Discrete q-distributions on Bernoulli trials with a geometrically varying success probability

Consider a sequence of independent Bernoulli trials and assume that the odds of success (or failure) or the probability of success (or failure) at the ith trial varies (increases or decreases) geometrically with rate (proportion) q, for increasing i=1,2,... Introducing the notion of a geometric sequence of trials as a sequence of Bernoulli trials, with constant probability, that is terminated with the occurrence of the first success, a useful stochastic model is constructed. Specifically, consider a sequence of independent geometric sequences of trials and assume that the probability of success at the jth geometric sequence varies (increases or decreases) geometrically with rate (proportion) q, for increasing j=1,2,... On both models, let Xn be the number of successes up the nth trial and Tk (or Wk) be the number of trials (or failures) until the occurrence of the kth success. The distributions of these random variables turned out to be q-analogues of the binomial and Pascal (or negative binomial) distributions. The distributions of Xn, for n→∞, and the distributions of Wk, for k→∞, can be approximated by a q-Poisson distribution. Also, as k→0, a zero truncated negative q-binomial distribution Uk=Wk|Wk>0 can be approximated by a q-logarithmic distribution. These discrete q-distributions and their applications are reviewed, with critical comments and additions. Finally, consider a sequence of independent Bernoulli trials and assume that the probability of success (or failure) is a product of two sequences of probabilities with one of these sequences depending only the number of trials and the other depending only on the number of successes (or failures). The q-distributions of the number Xn of successes up to the nth trial and the number Tk of trials until the occurrence of the kth success are similarly reviewed. © 2010 Elsevier B.V

On the distributions of absorbed particles in crossing a field containing absorption points

Consider a queue of particles that are required to cross a field containing a random number of absorption points (traps) acting independently. Suppose that if a particle clashes with (contacts) any of the absorption points, it is absorbed (trapped) with probability p and non absorbed with probability q = 1 - p. Let X n be the number of absorbed particles from a queue of n particles and T k the number of particles required to cross the field until the absorption of k particles. Assuming that the number Y of absorption points in the field obeys a q-Poisson distribution (Heine or Euler distribution), the distributions of X n and T k are obtained as q-binomial and q-Pascal distributions, respectively. Inversely, assuming that X n obeys a q-binomial distribution (or, equivalently, assuming that T k obeys a q-Pascal distribution), the distribution of Y is obtained as a q-Poisson distribution (Heine or Euler distribution)

Bernoulli related polynomials and numbers

The polynomials fn Δn(x; a, b) of degree n defined by the equations (formula presented) where (x)n, b= x(x-b)(x-2b)..(x-nb+b) is the generalized factorial and Δa f(x) = f(x + a) - f(x), are the subject of this paper. A representation of these polynomials as a sum of generalized factorials is given. The coefficients, B(n, s), s = a/b, of this representation are given explicitly or by a recurrence relation. The generating functions of Δn (x; a, b) and B(n, s) are obtained. The limits of Δn (x; a, b) as a → 1, b → 0 or a → 0, b → 1 and the limits of B(n, s) as s → ± °° or s → 0 are shown to be the Bernoulli polynomials and numbers of the first and second kind, respectively. Finally, the generalized factorial moments of a discrete rectangular distribution are obtained in terms of Bin, s) in a form similar to that giving its usual moments in terms of the Bernoulli numbers. © 1979 American Mathematical Society

Distributions of random partitions and their applications

Assume that a random sample of size m is selected from a population containing a countable number of classes (subpopulations) of elements (individuals). A partition of the set of sample elements into (unordered) subsets, with each subset containing the elements that belong to same class, induces a random partition of the sample size m, with part sizes {Z1, Z2,..., ZN} being positive integer-valued random variables. Alternatively, if Nj is the number of different classes that are represented in the sample by j elements, for j = 1, 2,..., m, then (N1, N2,..., Nm) represents the same random partition. The joint and the marginal distributions of (N1, N2,..., Nm), as well as the distribution of N = ∑j=1m Nj are of particular interest in statistical inference. From the inference point of view, it is desirable that all the information about the population is contained in (N1, N2,..., Nm). This requires that no physical, genetical or other kind of significance is attached to the actual labels of the population classes. In the present paper, combinatorial, probabilistic and compound sampling models are reviewed. Also, sampling models with population classes of random weights (proportions), and in particular the Ewens and Pitman sampling models, on which many publications are devoted, are extensively presented. © Springer Science+Business Media, LLC 2007

Combinatorial probability interpretation of certain modified orthogonal polynomials

A probabilistic interpretation of a modified Gegenbauer polynomial is supplied by its expression in terms of a combinatorial probability defined on a compound urn model. Also, a combinatorial interpretation of its coefficients is provided. In particular, probabilistic interpretations of a modified Chebyshev polynomial of the second kind and a modified Legendre polynomial together with combinatorial interpretations of their coefficients are deduced. Further, probabilistic interpretations of a modified Hermite and a modified Chebyshev polynomial of the first kind are supplied by their expressions in terms of combinatorial probability functions defined on two limiting forms of the compound urn model. Finally, combinatorial interpretations of their coefficients are obtained. © 2007 Elsevier Ltd. All rights reserved

q-Factorial moments of bivariate discrete q-distributions

The q-factorial moments of the q-trinomial and the negative q-trinomial distributions of the first and second kind are derived. Also, the q-factorial moments of the bivariate q-Pólya and inverse q-Pólya distributions are obtained. In particular, the q-covariance of the two random variables or suitable functions of these random variables are deduced. © 2022 Taylor & Francis Group, LLC

q-Multinomial and negative q-multinomial distributions

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 (Formula presented.) can be approximated by a multiple Heine or Euler (q-Poisson) distribution. © 2020 Taylor & Francis Group, LLC

The q-Bernstein basis as a q-binomial distribution

The q-Bernstein basis, used in the definition of the q-Bernstein polynomials, is shown to be the probability mass function of a q-binomial distribution. This distribution is defined on a sequence of zero-one Bernoulli trials with probability of failure at any trial increasing geometrically with the number of previous failures. A modification of this model, with the probability of failure at any trial decreasing geometrically with the number of previous failures, leads to a second q-binomial distribution that is also connected to the q-Bernstein polynomials. The q-factorial moments as well as the usual factorial moments of these distributions are derived. Further, the q-Bernstein polynomial Bn(f(t),q;x) is expressed as the expected value of the function f([Xn]q/[n]q) of the random variable Xn obeying the q-binomial distribution. Also, using the expression of the q-moments of Xn, an explicit expression of the q-Bernstein polynomial Bn(fr(t),q;x), for fr(t) a polynomial, is obtained. © 2010 Elsevier B.V

Multivariate q-Pólya and inverse q-Pólya distributions

An urn containing specified numbers of balls of distinct ordered colors is considered. A multiple q-Pólya urn model is introduced by assuming that the probability of q-drawing a ball of a specific color from the urn varies geometrically, with rate q, both with the number of drawings and the number of balls of the specific color, together with the total number of balls of the preceded colors, drawn in the previous q-drawings. Then, the joint distributions of the numbers of balls of distinct colors drawn (a) in a specific number of q-drawings and (b) until the occurrence of a specific number of balls of a certain color, are derived. These two distributions turned out to be q-analogues of the multivariate Pólya and inverse Pólya distributions, respectively. Also, the limiting distributions of the multivariate q-Pólya and inverse q-Pólya distributions, as the initial total number of balls in the urn tends to infinity, are shown to be q-multinomial and negative q-multinomial distributions, respectively. © 2020 Taylor & Francis Group, LLC