Question 12589: Both Coordinates Must Belong to a Rectangle Event

Original question and givens

Use the complete list of distinct integer pairs=[[1, -1], [2, 0], [3, 1], [4, 5]] 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. The joint table must qualify first. x_event=[1, 3], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.

• G001 — Use the complete list of distinct integer pairs=[[1, -1], [2, 0], [3, 1], [4, 5]] 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. The joint table must qualify first. x_event=[1, 3], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.

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=[[1, -1], [2, 0], [3, 1], [4, 5]];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

raw x event=[1, 3];raw y event=[-1, 1]

sorted distinct x values=[1, 3];y values=[-1, 1]

original atom=0;pair=[1, -1];x event=[1, 3];y event=[-1, 1]

both coordinate memberships=Yes

original atom=1;pair=[2, 0];x event=[1, 3];y event=[-1, 1]

both coordinate memberships=No

original atom=2;pair=[3, 1];x event=[1, 3];y event=[-1, 1]

both coordinate memberships=Yes

original atom=3;pair=[4, 5];x event=[1, 3];y event=[-1, 1]

both coordinate memberships=No

joint rectangle event mass := sum(1/8, 1/8)

joint rectangle event mass = 1/4

Reverse checks

• S008 · premise_and_result_audit — Recompute joint rectangle event mass with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

([1, 3], [-1, 1], 1/4)

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.