The Binomial and Negative Binomial Distribution in Discrete Time Markov Chains

L. Frank

2017, v.23, №3, 377-400


The classical results of the binomial and negative binomial probability distribution are generalized by means of homogeneous Discrete Time Markov Chains to series of stochastically independent random trials. These have not only two possible outcomes but two groups of them -- different kinds of successes and failures with occurrence probabilities depending on the outcome of the previous trial. This generalization allows a uniform view of occupation time, first passage time and recurrence time. Our results are consequently derived and presented in matrix form, the probabilities as well as the moments. They can be applied to all Discrete Time Markov Chains, especially in computer capacity planning, performability and economics.

Keywords: discrete time Markov Chains, binomial distribution, matrix representation, recurrence time, first passage time, occupation time.


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