The Expectation of $X^1$ as a Function of $\mathbb{E}(X)$ for an Exponential Family on the Positive Line

Abstract

If the distribution of $X$ belongs to a natural exponential family on the positive real line, this note studies the expectation of the reciprocal of $X$ as a function of the expectation $m$ of $X$ and characterizes the cases where this function is an affine function of $m^{-1}$ as gamma, inverse-Gaussian, Ressel or Abel families.