Symmetrization of Bernoulli
نویسنده
چکیده
Let X be a random variable. We shall call an independent random variable Y to be a symmetrizer forX , ifX+Y is symmetric around zero. If Y is independent copy of −X , it is obviously a symmetrizer. A random variable is said to be symmetry resistant if the variance of any symmetrizer Y , is never smaller than the variance of X itself. For example, let X be a Bernoulli(p) random variable. If p = 1/2, it is immediate that the degenerate random variable, Y ≡ −1/2, is a symmetrizer for X . Hence, X is not symmetry resistant. However, we shall show that if p 6= 1/2, for any symmetrizer Y , we have Var(Y ) ≥ pq,
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