A Measure of Variability Based on the Harmonic Mean, and Its Use in Approximations

Abstract

Let $X$ be a positive random variable and assume that both $a = EX^{-1}$ and $\mu = EX$ are finite. Define $c^2 = 1 - (a\mu)^{-1}$. This quantity serves as a measure of variability for $X$ which is reflected in the behavior of completely monotone functions of $X$. For $g$ completely monotone with $g(0) < \infty$: $0 \leq Eg(X) - g(EX) \leq c^2g(0) \text{and}\operatorname{Var} g(X) \leq c^2g^2(0).$