Recall this infernal distribution family that you met on Problem Set 6, Midterm 2, and again in lecture:

\[ f(x\mid \theta)=\theta(x+1)^{-(\theta+1)},\quad x>0. \]

We’ve done MLE for this, now let’s go Bayes. Consider this model:

\[ \begin{aligned} \theta&\sim\text{Gamma}(a_0,\,b_0)\\ X_{1:n}\mid\theta&\overset{\text{iid}}{\sim}f(x\mid \theta). \end{aligned} \]

1. What is the posterior distribution for \(\theta\) conditional on the data?
2. Derive the posterior mean and show that it is a convex combination of the prior mean and the MLE.