Thefamilyofprobabilitydistributionshaving pdf of form f(y; ) = B()h(y) exp[Q()R(y)] is calledthe exponentialfamily. The naturalexponentialfamily is thespecialcase Q()

Thefamilyofprobabilitydistributionshaving pdf of form f(y; θ) = B(θ)h(y) exp[Q(θ)R(y)]

is calledthe exponentialfamily. The naturalexponentialfamily is thespecialcase Q(θ) = θ

and R(y) = y, with θ called the naturalparameter.

(a) Showthatthebinomialdistributionisinthenaturalexponentialfamilywithnatural parameter θ = log[π~(1 − π)], the logit. (Otherdistributionsinthisfamilyincludethe Poisson,gammaandexponentialwithparameter λ, geometric,normalwithparameter

μ; onesnotinthefamilyincludetheuniformwithsupport [0, θ] and theCauchywith median θ, f(y; θ) = 1~{π[1 + (y − θ)2]}.)

(b) Forindependent {yi} from anaturalexponentialfamily,findthelikelihoodfunction.Show that asufficientstatisticfor θ is Σi Yi. (Itsdistributionisalsointheexponentialfamily.)25