A Note on Characterizations of the Exponential Distribution*

The following classical characterization of the exponential distribution is well known. Let X1,X2,.. Xn be independent and identically distributed random variables. Their common distribution is exponential if and only if random variables X1 and n min(X1,..,Xn) have the same distribution. In this note we show that the characterization can be substantially simplified if the exponentiality is characterized within a broad family of distributions that includes, in particular, gamma, Weibull and generalized exponential distributions. Then the necessary and sufficient condition is the equality only expectations of these variables. A similar characterization holds for the maximum. © 2016, Springer Science+Business Media New York.