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Let Y ⇠ Gamma(alpha,beta) and conditioned on Y = y, X ⇠ Poisson(y). Find the unconditional...

Let Y ⇠ Gamma(alpha,beta) and conditioned on Y = y, X ⇠ Poisson(y).
Find the unconditional distribution of X in the case that alpha = r is an integer and beta=1-p/p
for p in (0, 1). 
Find the conditional distribution of Y|X = x. (Use Bayes’ rule)

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