  This article presents and reviews several basic properties of the Cumulative
Ord family of distributions; this family contains all the commonly used
discrete distributions. A complete classification of the Ord family of
probability mass functions is related to the orthogonality of the corresponding
Rodrigues polynomials. Also, for any random variable $X$ of this family and for
any suitable function $g$ in $L^2(\mathbb{R},X)$, the article provides useful
relationships between the Fourier coefficients of $g$ (with respect to the
orthonormal polynomial system associated to $X$) and the Fourier coefficients
of the forward difference of $g$ (with respect to another system of
polynomials, orthonormal with respect to another distribution of the system).
Finally, using these properties, a class of bounds for the variance of $g(X)$
is obtained, in terms of the forward differences of $g$. These bounds unify and
improve several existing results.
