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Lattice Path Combinatorics and Applications pp 166–193Cite as

A Review of the Basic Discrete q-Distributions

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Part of the book series: Developments in Mathematics ((DEVM,volume 58))

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 Pólya urn model is constructed and q-Pólya and inverse q-Pólya 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Athens, 15784, Athens, Greece

    Ch. A. Charalambides

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  1. Ch. A. Charalambides
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Correspondence to Ch. A. Charalambides .

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Editors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, PA, USA

    George E. Andrews

  2. Fakultät für Mathematik, Universität Wien, Vienna, Austria

    Christian Krattenthaler

  3. Department of Mathematics and Statistics, California State Polytechnic University, Pomona, Pomona, CA, USA

    Alan Krinik

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Charalambides, C.A. (2019). A Review of the Basic Discrete q-Distributions. In: Andrews, G., Krattenthaler, C., Krinik, A. (eds) Lattice Path Combinatorics and Applications. Developments in Mathematics, vol 58. Springer, Cham. https://doi.org/10.1007/978-3-030-11102-1_9

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