The following classical characterization of the exponential distribution is well known. Let X 1 ,X 2 , . . . X n be independent and identically distributed random variables. Their common distribution is exponential if and only if random variables X 1 and n min(X 1 , . . .,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.
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*This research is supported by the Russian Scientific Foundation, project 14–11–00364
Proceedings of the XXXII International Seminar on Stability Problems for Stochastic Models, Trondheim, Norway, June 16–21, 2014.
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Ushakov, N.G., Ushakov, V.G. A Note on Characterizations of the Exponential Distribution*. J Math Sci 214, 132–138 (2016). https://doi.org/10.1007/s10958-016-2763-8
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DOI: https://doi.org/10.1007/s10958-016-2763-8