Characterization of distributions through failure rate and mean residual life functions

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 XX with finite mean μμ, E(1/r(X)]⩾1/μE(1/r(X)]⩾1/μ, where r(⋅)r(⋅) is the failure rate function of XX, with equality if and only if XX is exponentially distributed. Here we characterize exponential distribution and Rayleigh distribution through the expected values of r(X)r(X) and e(X)e(X), where e(⋅)e(⋅) is the mean residual life function of XX.