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Discrete distributions from a Markov chain

A discrete-time stochastic process derived from a model of basketball is used to generalize any discrete distribution. The generalized distributions can have one or two more parameters than the parent distribution. Those derived from binomial, Poisson and negative binomial distributions can be underdispersed or overdispersed. The mean can be simply expressed in terms of model parameters, thus making inference for the mean straightforward. Probabilities can be quickly computed, enabling likelihood-based inference. Random number generation is also straightforward. The properties of some of the new distributions are described and their use is illustrated with examples.