The Poisson process and associated probability distributions on time scales

Duals of probability distributions on continuous (R) domains exist on discrete (Z) domains. The Poisson distribution on R, for example, manifests itself as a binomial distribution on Z. Time scales are a domain generalization in which R and Z are special cases. We formulate a generalized Poisson process on an arbitrary time scale and show that the conventional Poisson distribution on R and binomial distribution on Z are special cases. The waiting times of the generalized Poisson process are used to derive the Erlang distribution on a time scale and, in particular, the exponential distribution on a time scale. The memoryless property of the exponential distribution on R is well known. We find conditions on the time scale which preserve the memorylessness property in the generalized case.