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 &ge; 0 is a random variable with finite variance, and if 0 < &theta; < 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-Schwarz 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.