An alternative negative binomial parameterization results from the gamma density formula,

for which E(λ) = µ, var(λ) = µ/k. Show that this gamma mixture of Poissons yields a negative binomial with

E(Y) = µ, var(Y) = µ(1 + k)/k.

For what limiting value of k does this reduce to the Poisson?

Transcribed Image Text:

(k)** r (kμ) f (A; k, μ) - exp ( -kλ) λέμ-1, λ2 0,