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.
^Petrov, Valentin V. (1 August 2007). "On lower bounds for tail probabilities". Journal of Statistical Planning and Inference. 137 (8): 2703–2705. doi:10.1016/j.jspi.2006.02.015.
Further reading[]
This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations.(November 2020) (Learn how and when to remove this template message)
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. Bibcode:1932PCPS...28..190P. doi:10.1017/S0305004100010860.
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. Bibcode:1932PCPS...28..266P. doi:10.1017/S0305004100010112.
Categories:
Probabilistic inequalities
Hidden categories:
Articles lacking in-text citations from November 2020