Let (X_{1}, ldots, X_{n}) be a set of independent and identically distributed random variables from a (operatorname{Poisson}(theta))

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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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