science

Paley–Zygmund inequality

Paley–Zygmund inequality is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Paley–Zygmund inequality rather than just read about it. In short: In mathematics, the Paley–Zygmund inequality bounds the probability that a positive random variable is small, in terms of its first two moments. The inequality was proved by Raymond Paley and Antoni Zygmund.

Key takeaways

Paley–Zygmund inequality belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Paley–Zygmund inequality to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Paley–Zygmund inequality from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Paley–Zygmund inequality bounds the probability that a positive random variable is small, in terms of 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 {\displaystyle 0\leq \theta \leq 1} , then

P ⁡ ( Z > θ E ⁡ [ Z ] ) ≥ ( 1 − θ ) 2 E ⁡ [ Z ] 2 E ⁡ [ Z 2 ] . {\displaystyle \operatorname {P} (Z>\theta \operatorname {E} [Z])\geq (1-\theta )^{2}{\frac {\operatorname {E} [Z]^{2}}{\operatorname {E} [Z^{2}]}}.}

Proof: First,

E ⁡ [ Z ] = E ⁡ [ Z 1 { Z ≤ θ E ⁡ [ Z ] } ] + E ⁡ [ Z 1 { Z > θ E ⁡ [ Z ] } ] . {\displaystyle \operatorname {E} [Z]=\operatorname {E} [Z\,\mathbf {1} _{\{Z\leq \theta \operatorname {E} [Z]\}}]+\operatorname {E} [Z\,\mathbf {1} _{\{Z>\theta \operatorname {E} [Z]\}}].}

The first addend is at most θ E ⁡ [ Z ] {\displaystyle \theta \operatorname {E} [Z]} , while the second is at most E ⁡ [ Z 2 ] 1 / 2 P ⁡ ( Z > θ E ⁡ [ Z ] ) 1 / 2 {\displaystyle \operatorname {E} [Z^{2}]^{1/2}\operatorname {P} (Z>\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

P ⁡ ( Z > θ E ⁡ [ Z ] ) ≥ ( 1 − θ ) 2 E ⁡ [ Z ] 2 Var ⁡ Z + E ⁡ [ Z ] 2 . {\displaystyle \operatorname {P} (Z>\theta \operatorname {E} [Z])\geq {\frac {(1-\theta )^{2}\,\operatorname {E} [Z]^{2}}{\operatorname {Var} Z+\operatorname {E} [Z]^{2}}}.}

This can be improved since, by the Cauchy–Schwarz inequality,

E ⁡ [ Z − θ E ⁡ [ Z ] ] ≤ E ⁡ [ ( Z − θ E ⁡ [ Z ] ) 1 { Z > θ E ⁡ [ Z ] } ] ≤ E ⁡ [ ( Z − θ E ⁡ [ Z ] ) 2 ] 1 / 2 P ⁡ ( Z > θ E ⁡ [ Z ] ) 1 / 2 {\displaystyle \operatorname {E} [Z-\theta \operatorname {E} [Z]]\leq \operatorname {E} [(Z-\theta \operatorname {E} [Z])\mathbf {1} _{\{Z>\theta \operatorname {E} [Z]\}}]\leq \operatorname {E} [(Z-\theta \operatorname {E} [Z])^{2}]^{1/2}\operatorname {P} (Z>\theta \operatorname {E} [Z])^{1/2}}

which, after rearranging, implies that

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Paley–Zygmund inequality

Start with the simplest possible case. Write down what Paley–Zygmund inequality claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Paley–Zygmund inequality before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Paley–Zygmund inequality ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Paley–Zygmund inequality

In research: Paley–Zygmund inequality appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Paley–Zygmund inequality in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Paley–Zygmund inequality is common in secondary-school and first-year university syllabi. It links to neighbouring topics Probabilistic inequalities, so understanding it makes those chapters shorter.
In everyday life: Look for Paley–Zygmund inequality outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Paley–Zygmund inequality in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Paley–Zygmund inequality means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Paley–Zygmund inequality out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Paley–Zygmund inequality in simple terms?

Why does Paley–Zygmund inequality matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Paley–Zygmund inequality?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Paley–Zygmund inequality.