# Symmetrization of binary random variables

Abram Kagan, Colin L. Mallows, Larry A. Shepp, Robert J. Vanderbei, Yehuda Vardi

Research output: Contribution to journalArticle

1 Scopus citations

### Abstract

A random variable Y is called an independent symmetrizer of a given random variable X if (a) it is independent of X and (b) the distribution of X+ Y is symmetric about 0. In cases where the distribution of X is symmetric about its mean, it is easy to see that the constant random variable Y = -EX is a minimum-variance independent symmetrizer. Taking Y to have the same distribution as -X clearly produces a symmetric sum, but it may not be of minimum variance. We say that a random variable X is symmetry resistant if the variance of any symmetrizer, Y, is never smaller than the variance of X. Let A' be a binary random variable: P{X = a} = p and P{X = b} = q, where ab, 0 < p < 1, and q = 1 -p. We prove that such a random variable is symmetry resistant if (and only if) p 1/2. Note that the minimum variance as a function of p is discontinuous at p = 1/2. Dropping the independence assumption, we show that the minimum variance reduces to pq -min(p, q)/2, which is a continuous function of p.

Original language English (US) 1013-1020 8 Bernoulli 5 6 https://doi.org/10.2307/3318557 Published - Jan 1 1999

### All Science Journal Classification (ASJC) codes

• Statistics and Probability

### Keywords

• Binary random variables
• Linear programming
• Symmctrization

• ## Cite this

Kagan, A., Mallows, C. L., Shepp, L. A., Vanderbei, R. J., & Vardi, Y. (1999). Symmetrization of binary random variables. Bernoulli, 5(6), 1013-1020. https://doi.org/10.2307/3318557