Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
$\E(z^{X_a})=f(az)/f(a)$. 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.

33
The categories of this paper are: plasm-ph
30
The categories of this paper are: q-alg
22
The categories of this paper are: chao-dyn
34
The categories of this paper are: chem-ph
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
17
The categories of this paper are: q-fin
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th