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 X with finite mean [mu], E(1/r(X)][greater-or-equal, slanted]1/[mu], where r([dot operator]) 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([dot operator]) is the mean residual life function of X.