# Question 12588: Both Coordinates Must Belong to a Rectangle Event [Return to catalogue page 63](https://ocme.evemisslab.com/en/questions/page/63/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations/#q-20261013-advance-20261026-r04-finite-joint-distribution-0063) 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. R0024-12588 · q-20261013-advance-20261026-r04-finite-joint-distribution-0063 ## Original question and givens Use the complete list of distinct integer pairs=[[0, -1], [1, 0], [2, 1], [3, 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. The joint table must qualify first. x_event=[0, 2], 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=[[0, -1], [1, 0], [2, 1], [3, 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. The joint table must qualify first. x_event=[0, 2], 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](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=[[0, -1], [1, 0], [2, 1], [3, 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 S003 · impliesBefore raw x event=[0, 2];raw y event=[-1, 1] After sorted distinct x values=[0, 2];y values=[-1, 1] Conditions: none These event labels are coordinate values,not support-row indices.Normalize them as sets;outside-support values remain but contribute no probability. Step identifiers and dependenciesRule: joint_value_events_not_carrier_indices · Uses: G001, S002 S004 · impliesBefore original atom=0;pair=[0, -1];x event=[0, 2];y event=[-1, 1] After both coordinate memberships=Yes Conditions: none Both coordinates of the same atom must belong to their respective events;one membership alone is insufficient and no independence is assumed. Step identifiers and dependenciesRule: rectangle_event_and_not_or · Uses: G001, S003 S005 · impliesBefore original atom=1;pair=[1, 0];x event=[0, 2];y event=[-1, 1] After both coordinate memberships=No Conditions: none Both coordinates of the same atom must belong to their respective events;one membership alone is insufficient and no independence is assumed. Step identifiers and dependenciesRule: rectangle_event_and_not_or · Uses: G001, S004 S006 · impliesBefore original atom=2;pair=[2, 1];x event=[0, 2];y event=[-1, 1] After both coordinate memberships=Yes Conditions: none Both coordinates of the same atom must belong to their respective events;one membership alone is insufficient and no independence is assumed. Step identifiers and dependenciesRule: rectangle_event_and_not_or · Uses: G001, S005 S007 · impliesBefore original atom=3;pair=[3, 4];x event=[0, 2];y event=[-1, 1] After both coordinate memberships=No Conditions: none Both coordinates of the same atom must belong to their respective events;one membership alone is insufficient and no independence is assumed. Step identifiers and dependenciesRule: rectangle_event_and_not_or · Uses: G001, S006 S008 · calculationBefore joint rectangle event mass := sum(1/8, 1/8) After joint rectangle event mass = 1/4 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, S007 ## 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 ([0, 2], [-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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: ada9d138dbc6457788f18df382500a8f04b785948375ab17318e0d59c0360060 - candidate record SHA-256: a5ce4737e8954a1f50b81301ede940b708279dda1b3255d20b44225fe8d3c2e0 - 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-0063.json)