CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Chebyshev's inequality for random variables

Statement

If $X$ is a square-integrable real random variable and $a > 0$ , then $P (∣ X - E [X] ∣ \geq a) \leq \frac{Var (X)}{a^{2}} .$

Facts & Assumptions

Given: A square-integrable real random variable $X$ and a real number $a > 0$ .

[L1]

Variance is the expectation of the squared centered variable (Moments, variance, and covariance on a probability space).

[L2]

Markov's inequality applies to every nonnegative random variable (Markov's inequality for random variables).

Proof

technique · direct

1.1

The random variable $Y : = (X - E [X])^{2}$ is nonnegative, and ${Y \geq a^{2}} = {∣ X - E [X] ∣ \geq a} .$

givenalgebra

2.1

Applying [L2] to $Y$ and using [L1] gives $P (∣ X - E [X] ∣ \geq a) = P (Y \geq a^{2}) \leq \frac{E [Y]}{a^{2}} = \frac{Var (X)}{a^{2}} .$

step 1.1L1L2∎

Depends on

Used by

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Rick Durrett, Probability: Theory and Examples, 5th ed., Section 1.6.1 (standard reference, not scraped)
J. R. Norris, Probability and Measure, Section 4.2 (standard reference, not scraped)