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In this paper, we investigate some new properties of the mean vitality function (MVF) of a random variable, proposed by A. Toomaj, M. Doostparast, J. Stat. Theory Appl. 13 (2013), 189–195. Specifically, we explore properties of MVF and study under what conditions the MVF of the first-order statistics can uniquely determines the parent distribution. We show that in all distributions the Weibull family, which is commonly used in several fields of applied probability, is characterized through the ratio of the MVF of the first-order statistics to its expectation.

Some Characterization Results Based on the Mean Vitality Function of the First-Order Statistics