False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck 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.
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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 · two levels

11 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

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