Paley–Zygmund inequality

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

$\Pr \lbrace Z\geq \theta \,\operatorname {E} (Z)\rbrace \geq (1-\theta )^{2}\,{\frac {\operatorname {E} (Z)^{2}}{\operatorname {E} (Z^{2})}}.$

Proof: First,

$\operatorname {E} Z=\operatorname {E} \lbrace Z\,\mathbf {1} _{Z<\theta \operatorname {E} Z}\rbrace +\operatorname {E} \lbrace Z\,\mathbf {1} _{Z\geq \theta \operatorname {E} Z}\rbrace ~.$

Obviously, the first addend is at most $\theta \operatorname {E} (Z)$. The second one is at most

$\lbrace \operatorname {E} Z^{2}\rbrace ^{1/2}\lbrace \operatorname {E} \mathbf {1} _{Z\geq \theta \operatorname {E} Z}\rbrace ^{1/2}={\Big (}\operatorname {E} Z^{2}{\Big )}^{1/2}{\Big (}\Pr \lbrace Z\geq \theta \,\operatorname {E} (Z)\rbrace {\Big )}^{1/2}$

according to the Cauchy-Schwartz inequality. ∎

Related inequalities

The right-hand side of the Paley - Zygmund inequality can be written as

$\Pr \lbrace Z\geq \theta \,\operatorname {E} (Z)\rbrace \geq {\frac {(1-\theta )^{2}\,\operatorname {E} (Z)^{2}}{\operatorname {E} (Z)^{2}+\operatorname {Var} Z}}.$

The one-sided Chebyshev inequality gives a slightly better bound:

$\Pr \lbrace Z\geq \theta \,\operatorname {E} (Z)\rbrace \geq {\frac {(1-\theta )^{2}\,\operatorname {E} (Z)^{2}}{(1-\theta )^{2}\,\operatorname {E} (Z)^{2}+\operatorname {Var} Z}}.$

The latter is sharp.

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