Bounds for factorial moments of discrete distributions

Abstract

The arithmetic mean $(A M)$ and geometric mean $(G M)$ of a set of non-negative real numbers $a_{1}, a_{2}, \dots, a_{n}$ are defined by $A M = \frac{1}{n} (a_{1} + a_{2} + \dots + a_{n}$ and $G M = {(a_{1}, a_{2} \dots a_{n})}^{\frac{1}{n}}$ with equality if and only if $a_{1} = a_{2} = \dots = a_{n}$ . This paper uses the well known algebraic inequality $G M ⩽ A M$ to derive bounds for factorial moments of a number of discrete probability distributions. As a consequence, a relationship is derived between various factorial moments and the factorial polynomial. We consider upper bounds for factorial moments. It is interesting to see that some of these bounds reduce to bounds for the factorial polynomial and bounds for $n!$ .

Keywords

Discrete probability distribution factorial moment factorial polynomial