Characterization of distributions through failure rate and mean residual life functions

Abstract

Abstract Makino [Makino, 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 μ , E ( 1 / r ( X ) ] ⩾ 1 / μ , 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 .