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The Poisson process and associated probability distributions on time scales

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3 Author(s)
Poulsen, D.R. ; Dept. of Math., Baylor Univ., Waco, TX, USA ; Spivey, M.Z. ; Marks, R.J.

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.

Published in:

System Theory (SSST), 2011 IEEE 43rd Southeastern Symposium on

Date of Conference:

14-16 March 2011

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