Question: Let Z Gamma(a, ), where a is an integer. Conditional on Z = z, let X Pois(z). Show that X has a negative

Let Z ∼ Gamma(a, λ), where a is an integer. Conditional on Z = z, let X ∼ Pois(z). Show that X has a negative binomial distribution with the following interpretation. For k = 0, 1, . . . , P(X = k) is equal to the number of failures before a successes in a sequence of i.i.d. Bernoulli trials with success parameter λ/(1 + λ).

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