Consider the following Bayesian model:

\[ \begin{aligned} p&\sim\text{Beta}(a_0,\,b_0)&& \text{(prior)}\\ X_1,\,X_2,\,...,\,X_n\,|\,p&\overset{\text{iid}}{\sim}\text{Geometric}(p).&& \text{(likelihood)} \end{aligned} \]

1. What is the posterior distribution?
2. Compute the posterior mean and show that it is a weighted average of the prior mean and the maximum likelihood estimator.