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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.
Date of Conference: 14-16 March 2011