A characterization of probability distributions similar to the exponential

Abstract

Abstract A concept of the lack‐of‐memory property at a given time point c > 0 is introduced. It is equivalent to the concept of the almost‐lack‐of‐memory (ALM) property of the random variables. A representation theorem is given for the cumulative distribution function of such random variables as well as for corresponding decompositions in terms of independent random variables. It is shown that a periodic failure rate for a random variable is equivalent to the ALM property. In addition some properties of the service time of an unreliable server are observed.