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Bayes Estimators Of Certain Probabilities

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Let \( \displaystyle X_{1}, \dots , X_{n} \) be a random sample from a Poisson distribution with parameter \( \theta \), P(\( \theta \)), where \( \theta \) is the value of a random variable \( \Theta \) with prior probability distribution Exp(1). If the squared-error loss function is our loss function, find the Bayes estimators \( \displaystyle d^{*}_{k} \) of the probabilities \( \displaystyle P_{X|\Theta}(X=k) , \, k \in \mathbb{N}_{0} \).

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