## DIMACS TR: 99-14

## 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

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