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. ∎
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:[1] 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.
Paley, R. E. A. C.; Zygmund, A. (April 1932). "On some series of functions, (3)". Mathematical Proceedings of the Cambridge Philosophical Society. 28 (2): 190–205.
Paley, R. E. A. C.; Zygmund, A. (July 1932). "A note on analytic functions in the unit circle". Mathematical Proceedings of the Cambridge Philosophical Society. 28 (3): 266–272.