Characterizations of the poisson process using,the variance

Let γ(t) be the residual life at time t of the renewal process {A(t), t > 0}, which has F as the common distribution function of the inter-arrival times. In this article we prove that if Var(γ(t)) is constant, then F will be exponentially or geometrically distributed under the assumption F is continuous or discrete respectively. An application and a related example also are given.