Linear combination of Gumbel random variables

Abstract

The recent paper by Loaiciga and Leipnik (Stoch Environ Res Risk Environ 13:251-259, 1999) derived the probability distribution of the sum of two independent Gumbel random variables. The results given are of little practical use because they are given in terms of characteristic functions. In this note, we consider the more general problem of deriving the linear combination of two independent Gumbel random variables. Explicit expressions are given for the probability density function and the cumulative distribution function of the linear combination. Various particular cases are also considered. [PUBLICATION ABSTRACT]