# Question 14085: Keep Verified Independence and Zero Covariance Distinct [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-0085) 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. R0027-14085 · q-20261014-advance-20261028-r06-random-variable-independence-0085 ## Original question and givens Use the complete joint grid with original x_values=[-3, 22], y_values=[-1, 29], and aligned row-by-column exact masses=[[10/49, 4/49], [25/49, 10/49]]. 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 (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication. - G001 — Use the complete joint grid with original x_values=[-3, 22], y_values=[-1, 29], and aligned row-by-column exact masses=[[10/49, 4/49], [25/49, 10/49]]. 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 (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication. ## 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(10/49, 4/49, 25/49, 10/49) 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=[[10, 49], [4, 49], [25, 49], [10, 49]] After normalized masses=[10/49, 4/49, 25/49, 10/49];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=[-3, 22];original y column labels=[-1, 29] After normalized aligned row-by-column joint grid=[[10/49, 4/49], [25/49, 10/49]] 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(10/49, 4/49) After X row0 marginal = 2/7 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(25/49, 10/49) After X row1 marginal = 5/7 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(10/49, 25/49) After Y column0 marginal = 5/7 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, 10/49) After Y column1 marginal = 2/7 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(2/7, 5/7) After joint cell row0 column0 marginal product = 10/49 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=[-3, -1];joint mass=10/49;marginal product=10/49 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(2/7, 2/7) 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=[-3, 29];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(5/7, 5/7) After joint cell row1 column0 marginal product = 25/49 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=[22, -1];joint mass=25/49;marginal product=25/49 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(5/7, 2/7) After joint cell row1 column1 marginal product = 10/49 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=[22, 29];joint mass=10/49;marginal product=10/49 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 S017 · calculationBefore moment row0 col0 weighted X := mul(10/49, -3) After moment row0 col0 weighted X = -30/49 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, S016 S018 · calculationBefore moment row0 col0 weighted Y := mul(10/49, -1) After moment row0 col0 weighted Y = -10/49 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, S017 S019 · calculationBefore moment row0 col0 XY := mul(-3, -1) After moment row0 col0 XY = 3 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, S018 S020 · calculationBefore moment row0 col0 weighted XY := mul(10/49, 3) After moment row0 col0 weighted XY = 30/49 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, S019 S021 · calculationBefore moment row0 col1 weighted X := mul(4/49, -3) After moment row0 col1 weighted X = -12/49 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, S020 S022 · calculationBefore moment row0 col1 weighted Y := mul(4/49, 29) After moment row0 col1 weighted Y = 116/49 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, S021 S023 · calculationBefore moment row0 col1 XY := mul(-3, 29) After moment row0 col1 XY = -87 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, S022 S024 · calculationBefore moment row0 col1 weighted XY := mul(4/49, -87) After moment row0 col1 weighted XY = -348/49 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, S023 S025 · calculationBefore moment row1 col0 weighted X := mul(25/49, 22) After moment row1 col0 weighted X = 550/49 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, S024 S026 · calculationBefore moment row1 col0 weighted Y := mul(25/49, -1) After moment row1 col0 weighted Y = -25/49 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, S025 S027 · calculationBefore moment row1 col0 XY := mul(22, -1) After moment row1 col0 XY = -22 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, S026 S028 · calculationBefore moment row1 col0 weighted XY := mul(25/49, -22) After moment row1 col0 weighted XY = -550/49 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, S027 S029 · calculationBefore moment row1 col1 weighted X := mul(10/49, 22) After moment row1 col1 weighted X = 220/49 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, S028 S030 · calculationBefore moment row1 col1 weighted Y := mul(10/49, 29) After moment row1 col1 weighted Y = 290/49 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, S029 S031 · calculationBefore moment row1 col1 XY := mul(22, 29) After moment row1 col1 XY = 638 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, S030 S032 · calculationBefore moment row1 col1 weighted XY := mul(10/49, 638) After moment row1 col1 weighted XY = 6380/49 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, S031 S033 · calculationBefore grid E_X := sum(-30/49, -12/49, 550/49, 220/49) After grid E_X = 104/7 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, S032 S034 · calculationBefore grid E_Y := sum(-10/49, 116/49, -25/49, 290/49) After grid E_Y = 53/7 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, S033 S035 · calculationBefore grid E_XY := sum(30/49, -348/49, -550/49, 6380/49) After grid E_XY = 5512/49 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, S034 S036 · calculationBefore grid product of means := mul(104/7, 53/7) After grid product of means = 5512/49 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, S035 S037 · calculationBefore grid covariance := sub(5512/49, 5512/49) After grid covariance = 0 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, S036 S038 · impliesBefore verified independence=Yes;E_X=104/7;E_Y=53/7;E_XY=5512/49;covariance=0 After source implication_pass is conjunction(independent,covariance0)=Yes,not material implication Conditions: none Verified all-cell independence distributes the finite double sum to E[XY]=E[X]E[Y],hence Cov=0.Despite the field name implication_pass,the protocol asks both facts to hold.A dependent zero-covariance model still returns No;this neither refutes the theorem nor proves the converse. Step identifiers and dependenciesRule: RV_zero_covariance_conjunction_not_converse · Uses: G001, S037 ## Reverse checks - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Recompute grid covariance with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute grid product of means with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute grid E_XY with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute grid E_Y with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute grid E_X with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute moment row1 col1 weighted XY with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute moment row1 col1 XY with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute moment row1 col1 weighted Y with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute moment row1 col1 weighted X with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute moment row1 col0 weighted XY with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute moment row1 col0 XY with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute moment row1 col0 weighted Y with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute moment row1 col0 weighted X with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute moment row0 col1 weighted XY with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute moment row0 col1 XY with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute moment row0 col1 weighted Y with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute moment row0 col1 weighted X with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute moment row0 col0 weighted XY with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute moment row0 col0 XY with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute moment row0 col0 weighted Y with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute moment row0 col0 weighted X 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 — 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 (Yes, 104/7, 53/7, 5512/49, 0, Yes, zero_covariance_converse_not_claimed) 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: 22be1f6a2deb2867022c9fe9ac2556c9d112c18808aca1ee3f93824ff735803d - candidate record SHA-256: 09ec126f800b1d27993c7b554f8dd70c62c5f2fb847fee9d2e6d3e8bb790d8de - 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-0085.json)