HISTABRUT: A Maple Package for Symbol-Crunching in Probability theory

AMaple package HISTABRUT (available from http://www.math.rutgers.edu/~zeilberg/tokhniot/HISTABRUT) is presented and briefly described. It uses the polynomial ansatz to discover (often fully rigorously, but in some cases only semi-rigorously (yet rigorizably!)) explicit asymptotic formulas for the moments of uni-variate and, more impressively, bi-variate, discrete probability random variables. It would be hopefully extended, in the future, to multi-variate random variables. Many sequences of discrete random variables (e.g. tossing a (fair or loaded) coin n times, and keeping track of the number of Heads minus the number of Tails) are asymptotically normal. In [Z1], I introduced and described Maple packages, CLT and AsymptoticMoments, that empirically-yetrigorously prove asymptotic normality for a wide class of sequences of discrete random variables. They used the method of moments. Furthermore, they are able to prove much stronger theorems than mere “asymptotic normality” by finding the asymptotics (to any desired order!) of the (normalized) moments, rather than only the leading terms (that should be those of the normal distribution e−x 2/2/ √ 2π, namely 1 · 3 · 5 · · · (2r − 1) for the even 2r-th moment, and 0 for the odd moments). But not all discrete probability random variables are asymptotically normal! For example, the number of times that your current capital is positive, upon tossing a fair coin n times and winning a dollar if it is Heads and losing a dollar it is Tails, that converges to Paul Lévy’s arcsine distribution (see [Z2]), and the other random variables considered by Feller (see [Z3]). Another intriguing random variable is the duration of a gambler’s ruin considered in [Z4]. The much larger Maple package HISTABRUT available from http://www.math.rutgers.edu/~zeilberg/tokhniot/HISTABRUT can handle any sequence of discrete probability distributions, that the users have to program themselves. There are quite a few ones pre-programmed, (type EzraPGF(); in the Maple package HISTABRUT for a list). It can also sketch the limiting distributions, using procedure plotDist (see the on-line help). Another new feature is that it can handle directly sequences of probability distributions defined in terms of rational generating functions, R(t, s), where the coefficient of s in the power-series expansion of R(t, s) in terms of s is the probability generating function (in t) for a typical memebr 1 Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA. zeilberg at math dot rutgers dot edu , http://www.math.rutgers.edu/~zeilberg/ . First version: Aug. 25, 2010. This article accompanies the Maple package http://www.math.rutgers.edu/~zeilberg/tokhniot/HISTABRUT . Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger (http://www.math.rutgers.edu/~zeilberg/pj.html). Supported in part by the NSF.