Characterization of distributions through failure rate and mean residual life functions

Makino [Makin, T., 1984. Mean hazard rate and its applications to the normal approximation of the Weibull distribution. Naval Research Logistics Quarterly 31, 1-8] proves that, for any random variable X with finite mean mu, E(1/r(X)] >= 1/mu, where r(.) is the failure rate function of X, with equality if and only if X is exponentially distributed. Here we characterize exponential distribution and Rayleigh distribution through the expected values of r(X) and e(X), where e(.) is the mean residual life function of X. (C) 2010 Elsevier B.V. All rights reserved.