For discrete distributions, Jrgensen (1987) showed that it is natural to define the exponential dispersion family as

For discrete distributions, Jørgensen (1987) showed that it is natural to define the exponential dispersion family as f(yi; ????i, ????) = exp[yi????i − b(????i)∕a(????) + c(yi, ????)].

a. For fixed k, show that the negative binomial distribution (7.8) has this form with ????i = log[????i∕(????i + k)], b(????i) = − log(1 − e????i), and a(????) = 1∕k.

b. For this version, show that xi = yia(????) has the usual exponential dispersion family form (4.1).