In mathematics, the Paley–Zygmund inequality bounds the probability that a positive random variable is small, in terms of 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 , then Proof: First, The first addend is at most , while the second is at most by the Cauchy–Schwarz inequality. The desired inequality then follows. ∎ Related inequalities The Paley–Zygmund inequality can be written as This can be improved. By the Cauchy–Schwarz inequality, which, after rearranging, implies that This inequality is sharp; equality is achieved if Z almost surely equals a positive constant. In turn, this implies another convenient form (known as Cantelli's inequality) which is where and . This follows from the substitution valid when . A strengthened form of the Paley-Zygmund inequality states that if Z is a non-negative random variable then for every . This inequality follows by applying the usual Paley-Zygmund inequality to the conditional distribution of Z given that it is positive and noting that the various factors of cancel. Both this inequality and the usual Paley-Zygmund inequality also admit versions: If Z is a non-negative random variable and then for every . This follows by the same proof as above but using Hölder's inequality in place of the Cauchy-Schwarz inequality. See also Cantelli's inequality Second moment method Concentration inequality – a summary of tail-bounds on random variables. References Further reading Probabilistic inequalities