  Suppose that the distribution of $X_a$ belongs to a natural exponential
family concentrated on the nonegative integers and is such that
$\E(z^{X_a})=f(az)/f(a)$. Assume that $\Pr(X_a\leq k)$ has the form $c_k\int_a
^{\infty}u^k\mu(du)$ for some number $c_k$ and some positive measure $\mu,$
both independent of $a.$ 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,
  \textsc{Keywords:} Binomial, Poisson and negative binomial distributions.
Gaussian and Gamma distributions. Moebius transforms. Cross ratio.
