5.21 The negative binomial distribution is often used to model the number of trials until the r th success. However, this distribution provides an interesting alternative to the Poisson distribution in the presence of overdispersion. The probability function for the negative binomial distribution is />Ο = ('Ι 1 1 )*'( -*Γ Γ

(a) Show that this distribution belongs to the exponential family.

(b) Derive the mean and the variance for this distribution.

(c) Derive the canonical link for this distribution.

(d) Assume that the Poisson mean follows a gamma distribution. Show that the compound distribution becomes the negative binomial.