A Review of the Basic Discrete q-Distributions

Abstract

Consider a sequence of independent Bernoulli trials and assume that the probability (or odds) of success (or the probability (or odds) of failure) at a trial varies (increases or decreases) geometrically, with rate q, either with the number of trials or with the number of successes. Let \(X_n\) be the number of successes up the nth trial and \(W_n\) (or \(T_k\)) be the number of failures (or trials) until the occurrence of the nth (or kth) success. The distributions of these random variables turned out to be q-analogues of the binomial and negative binomial (or Pascal) distributions. The Heine and Euler distributions, which are q-analogues of the Poisson distribution, are obtained as limiting distributions of q-binomial distributions (or negative q-binomial distributions), as the number of trials (or the number of successes) tends to infinity. Also, introducing the notion of a q-drawing of a ball from an urn containing balls of various kinds, a q-analogue of the Polya urn model is constructed and q-Polya and inverse q-Polya distributions are examined. Finally, considering a stochastic model that is developing in time or space, in which events (successes) may occur at continuous points, a Heine and an Euler stochastic processes are presented.