Let \(X_{1}, \ldots, X_{n}\) be a set of independent and identically distributed random variables from a \(\operatorname{Poisson}(\theta)\) distribution and let \(\theta\) have a \(\operatorname{Gamma}(\alpha, \beta)\) prior distribution where \(\alpha\) and \(\beta\) are known.

a. Prove that the posterior distribution of \(\theta\) is a \(\operatorname{Gamma}(\tilde{\alpha}, \tilde{\beta})\) distribution where

\[\tilde{\alpha}=\alpha+\sum_{i=1}^{n} Y_{i}\]

and \(\tilde{\beta}=\left(\beta^{-1}+night)^{-1}\).

b. Compute the Bayes estimator of \(\theta\) using the squared error loss function. Is this estimator consistent and asymptotically NORMAL in accordance with Theorem 10.15?

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Theorem 10.15. Suppose that X' = (X1,...,xn) is a vector of independent and identically distributed random variables from a distribution f(x|0), con- ditional on 0, where the prior distribution of 8 is (0) for E. Let r(tx) be the posterior density of n/2 (0-00,n), where 8o,n =00+n-1[1(00)]-11'(00) and is the true value of 0. Suppose that the conditions of Theorem 10.14 hold, then when the loss function is squared error loss, it follows that n1/2(n-00) Zas n where Z is a N{0, [1(00)]} random variable.