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

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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.
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False: linearity of expectation requires independence

Statement

The identity $E [X + Y] = E [X] + E [Y]$ is valid only when $X$ and $Y$ are independent.

Facts & Assumptions

Given: Finite real random variables on one finite probability space.

[L1]

Expectation is linear for every finite family of random variables, without any independence hypothesis (Expectation is linear for every finite family of random variables, without any independence hypothesis).

[L2]

A uniform two-point space assigns each outcome probability $1 / 2$ (The uniform probability space on a nonempty finite set).

[L3]

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

Refutation

technique · direct

1.1

On the uniform random-sign space, take $X (ω) = ω$ and $Y = X$ . Then $P (X = 1, Y = 1) = 1 / 2 \neq 1 / 4 = P (X = 1) P (Y = 1)$ , so [L3] shows that $X$ and $Y$ are dependent.

L2L3construct

2.1

Nevertheless [L1] gives $E [X + Y] = E [X] + E [Y]$ . Therefore independence is not necessary for linearity, so the statement is false.

step 1.1L1∎

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: 40 results over 14 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

C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Theorem 6.2 (standard reference, not scraped)