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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

$\mathrm{P}(Z\ge \theta \mathrm{E}[Z])\ge (1-\theta {)}^2\frac{\mathrm{E}[Z{]}^2}{\mathrm{E}[Z^2]}.$

Proof: First,

$\mathrm{E}[Z]=\mathrm{E}[Z{\mathbf{1}}_{\{Z<\theta \mathrm{E}[Z]\}}]+\mathrm{E}[Z{\mathbf{1}}_{\{Z\ge \theta \mathrm{E}[Z]\}}].$

The first addend is at most $\theta \mathrm{E}[Z]$, while the second is at most $\mathrm{E}[Z^2{]}^{1/2}\mathrm{P}(Z\ge \theta \mathrm{E}[Z]{)}^{1/2}$ by the Cauchy–Schwarz inequality. The desired inequality then follows. ∎

Related inequalities

The Paley–Zygmund inequality can be written as

$\mathrm{P}(Z\ge \theta \mathrm{E}[Z])\ge \frac{(1-\theta {)}^2\mathrm{E}[Z{]}^2}{\mathrm{var}Z+\mathrm{E}[Z{]}^2}.$

This can be improved. By the Cauchy–Schwarz inequality,

$\mathrm{E}[Z-\theta \mathrm{E}[Z]]\le \mathrm{E}[(Z-\theta \mathrm{E}[Z]){\mathbf{1}}_{\{Z\ge \theta \mathrm{E}[Z]\}}]\le \mathrm{E}[(Z-\theta \mathrm{E}[Z]{)}^2{]}^{1/2}\mathrm{P}(Z\ge \theta \mathrm{E}[Z]{)}^{1/2}$

which, after rearranging, implies that

$\mathrm{P}(Z\ge \theta \mathrm{E}[Z])\ge \frac{(1-\theta {)}^2\mathrm{E}[Z{]}^2}{\mathrm{E}[(Z-\theta \mathrm{E}[Z]{)}^2]}=\frac{(1-\theta {)}^2\mathrm{E}[Z{]}^2}{\mathrm{var}Z+(1-\theta {)}^2\mathrm{E}[Z{]}^2}.$

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