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 < θ < 1, then
The first addend is at most , while the second is at most by the Cauchy–Schwarz inequality. The desired inequality then follows. ∎
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, for example.
- 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