Covariance

Mathematical statement

For X,Y on the same finite joint law, Cov(X,Y)=E[XY]−E[X]E[Y]. Covariance is symmetric in the two variables, and affine transformations satisfy Cov(aX+b,cY+d)=ac Cov(X,Y).

Representative formula (exact TeX)

Cov(X,Y)

Assumptions

• Every expectation is calculated from the same exact finite joint law.
• Nonzero covariance can refute independence; zero covariance cannot in general prove it.

Symbol scope

• Cov — covariance
• X — first variable
• Y — second variable

Intuition, definition, and misconceptions

Calculate E[X], E[Y] and E[XY] before subtracting.

Swapping X and Y does not change their covariance.

Taking X uniformly on −1,0,1 and Y=X² gives a finite example of zero covariance with dependence.

Do not infer independence from Cov=0 or mix expectations from different joint models.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Finite joint distribution — Covariance requires a common joint law.
• Expectation in finite models — The formula contains three expectations.
• Variance in finite probability models — Covariance and variance share centered second-order quantities.
• Real random variables on finite sample spaces — X and Y belong to a common probability space.

Verification boundary

Zero covariance is not an independence certificate.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add independence of random variables, conditional expectation and correlation coefficients.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.