On the dependence of X1+ X2and X1/X2

Abstract

A well-known characterization of the family of gamma distributions is that if X1 and X2 are Independent positive random variables, then X1 + X2 and X1/X2 are independent if and only if X1 and X2 have gamma distributions with the same scale parameter. We describe the dependence of X1 + X2 and X1/X2 for certain classes of distributions, which include the common models used for positive random variables of the continuous type. A method for testing the hypothesis that a random sample comes from a gamma distribution is proposed.