Independence of random variables

Mathematical statement

Finite-valued X,Y are independent if every pair (x,y) satisfies P(X=x,Y=y)=P(X=x)P(Y=y). Equivalently, all inverse images of value events are independent. This slice checks the complete finite joint table; zero covariance is only a consequence, never a converse certificate.

Representative formula (exact TeX)

K(X,Y)

Assumptions

• The complete finite joint support and exact rational masses belong to the same probability model.
• Primarily two variables are handled; pairwise independence does not establish mutual independence of three or more variables.

Symbol scope

• K — random-variable independence record
• X — first variable
• Y — second variable

Intuition, definition, and misconceptions

First aggregate both marginals from the joint table.

Every value pair must pass factorization.

Independence implies zero covariance with finite second moments, but zero covariance does not generally imply independence.

Checking one pair, looking only at marginals, or observing zero covariance does not establish independence or IID.

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

• Event independence — Inverse images of value events must satisfy the event-product obligation.
• Finite joint distribution — Every joint and marginal mass comes from the same complete law.
• Real random variables on finite sample spaces — X and Y are variables on the same probability space.
• Measurable function — Event inverse images must be measurable events.

Verification boundary

Only two variables under the same complete finite joint law are checked; identical distribution, IID, Markov behavior or three-variable mutual independence is not inferred.

not_formalized

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

Next obligation

Add independent identically distributed sequences.

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.