A Note on Characterizations of the Exponential Distribution*

Аннотация

The following classical characterization of the exponential distribution is well known. Let X₁,X₂, . . . X_n be independent and identically distributed random variables. Their common distribution is exponential if and only if random variables X₁ and n min(X₁, . . .,X_n) 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.