Conditional on , Y has a Poisson distribution with mean Values of vary according to gamma destiny, which has E() , var() 2 k Show that marginally Y has the negative binomial distribution Explain why the negative binomial model is a way to handle overdispersion for the Poisson

Question: Conditional on , Y has a Poisson distribution with mean . Values of vary according to gamma destiny, which has E() = , var()

Conditional on λ, Y has a Poisson distribution with mean λ. Values of λ vary according to gamma destiny, which has E(λ) = µ, var(λ) = µ²/k. Show that marginally Y has the negative binomial distribution. Explain why the negative binomial model is a way to handle overdispersion for the Poisson.

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