Symmetrization of binary random variables

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 $Y=\mathrm{EX}$ 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 $X$ be a binary random variable: $P\{X=a\}=p$ and $P\{X=b\}=q$ 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 $\mathrm{pq}-min(p,q)/2$, which is a continuous function of $p$.