Question: Suppose that \(X_{1}, \ldots, X_{n}\) is an iid random sample from the uniform distribution on the interval \((0, \theta), \theta>0\). Consider the following prior distribution for \(\theta\) :

\(\pi_{a, b}(\theta)=\left\{\begin{array}{ll}b a^{b} \theta^{-(b+1)} & \theta>a \\ 0 & \theta \leq a\end{array},\right.\),

where \(a, b>0\).

1. Show that the family of prior distributions \(\left\{\pi_{a, b}(\theta), a, b>0\right\}\) is a conjugate family of prior distributions.
2. Find the posterior mean and posterior variance for \(\theta\) given the prior \(\pi_{a, b}(\theta)\) and the sample \(X_{1}, \ldots, X_{n}\).

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