CounterexampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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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.

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Markov's conclusion can fail without nonnegativity

Statement refuted

The conclusion of Markov's inequality remains valid when the hypothesis $X \geq 0$ is removed.

Facts & Assumptions

Given: The uniform two-point probability space and a random variable $X$ taking values $1$ and $- 3$ .

[L1]

Each point in a uniform two-point space has probability $1 / 2$ (The uniform probability space on a nonempty finite set).

[L2]

A real random variable and its expectation are a function and its weighted finite sum (Real random variables on finite probability spaces and their finite distributions, Expectation of a real random variable on a finite probability space).

[L3]

Markov's theorem assumes that $X$ is nonnegative (Markov's inequality on a finite probability space).

Counterexample

technique · constructive

1.1

For the constructed two-valued variable, direct calculation gives $E [X] = (1 - 3) / 2 = - 1$ .

givenL1L2algebraconstruct

2.1

At threshold $a = 1$ , $P (X \geq 1) = 1 / 2$ , while $E [X] / a = - 1$ .

step 1.1L1algebra

3.1

The purported conclusion would be $1 / 2 \leq - 1$ , which is false. This refutes removal of the nonnegativity hypothesis, not [L3].

step 2.1L3discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 6.2.2 (standard reference, not scraped)