Symmetrization of Binary Random Variables

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 X be a binary random variable: P{X = a} = p and P{X = b} = q where a 6= b, 0 < p < 1, and q = 1 − p. We prove that such a binary random variable is symmetry resistant if (and only if) p 6= 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.