A formulation of the lifetime distribution and the existence of its moments.

The lifetime distribution is formulated in terms of g(t), defined as the ratio of the hazard function to the tail probability function, to study the properties of the lifetime distribution. A criterion is provided for the asymptotic behaviour of g(t) and the hazard function. Criteria for the existence and non-existence of the moments of any probability distribution of a non-negative random variable are obtained in terms of the derivatives of g(t). Examples are given to illustrate the use of the criteria and applications made to stochastic models of population growth as well as other lifetime distributions.