CounterexampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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.

For dependent variables, $E [X Y]$ need not equal $E [X] E [Y]$

Statement refuted

Every pair of finite random variables satisfies $E [X Y] = E [X] E [Y]$ .

Facts & Assumptions

Given: A uniform random sign $X$ on ${- 1, 1}$ and the variable $Y = X$ .

[L1]

A uniform two-point space gives both outcomes probability $1 / 2$ (The uniform probability space on a nonempty finite set).

[L2]

Independence requires joint attained-value probabilities to factor (Pairwise and mutual independence of finite-valued random variables).

[L3]

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

Counterexample

technique · constructive

1.1

For the constructed pair $Y = X$ , directly $E [X] = E [Y] = 0$ , while $X Y = X^{2} = 1$ everywhere and hence $E [X Y] = 1$ .

givenL1L3algebraconstruct

1.2

Also $P (X = 1, Y = 1) = 1 / 2 \neq 1 / 4 = P (X = 1) P (Y = 1)$ , so the variables are dependent.

L1L2algebra

2.1

Thus $E [X Y] = 1 \neq 0 = E [X] E [Y]$ , refuting the statement.

step 1.1step 1.2discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 32 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Matousek and J. Vondrak, The Probabilistic Method, Section 1.1 (standard reference, not scraped)

For dependent variables, E[XY]E[XY] need not equal E[X]E[Y]E[X]E[Y]