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
![{\displaystyle \operatorname {P} (Z>\theta \operatorname {E} [Z])\geq (1-\theta )^{2}{\frac {\operatorname {E} [Z]^{2}}{\operatorname {E} [Z^{2}]}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d215506063bd4c6d27d04808bcb94387d7931d7)
Proof: First,
![{\displaystyle \operatorname {E} [Z]=\operatorname {E} [Z\,\mathbf {1} _{\{Z\leq \theta \operatorname {E} [Z]\}}]+\operatorname {E} [Z\,\mathbf {1} _{\{Z>\theta \operatorname {E} [Z]\}}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1771609dffe911af2dcaa45558284564b75f5ef)
The first addend is at most
, while the second is at most
by the Cauchy–Schwarz inequality. The desired inequality then follows. ∎
Related inequalities
The Paley–Zygmund inequality can be written as
![{\displaystyle \operatorname {P} (Z>\theta \operatorname {E} [Z])\geq {\frac {(1-\theta )^{2}\,\operatorname {E} [Z]^{2}}{\operatorname {Var} Z+\operatorname {E} [Z]^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03eb36ae16ffe077072e03d9db8855632f5dd47e)
This can be improved[citation needed]. By the Cauchy–Schwarz inequality,
![{\displaystyle \operatorname {E} [Z-\theta \operatorname {E} [Z]]\leq \operatorname {E} [(Z-\theta \operatorname {E} [Z])\mathbf {1} _{\{Z>\theta \operatorname {E} [Z]\}}]\leq \operatorname {E} [(Z-\theta \operatorname {E} [Z])^{2}]^{1/2}\operatorname {P} (Z>\theta \operatorname {E} [Z])^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9dea4705e5d7f39aaa9765aa7b89f47353559b)
which, after rearranging, implies that
![{\displaystyle \operatorname {P} (Z>\theta \operatorname {E} [Z])\geq {\frac {(1-\theta )^{2}\operatorname {E} [Z]^{2}}{\operatorname {E} [(Z-\theta \operatorname {E} [Z])^{2}]}}={\frac {(1-\theta )^{2}\operatorname {E} [Z]^{2}}{\operatorname {Var} Z+(1-\theta )^{2}\operatorname {E} [Z]^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bce3bc5bd01a6479c20603a21fd1daa8162c826)
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
![{\displaystyle \operatorname {P} (Z>\mu -\theta \sigma )\geq {\frac {\theta ^{2}}{1+\theta ^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9b88de7aaa1c10ec0801f37d9f06496b1b3cf88)
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
![{\displaystyle \operatorname {P} (Z>\theta \operatorname {E} [Z\mid Z>0])\geq {\frac {(1-\theta )^{2}\,\operatorname {E} [Z]^{2}}{\operatorname {E} [Z^{2}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/050239d865ee5225d6950e655f86a671ac5183cc)
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
![{\displaystyle \operatorname {P} (Z>\theta \operatorname {E} [Z\mid Z>0])\geq {\frac {(1-\theta )^{p/(p-1)}\,\operatorname {E} [Z]^{p/(p-1)}}{\operatorname {E} [Z^{p}]^{1/(p-1)}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/340272ea66f5f6253d4c426fd7e75124c51a9083)
for every
. This follows by the same proof as above but using Hölder's inequality in place of the Cauchy-Schwarz inequality.
See also
- Cantelli's inequality
- Second moment method
- Concentration inequality – a summary of tail-bounds on random variables.
References
- ^ 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
- 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. S2CID 178702376.
- 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. S2CID 122832495.