---
tag: 211PJYW8W6
kind: statement
role: theorem
name: Paley–Zygmund inequality
msc: ["60E15"]
uses: [0PNRKMRC55, 0WQNRCNY3M, 0RFWW56Y22]
about: [0WQNRCNY3M]
sources:
  - url: https://en.wikipedia.org/wiki/Paley%E2%80%93Zygmund_inequality
same_as:
  wikidata: Q2718608
  wikipedia: Paley–Zygmund inequality
---

## Statement

Let $Z \ge 0$ be a real [[random variable|0PNRKMRC55]] with $0 < E[Z^2] < \infty$, and let $\theta \in [0, 1]$.
Then
$$P(Z > \theta \, E[Z]) \ge (1 - \theta)^2 \frac{E[Z]^2}{E[Z^2]}.$$

## Remarks

Proof: split $E[Z] = E[Z \mathbf{1}_{\set{Z \le \theta E[Z]}}] + E[Z \mathbf{1}_{\set{Z > \theta E[Z]}}]$.
The first term is at most $\theta E[Z]$, and by the [[Cauchy–Schwarz inequality|0RFWW56Y22]] the second is at most $\sqrt{E[Z^2] \, P(Z > \theta E[Z])}$.
Rearranging gives the bound.
With $\theta = 0$ it becomes the second moment method: $P(Z > 0) \ge E[Z]^2 / E[Z^2]$.
Example: if $Z$ counts the triangles in an [[Erdős–Rényi random graph|6E83YKANV3]] and $E[Z^2] / E[Z]^2 \to 1$, then $P(Z > 0) \to 1$, so a triangle exists with high probability.
The bound is a lower bound on the probability of being large, while [[Markov's inequality|05PWWHYBX5]] and [[Chebyshev's inequality|02HZ1BK3MW]] give upper bounds.
The hypothesis $Z \ge 0$ is needed: the constant $Z = -1$ has $E[Z]^2 / E[Z^2] = 1$, but with $\theta = 0$ the event $\set{Z > 0}$ is empty.

## Explanations

### L1 human

Take a nonnegative quantity whose second moment is not much larger than the square of its mean.
Then the quantity cannot be small most of the time.
It must exceed a fixed fraction of its mean with a probability bounded away from $0$.

### L2 human

The inequality is the basis of the second moment method in combinatorics and random graphs, where one shows that a count $Z$ is positive with high probability.
In analysis, Paley and Zygmund used it for random series, to show that a random trigonometric series is large with positive probability.
It complements the first moment method, which uses $P(Z > 0) \le E[Z]$ for integer-valued $Z \ge 0$.