16-cell honeycomb

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

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

Proof: First,

$\operatorname {E} [Z]=\operatorname {E} [Z\,\mathbf {1} _{\{Z<\theta \operatorname {E} [Z]\}}]+\operatorname {E} [Z\,\mathbf {1} _{\{Z\geq \theta \operatorname {E} [Z]\}}].$

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

Related inequalities

The Paley–Zygmund inequality can be written as

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

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

$\operatorname {E} [Z-\theta \operatorname {E} [Z]]\leq \operatorname {E} [(Z-\theta \operatorname {E} [Z])\mathbf {1} _{\{Z\geq \theta \operatorname {E} [Z]\}}]\leq \operatorname {E} [(Z-\theta \operatorname {E} [Z])^{2}]^{1/2}\operatorname {P} (Z\geq \theta \operatorname {E} [Z])^{1/2}$

which, after rearranging, implies that

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

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