ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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 and Chebyshev sharpness

Example

Both Markov's and Chebyshev's inequalities can be sharp.

If $X = a 1_{A}$ with $a > 0$ , then $P (X \geq a) = \frac{E [X]}{a} = P (A) .$
If $Y$ takes the values $\pm a$ with probability $1 / 2$ each, then $P (∣ Y - E [Y] ∣ \geq a) = 1 = \frac{Var (Y)}{a^{2}} .$

These equality examples do not mean the bounds are always informative: when a distribution has much lighter tails, the same inequalities may be very far from equality.

Facts & Assumptions

Given: The two-point random variables described above.

[L1]

Markov's and Chebyshev's inequalities are the probability-space bounds already proved on the A page (Markov's inequality for random variables, Chebyshev's inequality for random variables).

Verification

technique · direct

1.1

For $X = a 1_{A}$ , one has $E [X] = a P (A)$ and ${X \geq a} = A$ , so $P (X \geq a) = P (A) = \frac{E [X]}{a} .$

L1algebra

1.2

For $Y = \pm a$ with equal probabilities, $E [Y] = 0$ and $Var (Y) = a^{2}$ , while $∣ Y - E [Y] ∣ = a$ always. Hence $P (∣ Y - E [Y] ∣ \geq a) = 1 = \frac{Var (Y)}{a^{2}} .$

L1algebra

2.1

Thus both inequalities admit equality, even though other laws can make the bounds much weaker.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

J. R. Norris, Probability and Measure, Section 4.2 (standard reference, not scraped)