# Finite joint distribution

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

Value tuples of random variables on one common finite probability space, together with exact masses, specify a finite joint distribution.

mko-finite-joint-distribution · definition · v0.10.0



## 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

[Finite joint distribution: validating a joint mass table and aggregating marginals and events from one joint law](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-joint-distribution-01/)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](https://ocme.evemisslab.com/en/mko/mko-finite-discrete-probability-distribution/) — Joint value pairs form a finite support and probability mass function.
- [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-random-variable/) — Coordinate values come from variables on a common sample space.
- [Probability on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-probability/) — Event masses follow the probability axioms.
- [Sample space](https://ocme.evemisslab.com/en/mko/mko-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.

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