[X]$ and $\text{Var}(X)$ via the tower property (law of total expectation and law of total variance), without computing the PMF.

(d) Verify numerically: $r = 3$, $\beta = 4$. Compute $P(X = 2)$ and $E[X]$.