# Question 12525: Aggregate Both Marginals from the Same Qualified Joint Law [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-0050) 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. R0024-12525 · q-20261013-advance-20261026-r04-finite-joint-distribution-0050 ## Original question and givens Use the complete list of distinct integer pairs=[[12, -1], [13, 0], [14, 1], [15, 6]] 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. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero. - G001 — Use the complete list of distinct integer pairs=[[12, -1], [13, 0], [14, 1], [15, 6]] 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. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero. ## 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=[[12, -1], [13, 0], [14, 1], [15, 6]];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 · calculationBefore marginal coordinate0 label12 := sum(1/8) After marginal coordinate0 label12 = 1/8 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 S004 · impliesBefore coordinate=0;label=12;original support indices=[0] After aggregated marginal mass=1/8;listed label retained even at mass0 Conditions: none Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law. Step identifiers and dependenciesRule: joint_marginal_all_original_atoms · Uses: G001, S003 S005 · calculationBefore marginal coordinate0 label13 := sum(1/4) After marginal coordinate0 label13 = 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, S004 S006 · impliesBefore coordinate=0;label=13;original support indices=[1] After aggregated marginal mass=1/4;listed label retained even at mass0 Conditions: none Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law. Step identifiers and dependenciesRule: joint_marginal_all_original_atoms · Uses: G001, S005 S007 · calculationBefore marginal coordinate0 label14 := sum(1/8) After marginal coordinate0 label14 = 1/8 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, S006 S008 · impliesBefore coordinate=0;label=14;original support indices=[2] After aggregated marginal mass=1/8;listed label retained even at mass0 Conditions: none Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law. Step identifiers and dependenciesRule: joint_marginal_all_original_atoms · Uses: G001, S007 S009 · calculationBefore marginal coordinate0 label15 := sum(1/2) After marginal coordinate0 label15 = 1/2 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, S008 S010 · impliesBefore coordinate=0;label=15;original support indices=[3] After aggregated marginal mass=1/2;listed label retained even at mass0 Conditions: none Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law. Step identifiers and dependenciesRule: joint_marginal_all_original_atoms · Uses: G001, S009 S011 · calculationBefore marginal coordinate1 label-1 := sum(1/8) After marginal coordinate1 label-1 = 1/8 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, S010 S012 · impliesBefore coordinate=1;label=-1;original support indices=[0] After aggregated marginal mass=1/8;listed label retained even at mass0 Conditions: none Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law. Step identifiers and dependenciesRule: joint_marginal_all_original_atoms · Uses: G001, S011 S013 · calculationBefore marginal coordinate1 label0 := sum(1/4) After marginal coordinate1 label0 = 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, S012 S014 · impliesBefore coordinate=1;label=0;original support indices=[1] After aggregated marginal mass=1/4;listed label retained even at mass0 Conditions: none Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law. Step identifiers and dependenciesRule: joint_marginal_all_original_atoms · Uses: G001, S013 S015 · calculationBefore marginal coordinate1 label1 := sum(1/8) After marginal coordinate1 label1 = 1/8 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, S014 S016 · impliesBefore coordinate=1;label=1;original support indices=[2] After aggregated marginal mass=1/8;listed label retained even at mass0 Conditions: none Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law. Step identifiers and dependenciesRule: joint_marginal_all_original_atoms · Uses: G001, S015 S017 · calculationBefore marginal coordinate1 label6 := sum(1/2) After marginal coordinate1 label6 = 1/2 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, S016 S018 · impliesBefore coordinate=1;label=6;original support indices=[3] After aggregated marginal mass=1/2;listed label retained even at mass0 Conditions: none Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law. Step identifiers and dependenciesRule: joint_marginal_all_original_atoms · Uses: G001, S017 ## Reverse checks - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Recompute marginal coordinate1 label6 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Recompute marginal coordinate1 label1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Recompute marginal coordinate1 label0 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute marginal coordinate1 label-1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute marginal coordinate0 label15 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute marginal coordinate0 label14 with exact rational arithmetic, not rounded decimals. - 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 — Recompute marginal coordinate0 label13 with exact rational arithmetic, not rounded decimals. - 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 — Recompute marginal coordinate0 label12 with exact rational arithmetic, not rounded decimals. - 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 ([[12, 1/8], [13, 1/4], [14, 1/8], [15, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]]) 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: 8f387424391aec0a63dcd04a7bb3728e7eec978663befda39f8e5499bdff47cc - candidate record SHA-256: dc26f9d120843445795b019d08df2d9dff77a5faebebe0db12c4f21591ba977e - 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-0050.json)