On the characterization of survival distributions in reliability by properties of their renewal densities
Let X be a random variable representing the life of a component operating in a system, the sequence of component life lengths forming a renewal process. Then for large values of t, the residual time of faultless operation of the component is given by f (y) = F¯(y)/μ, where F(·) is the distribution function of X, , μ=E(x)<∞ and F(0)=0. Let λ(t) and λ2(t) be the failure rates of X and Y, respectively. In this paper we exhibit an interpretation of λ1(t), in the continuous as well as in the discrete case, and show that if X has an increasing failure rate distribution, so does Y. In addition, some characterizations of the exponential and the geometric distributions are obtained by certain properties of failure rates of X and Y.