AbstractA distribution admitting the moment generating function (mgf) is a member of one and only one linear exponential family Pμ= {Pω:ω ϵ Ωμ} withdPω(x) = {eω'x/f(ω)}dμ(x). Factorization of the mgf of a member of a linear exponential family into a finite number ofmgf's of linear exponential distributions, implies such a factorization for all members of the family. Consequently, infinite divisibility of themgfof a member of a linear exponential family implies infinite divisibility for all members of t