Question 12572: Qualify a Whole Exact Joint Probability Table

Original question and givens

Use the complete list of distinct integer pairs=[[9, -1], [10, 0], [11, 1], [12, 3]] 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=[[9, -1], [10, 0], [11, 1], [12, 3]] 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

Step-by-step derivation

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

joint law exact total = 1

original ordered support=[[9, -1], [10, 0], [11, 1], [12, 3]];raw aligned masses=[[1, 8], [1, 4], [1, 8], [1, 2]]

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

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

([[9, -1], [10, 0], [11, 1], [12, 3]], [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.