# Question 14053: Every Joint Grid Entry Must Factorize [Return to catalogue page 71](https://ocme.evemisslab.com/en/questions/page/71/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261028-r06-random-variable-independence-foundations/#q-20261014-advance-20261028-r06-random-variable-independence-0003) 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. R0027-14053 · q-20261014-advance-20261028-r06-random-variable-independence-0003 ## Original question and givens Use the complete joint grid with original x_values=[-10, 15], y_values=[-1, 22], and aligned row-by-column exact masses=[[24/49, 4/49], [18/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?). - G001 — Use the complete joint grid with original x_values=[-10, 15], y_values=[-1, 22], and aligned row-by-column exact masses=[[24/49, 4/49], [18/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?). ## 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-random-variable-independence](https://ocme.evemisslab.com/en/mko/mko-random-variable-independence/) ## Step-by-step derivation S001 · calculationBefore joint grid model whole mass sum := sum(24/49, 4/49, 18/49, 3/49) After joint grid model whole mass sum = 1 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · impliesBefore joint grid model:raw aligned masses=[[24, 49], [4, 49], [18, 49], [3, 49]] After normalized masses=[24/49, 4/49, 18/49, 3/49];nonnegative=Yes;total=1 Conditions: none First qualify nonnegativity and exact unit total on one complete finite probability model.Retain zero-mass atoms;do not assume uniformity or substitute another model. Step identifiers and dependenciesRule: whole_finite_probability_law_not_uniform_assumption · Uses: G001, S001 S003 · impliesBefore original x row labels=[-10, 15];original y column labels=[-1, 22] After normalized aligned row-by-column joint grid=[[24/49, 4/49], [18/49, 3/49]] Conditions: none Each value list is distinct but need not be numerically sorted.Pair every mass with its original row/column,including zero cells;sorting values must not misalign the grid. Step identifiers and dependenciesRule: RV_joint_grid_original_order_not_value_sort · Uses: G001, S002 S004 · calculationBefore X row0 marginal := sum(24/49, 4/49) After X row0 marginal = 4/7 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore X row1 marginal := sum(18/49, 3/49) After X row1 marginal = 3/7 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Y column0 marginal := sum(24/49, 18/49) After Y column0 marginal = 6/7 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Y column1 marginal := sum(4/49, 3/49) After Y column1 marginal = 1/7 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore joint cell row0 column0 marginal product := mul(4/7, 6/7) After joint cell row0 column0 marginal product = 24/49 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore original row=0;column=0;value labels=[-10, -1];joint mass=24/49;marginal product=24/49 After cell factorizes=Yes Conditions: none Audit every cell's singleton-event product,including zeros.One failed cell gives a singleton-preimage counterexample.Retain original row-major failure order. Step identifiers and dependenciesRule: RV_independence_every_joint_cell · Uses: G001, S008 S010 · calculationBefore joint cell row0 column1 marginal product := mul(4/7, 1/7) After joint cell row0 column1 marginal product = 4/49 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore original row=0;column=1;value labels=[-10, 22];joint mass=4/49;marginal product=4/49 After cell factorizes=Yes Conditions: none Audit every cell's singleton-event product,including zeros.One failed cell gives a singleton-preimage counterexample.Retain original row-major failure order. Step identifiers and dependenciesRule: RV_independence_every_joint_cell · Uses: G001, S010 S012 · calculationBefore joint cell row1 column0 marginal product := mul(3/7, 6/7) After joint cell row1 column0 marginal product = 18/49 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore original row=1;column=0;value labels=[15, -1];joint mass=18/49;marginal product=18/49 After cell factorizes=Yes Conditions: none Audit every cell's singleton-event product,including zeros.One failed cell gives a singleton-preimage counterexample.Retain original row-major failure order. Step identifiers and dependenciesRule: RV_independence_every_joint_cell · Uses: G001, S012 S014 · calculationBefore joint cell row1 column1 marginal product := mul(3/7, 1/7) After joint cell row1 column1 marginal product = 3/49 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore original row=1;column=1;value labels=[15, 22];joint mass=3/49;marginal product=3/49 After cell factorizes=Yes Conditions: none Audit every cell's singleton-event product,including zeros.One failed cell gives a singleton-preimage counterexample.Retain original row-major failure order. Step identifiers and dependenciesRule: RV_independence_every_joint_cell · Uses: G001, S014 S016 · impliesBefore all joint factorization failures=[] After random variables independent=Yes;all-cell factorization covers every value-event rectangle Conditions: none If every cell factorizes,any value-event rectangle sum splits into a product of marginal sums,covering every preimage-event choice.This implies no identical distribution,IID,Markov behavior or larger-family mutual independence. Step identifiers and dependenciesRule: RV_all_cells_to_all_event_preimages · Uses: G001, S015 ## Reverse checks - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Recompute joint cell row1 column1 marginal product with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute joint cell row1 column0 marginal product with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute joint cell row0 column1 marginal product with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute joint cell row0 column0 marginal product with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Y column1 marginal with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Y column0 marginal with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute X row1 marginal with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute X row0 marginal with exact rational arithmetic, not rounded decimals. - 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 grid model whole mass sum with exact rational arithmetic, not rounded decimals. ## Result ([-10, 15], [-1, 22], [4/7, 3/7], [6/7, 1/7], [], 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: 07471d5f37ba9bcd3331a8277b7ba297527d28cc3e95ebe93450eccad86ee77d - candidate record SHA-256: edf6de70ca241474946009b5b841b449a4b9c868ffd64d5da4f97db414382573 - semantic review receipt SHA-256: e49bf5eaa968c09ebac4ad538ce58251671ed922c957cee1471cfc7b68934bfe - question-first witness SHA-256: a7976057c26051fab92342fc95d48af224466c0e1e4ab6ae4c8811c02772e6da - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261028-r06-random-variable-independence-0003.json)