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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Expectation equal to $1$ does not force a nonnegative integer-valued variable to vanish somewhere

Statement refuted

If a nonnegative integer-valued random variable satisfies $E [X] \leq 1$ , then some outcome has $X = 0$ .

Facts & Assumptions

Given: The uniform probability space on a singleton and its constant random variable $X = 1$ .

[L1]

A nonempty singleton carries a uniform finite probability space (The uniform probability space on a nonempty finite set).

[L2]

Expectation is the finite weighted sum of values (Expectation of a real random variable on a finite probability space).

[L3]

The first-moment avoidance conclusion assumes the strict inequality $E [X] < 1$ (The first-moment method for avoiding or forcing a finite count of bad events).

Counterexample

technique · constructive

1.1

Construct $X$ to equal $1$ on the unique outcome. That outcome has weight $1$ , so $E [X] = 1$ .

givenL1L2construct

1.2

The event ${X = 0}$ is empty.

given

2.1

Thus the weak threshold $E [X] \leq 1$ does not force a zero outcome; the strict hypothesis in [L3] is necessary.

step 1.1step 1.2L3discharge-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

J. Matousek and J. Vondrak, The Probabilistic Method, Chapter 2 (standard reference, not scraped)

Expectation equal to 11 does not force a nonnegative integer-valued variable to vanish somewhere