Characterization results have great importance in statistics and probability applications. New characterizations of Exponential and Power Function distributions are presented using the s th conditional expectation of order statistics in
terms of their failure (hazard) rate. Our results generalize some of the known results of Ahsanullah (2009). A simulation study has been conducted to help an engineer or a practitioner to check whether the underlying distribution belongs to the
hypothesized family.
Keywords: Characterization; Failure Rate; Conditional Expectation; Order Statistics; Exponential; Power Function Distributions.