Paley–Zygmund inequality

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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 , 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[edit]

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.


  • R. E. A. C. Paley and A. Zygmund, "On some series of functions, (3)," Proc. Camb. Phil. Soc. 28 (1932), 190-205, (cf. Lemma 19 page 192).
  • R. E. A. C. Paley and A. Zygmund, A note on analytic functions in the unit circle, Proc. Camb. Phil. Soc. 28 (1932), 266–272