CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Independence forces covariance to vanish

Statement

If $X$ and $Y$ are independent square-integrable real random variables, then $Cov (X, Y) = 0.$

Thus independence implies zero covariance. The converse is false in general.

Facts & Assumptions

Given: Independent square-integrable real random variables $X$ and $Y$ .

[L1]

Expectations factor for products of integrable independent random variables. (Expectations factor over finite products of independent random variables)

[L2]

Covariance satisfies $Cov (X, Y) = E [X Y] - E [X] E [Y]$ . (Moments, variance, and covariance on a probability space, Variance and covariance identities for random variables)

Proof

technique · direct

1.1

Since $X$ and $Y$ are square-integrable, they are integrable. Applying [L1] with $g_{0} (x) = x$ and $g_{1} (y) = y$ gives $E [X Y] = E [X] E [Y] .$

givenL1

2.1

Substituting step 1.1 into [L2] yields $Cov (X, Y) = E [X Y] - E [X] E [Y] = 0.$

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Rick Durrett, Probability: Theory and Examples, 5th ed., Section 2.1.2 (standard reference, not scraped)