# Question 12558: Qualify a Whole Exact Joint Probability Table

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0024-12558 · q-20261013-advance-20261026-r04-finite-joint-distribution-0008



## Original question and givens

Use the complete list of distinct integer pairs=[[-5, -1], [-4, 0], [-3, 1], [-2, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).


- G001 — Use the complete list of distinct integer pairs=[[-5, -1], [-4, 0], [-3, 1], [-2, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-finite-joint-distribution](https://ocme.evemisslab.com/en/mko/mko-finite-joint-distribution/)





## Step-by-step derivation

S001 · calculationBefore

joint law exact total := sum(1/8, 1/4, 1/8, 1/2)

After

joint law exact total = 1

Conditions: none

Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001





S002 · impliesBefore

original ordered support=[[-5, -1], [-4, 0], [-3, 1], [-2, 4]];raw aligned masses=[[1, 8], [1, 4], [1, 8], [1, 2]]

After

normalized aligned masses=[1/8, 1/4, 1/8, 1/2];nonnegative=Yes;total=1;qualified=Yes

Conditions: none

Keep the complete joint table's original pair-mass alignment with distinct support pairs.Zero masses are allowed.Check nonnegativity and exact unit total separately;do not substitute marginals or a uniform law.

Step identifiers and dependenciesRule: whole_joint_table_not_marginal_substitution · Uses: G001, S001









## Reverse checks


- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Recompute joint law exact total with exact rational arithmetic, not rounded decimals.





## Result

([[-5, -1], [-4, 0], [-3, 1], [-2, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: a69d638af0f5079dfa3f39adfd92d95748b24d63371cbc76a9f9b0bea09eee88
- candidate record SHA-256: 2dee0eb872333aabfde0fc34efb0a17d884a79dbdcc9ed22f4c9e66cc9dfd825
- semantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35
- question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0008.json)