  By using the point model of reaction kinetics we have studied the stochastic
properties of the lifetime of small systems controlled by autocatalytic
reaction A+X -> X+X -> A+X, X -> B. Assuming that a system is living only when
the number of autocatalytic particles is larger than zero but smaller than a
positive integer N, we have calculated the probability of the lifetime provided
that the number of substrate particles A is kept constant by a suitable
reservoir, and the end-products B do not take part in the reaction. We have
shown that the density function of the lifetime is strongly asymmetric and in
certain cases it has a well-defined minimum at the beginning of the process. It
has been also proven that the extinction probability of systems of this type is
exactly 1.
