Cumulative distribution functions for the five simplest natural exponential families

Suppose that the distribution of $X_a$ belongs to a natural exponential family concentrated on the nonegative integers and is such that $\text{E}(z^{X_a})=f(az)\mathrm{/}f(a)$. Assume that $\mathrm{Pr}(X_a\le k)$ has the form $c_k\int_a ^{\infty}u^k\mu(du)$forsomenumber$c_k$andsomepositivemeasure$\mu,$bothindependentof$a.$ We show that this asumption 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 Moebius functions preserve the cross ratio, $\text{textsc}Keywords:$ Binomial, Poisson and negative binomial distributions. Gaussian and Gamma distributions. Moebius transforms. Cross ratio.