Orthogonal polynomials in the Cumulative Ord family and its application to variance bounds

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.