2. (26 pts) Consider tracking n vaccinated individuals, where each individual is independent. The outcome $X_i$ for the $i^{th}$ individual is either contracting the disease or remaining disease-free. Let the probability of contracting the disease after vaccination be $p$.
(a) Find the distribution of $\sum_{i=1}^{n} X_i$, the total number of individuals who contract the disease. (4 pts)
(b) Show that $\sum_{i=1}^{n} X_i$ is a sufficient statistic of $p$. (5 pts)
(c) Argue that the sample proportion of individuals who contract the disease, denoted by $\hat{p}$, is also a sufficient statistic for $p$. (3 pts)
(d) Assuming the prior distribution of $p$ follows a Beta($\alpha$, $\beta$), derive the posterior distribution of $p$. (5 pts)
(e) Using a squared error loss function, determine the Bayesian estimation of $p$. (4 pts)
(f) Assume $p = 0.00001$ and $n = 1,000$. Use Poisson distribution to approximate the probability that exactly one individual out of 1,000 contracts the disease. (5 pts)