The Paley–Zygmund inequality is a result in probability theory, specifically in the context of the study of random variables and their moments. It provides a bound on the probability that a non-negative random variable is significantly greater than its expected value.
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For a nonnegative random variable with and , the Paley-Zygmund inequality givesThe truncated first moment bound gives for . Applying the Cauchy-Schwarz inequality to and its indicator variable proves the result. For a Bernoulli random variable of mean and , the bound is exactly . It is useful for proving a positive chance of a nonzero count from only two moments.