# Covariance

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Under one finite joint distribution, covariance is E[XY]−E[X]E[Y]; zero covariance does not in general imply independence.

mko-covariance · definition · v0.10.0



## 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

[Covariance: symmetry, the affine rule, and why zero covariance does not imply independence](https://ocme.evemisslab.com/en/derivations/derivation-mko-covariance-01/)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](https://ocme.evemisslab.com/en/mko/mko-finite-joint-distribution/) — Covariance requires a common joint law.
- [Expectation in finite models](https://ocme.evemisslab.com/en/mko/mko-expectation/) — The formula contains three expectations.
- [Variance in finite probability models](https://ocme.evemisslab.com/en/mko/mko-variance/) — Covariance and variance share centered second-order quantities.
- [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-random-variable/) — 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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/covariance.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-covariance/index.md)