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Abstract
In a random sample of size two, we use the relation between the conditional expectation of order statistics and the hazard rate function of the distribution to characterize the discrete mixtures of exponential distribution. In addition, we also n ention some related theorems to characterize the discrete mixtures of exponential distribution. Moreover, when the sample size is n, the above results are also valid. Finally, we give a relative result of the different formula for Nassar [14] and an application to parallel system in engineering practice.