## Symmetrization of Binary Random Variables

### Authors: Abram Kagan, Colin L. Mallows, Larry A. Shepp, Robert J. Vanderbei and Yehuda Vardi

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 = - \Exp X\$ 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 \$X\$ be a binary random variable: \$\Prob \{ X = a \} = p\$ and \$\Prob \{ X = b \} = q\$ where \$a \ne b\$, \$0 < p < 1\$, and \$q = 1-p\$. We prove that such a binary random variable is symmetry resistant if (and only if) \$p \ne 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\$.

Paper Available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1999/99-14.ps.gz