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.

Markov's inequality for random variables

Statement

If $X : Ω \to [0, + \infty]$ is a nonnegative random variable on a probability space and $a > 0$ , then $P (X \geq a) \leq \frac{E [X]}{a} .$

Facts & Assumptions

Given: A nonnegative random variable $X$ and a real number $a > 0$ .

[L1]

Expectation is integration against the probability measure (Expectation of a nonnegative or integrable random variable).

[L2]

The integral Markov inequality states $μ ({f \geq t}) \leq t^{- 1} \int f d μ$ for nonnegative measurable $f$ and $t > 0$ (Chebyshev-Markov inequality for the integral).

Proof

technique · direct

1.1

Apply [L2] to the probability measure $P$ , the function $X$ , and the threshold $a > 0$ . Rewriting the integral by [L1] gives $P (X \geq a) \leq \frac{E [X]}{a} .$

L1L2

2.1

Step 1.1 is exactly Markov's inequality for random variables.

step 1.1∎

Depends on

Used by

Chebyshev's inequality for random variables Corollary
Markov and Chebyshev sharpness Example
The general inequalities compare cleanly with the published finite ones Remark

Dependency tree · two levels

8 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)