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

Abstract

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.