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Consider an exponential family F on the set of non-negative integers indexed by the parameter a. The cumulative distribution function of an element of F estimated on k is both a function of a and k. Assume that the derivative of this function with respect to a is the product of three things: a function of k,  a function of a and the function a to the power k. We show that this assumption implies that the exponential family is either a binomial, or the Poisson, or a negative binomial family. Next, we study an analogous property for continuous distributions and we find that it is satisfied if and only the families are either Gaussian or Gamma. Ultimately, the proofs rely on the fact that only Möbius functions preserve the cross ratio.

Cumulative distribution functions for the five simplest natural exponential families