A Polya Distribution for Teaching

Abstract

In teaching elementary pre-calculus statistics, we often concentrate on two parent distributions. First, we consider a dichotomous distribution, i.e., a distribution associated with a random variable which can take on only two possible values. In addition to its usefulness in application, this distribution has a distinct pedagogical virtue: it may readily be explored in depth by mathematically unsophisticated students. We usually then make the mighty leap to the normal distribution which, for precalculus students, because of its being continuous, of necessity can only be treated in a superficial cookbook fashion (look up some numbers in a table). For teaching purposes it would seem desirable to have a parent distribution intermediate between these two, a distribution which is discrete, so that it may be explored in depth, but a distribution which is more challenging than a dichotomous distribution. Additionally, we would like our distribution to be unimodal and symmetric, and to have easy arithmetic associated with it (both for the students' and the instructor's sakes). Thus, let our distribution take on single-digit integers. (Limiting the values to one digit is not only to make arithmetic easy but also to take up only one column on an IBM card.) We first looked at binomial distributions and hypergeometric distributions. We rejected these because their tails are very slim (and, consequently, their variances small). Let us find a similar distribution with fatter tails and a larger variance (in particular o2(X) = 4 and o(X) = 2). To accomplish this we turn to Polya's Urn Scheme (Feller, 1950, ch. 5). Let an urn contain 0 black balls and 0 white balls, and draw a ball with extra replacement, i.e., if a black ball is drawn replace it together with another black ball, and similarly for a white ball. Do this eight successive times and let X be the number of black balls drawn. If we start with 0 = 3 black balls and 0 = 3 white balls in the urn, the distribution of X is given by: (i + 2)! (10-i)! /8 2! 2! P(X=i)=K 2! 2! 13!