P. Feinsilver

Mar 1, 1983

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Abstract

One-parameter extensions of the Binomial, Negative Binomial, Poisson. Geometric and Gamma distributions are derived via an extension of the standard Bernoulli counting scheme. The method is to examine the elementary "potential theory" for the basic process. Some interesting extensions to continuous time are mentioned also. I. The basic process. We will consider a counting process on the nonnegative integers such that as time goes on the probability of increasing decreases. The related probability distributions are studied using "basic" or "Eulerian" mathematics [8]. We will explain the "q" theory needed as it arises. Throughout, x is a fixed number, 0 , K )a denoting expected value with respect to the measure on paths starting at a. We derive from (1.1) and (1.2), (1.3) Pf(n + 1, a) Pf(n, a) = (qnS-ix[f(S + 1) -f(Sn)])a, Pf(O, a) =-f(a ). In this context the useful functions are of the formfv(k) = Il-(l vq) -(v)k, using the abbreviated standard notation, for 0 0, (1.4) (V)k+1 (V)k = -q kV(V)k yields Pf,(n + 1, a) = (1xvqn)Pfv(n, a), Pf,(0, a) = (v)a. Thus, Pfv,(n, a) = (V)a(XV)n . Received by the editors February 27, 1981 and, in revised form, June 25, 1982. 1980 Mathematics Subject Classification. Primary 60J75; Secondary 60J15. ( 1983 American Mathematical Society 0002-9939/82/0000-0721 /$02.50 508 This content downloaded from 207.46.13.168 on Wed, 22 Feb 2017 19:15:55 UTC All use subject to http://about.jstor.org/terms q-PROBABILITY DISTRIBUTION 509 For the process starting at 0 (we drop the subscript 0) this yields (1.5) ((V)Sn ) = (XV),. It is useful to consider briefly generating functions of the form +(v) = v) Pk for probability distributions Pk* We denote an " Eulerian derivative" by (1.6) Wf(V) = [f(v) f(QV)7/VQ where Q = q '. Note that (q)n =l>i(I qJ). Then we have PROPOSITION 1. 1. Map the distribution {pn) + 0' Pk(V) )k Then: (a) 4 is holomorphic on 6D, (b) pn = (q)i'WnO(v) 1v= I is the inverse transform. PROOF. (a) Observe the estimate sup | (V )| n sup 111expl ) K,n lq K IV I q for any compact subset K of 6D. (b) Inductively, check that Wm(V)k = (q)k(V)k-m/(q)k-m Applying Proposition 1.1 to (1.5) yields the q-binomial distribution (1.7) P(Sn = k) ( k) Xk(X)n-ki where (n )q denotes the q-binomial coefficient k)q ~ ~ ~ ~ ! ( 1 .8) ( ~k )q ( q)k (q )n-k' The identity P(Sn < xo) = 1 yields a q-binomial theorem

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