Extended Poisson process modelling and analysis of count data

It is shown that any discrete distribution with non-negative support has a representation in terms of an extended Poisson process (or pure birth process). A particular extension of the simple Poisson process is proposed: one that admits a variety of distributions; the equations for such processes may be readily solved numerically. An analytical approximation for the solution is given, leading to approximate mean-variance relationships. The resulting distributions are then applied to analyses of some biological data-sets.