DIMACS TR: 97-24

Symmetrization of Binary Random Variables

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


A random variable $Y$ is called an {\em 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 {\em 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 $p$.

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