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
![{\displaystyle \operatorname {P} (Z\geq \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/e78672f6d01cbb6df23f5ddb589cb2b69e16db82)
Proof: First,
![\operatorname {E}[Z]=\operatorname {E}[Z\,{\mathbf {1}}_{{\{Z<\theta \operatorname {E}[Z]\}}}]+\operatorname {E}[Z\,{\mathbf {1}}_{{\{Z\geq \theta \operatorname {E}[Z]\}}}].](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0af39f95e49ac20529bef7243415c50622f354a)
The first addend is at most ![{\displaystyle \theta \operatorname {E} [Z]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5ac7cb0d70052cc97d4764e91cc2f0bc62bbd7)
, while the second is at most ![{\displaystyle \operatorname {E} [Z^{2}]^{1/2}\operatorname {P} (Z\geq \theta \operatorname {E} [Z])^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60ff048002ca23de63c88c158d8b2b88c43705c6)
by the Cauchy–Schwarz inequality. The desired inequality then follows. ∎
Related inequalities
The Paley–Zygmund inequality can be written as
![{\displaystyle \operatorname {P} (Z\geq \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/9485e75ce47b9e4298133089a1ea11f3b45a9777)
This can be improved. By the Cauchy–Schwarz inequality,
![{\displaystyle \operatorname {E} [Z-\theta \operatorname {E} [Z]]\leq \operatorname {E} [(Z-\theta \operatorname {E} [Z])\mathbf {1} _{\{Z\geq \theta \operatorname {E} [Z]\}}]\leq \operatorname {E} [(Z-\theta \operatorname {E} [Z])^{2}]^{1/2}\operatorname {P} (Z\geq \theta \operatorname {E} [Z])^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9740bc9fcef67a2c8e44de4148c00601051b742d)
which, after rearranging, implies that
![\operatorname {P}(Z\geq \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/074d03f9e03c1fead0658b4b9dd2097f17bb1d72)
This inequality is sharp; equality is achieved if Z almost surely equals a positive constant, for example.
References
- 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