Finite joint distribution

Mathematical statement

A two-dimensional finite joint distribution lists distinct pairs (x,y) and nonnegative exact masses summing exactly to 1. Marginals, rectangle events and single-pair masses all aggregate from that same joint law.

Representative formula (exact TeX)

J(X,Y)

Assumptions

• The listed support consists of distinct integer pairs and the masses are exact rational numbers.
• The joint law is a mathematical model, not a claim of empirical data fitting or a continuous density.

Symbol scope

• J — finite joint distribution
• X — first random variable
• Y — second random variable

Intuition, definition, and misconceptions

The two marginals aggregate the same listed support separately by coordinate.

A rectangle event requires both coordinates to belong to their specified sets.

Marginals generally do not determine the complete joint law.

Do not treat two marginal tables as a joint distribution or allow masses that sum only approximately to 1.

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 discrete probability distribution — Joint value pairs form a finite support and probability mass function.
• Real random variables on finite sample spaces — Coordinate values come from variables on a common sample space.
• Probability on finite sample spaces — Event masses follow the probability axioms.
• Sample space — A common probability space grounds the joint semantics.

Verification boundary

Only complete finite joint PMFs are handled; no continuous-distribution conclusion is 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 covariance, independence of random variables and conditional expectation.

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.