A Note on Showing a Lack of Ties in a Set of s Observations

Abstract

Suppose that Xl, X2, . .. , X, is a random sample from some population which has a continuous cumulative distribution function F(x). Let Y1, Y2, . -.1 , be the corresponding set of order statistics, that is, let Y, equal the smallest observation among the Xi's, Y2 the next smallest, ... , Y, the largest observation. In the usual course in mathematical statistics it is assumed that a density f(x) exists for the population. Using the joint densityf(x1)f(x2) . .. f(x,) of the random sample it is shown or indicated through the Riemann integral that the probability of a tie among the observationis is zero, and thus that the order statistics are well defined, satisfying Yl < Y2 < ... < Y, (in a more advanced course using measure-theoretic concepts the zero probability of a tie would follow from absolute continuity and the notion that the subset of Rs corresponding to a tie has s-dimensional Lebesgue measure zero). In this note we show that the probability of a tie among the observations is zero, using only the assumed continuity of the cdf F(x), thus demonstrating that we perhaps depend on the existence of a density function more than we should, pedagogically speaking. For simplicity of presentation we prove the claim first for the case s = 2. Suppose X1, X2 are two independent, identically distributed random variables with common cdf F(x). We show Pr[X1 = X2] = 0 by developing a monotone decreasing sequence I p, }o=o of positive numbers which approach zero and all of which dominate Pr[Xi = X21. Although many possible sequences would work, we choose the sequence { p} '=0 where pn = 2-n. Thus we show