q-Multinomial and negative q-multinomial distributions

Abstract

The notion of a Bernoulli trial is extended, by introducing recursively different kinds (ranks, levels) of success and failure, to a Bernoulli trial with chain-composite success (or failure). Then, a stochastic model of a sequence of independent Bernoulli trials with chain-composite successes (or failures) is considered, where the odds (or probability) of success of a certain kind at a trial is assumed to vary geometrically, with rate q, with the number of trials or the number of successes. In this model, the joint distributions of the numbers of successes (or failures) of k kinds up to the nth trial and the joint distributions of the numbers of successes (or failures) of k kinds until the occurrence of the nth failure (or success) of the kth kind, are examined. These discrete q-distributions constitute multivariate extensions of the q-binomial and negative q-binomial distributions of the first and second kind. The q-multinomial and the negative q-multinomial distributions of the first and second kind, for $n\to \infty ,$ can be approximated by a multiple Heine or Euler (q-Poisson) distribution.

Keywords:

• Multiple q-Poisson distributions
• multivariate absorption distributions
• q-multinomial formula
• negative q-multinomial formula

AMS (2000) SUBJECT CLASSIFICATION:

• Primary 60C05
• Secondary 05A30

Acknowledgments

The author expresses his thanks to the referee for his comments and suggestions toward revising this paper.