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 by a random variable that almost surely equals 1, for example.
- 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