Paley Zygmund Inequality

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the Paley Zygmund inequality bounds the probability that a positive random variable is small, in terms of its mean and variance (i.e., its first two moments). The inequality was proved by Raymond Paley and Antoni Zygmund.Theorem: If Z 0 is a random variable with finite variance, and if 0 1, thenThe right-hand side of the Paley - Zygmund inequality can be written asThe one-sided Chebyshev inequality gives a slightly better bound: