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
The desired inequality then follows.
∎ The Paley–Zygmund inequality can be written as This can be improved[citation needed].
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.
In turn, this implies another convenient form (known as Cantelli's inequality) which is where
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
Both this inequality and the usual Paley-Zygmund inequality also admit
versions:[1] If Z is a non-negative random variable and
This follows by the same proof as above but using Hölder's inequality in place of the Cauchy-Schwarz inequality.