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Author Zeilberger, Doron
Source CiteSeerX
Content type Text
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Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Maple Package ♦ Probability Theory ♦ Discrete Random Variable ♦ Asymptotic Normality ♦ Discrete Probability Random Variable ♦ Multi-variate Random Variable ♦ Many Sequence ♦ Odd Moment ♦ Random Variable ♦ Briefly Described ♦ Fair Coin Time ♦ Wide Class ♦ Coin Time ♦ Polynomial Ansatz ♦ Desired Order ♦ Paul Vy Arcsine Distribution ♦ 2r-th Moment ♦ Intriguing Random Variable ♦ Asymptotic Formula ♦ Normal Distribution ♦ Gambler Ruin ♦ Maple Package Histabrut ♦ Current Capital
Abstract 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−x2 √ /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
Educational Role Student ♦ Teacher
Age Range above 22 year
Educational Use Research
Education Level UG and PG ♦ Career/Technical Study