(Solution) - Suppose that a random sample of size n is taken -(2025 Original AI-Free Solution)

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Suppose that a random sample of size n is taken from a Poisson distribution for which the value of the mean ? is unknown, and the prior distribution of ? is a gamma distribution for which the mean is ?0. Show that the mean of the posterior distribution of ? will be a weighted average having the form ?nn + (1? ?n)?0, and show that ?n ?1 as n??.