# Question 12295: Zero Covariance Can Coexist with Dependence [Return to catalogue page 62](https://ocme.evemisslab.com/en/questions/page/62/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261026-r05-covariance-foundations/#q-20261013-advance-20261026-r05-covariance-0095) 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. R0023-12295 · q-20261013-advance-20261026-r05-covariance-0095 ## Original question and givens Take X uniformly on [−scale,0,scale] for nonzero positive scale=20, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers. - G001 — Take X uniformly on [−scale,0,scale] for nonzero positive scale=20, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers. ## 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-covariance](https://ocme.evemisslab.com/en/mko/mko-covariance/) ## Step-by-step derivation S001 · calculationBefore zero covariance scale squared := mul(20, 20) After zero covariance scale squared = 400 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore joint law total mass := sum(1/3, 1/3, 1/3) After joint law total mass = 1 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore original distinct joint pairs=[[-20, 400], [0, 0], [20, 400]];raw aligned masses=[[1, 3], [1, 3], [1, 3]] After normalized aligned masses=[1/3, 1/3, 1/3];nonnegative=Yes;total=1 Conditions: none Use the complete original joint table on one common probability space,preserving order,pairing and zero masses.Covariance requires nonnegative masses with exact total1. Step identifiers and dependenciesRule: qualify_whole_joint_law_preserve_alignment · Uses: G001, S002 S004 · calculationBefore original atom0 weighted X := mul(1/3, -20) After original atom0 weighted X = -20/3 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore original atom0 weighted Y := mul(1/3, 400) After original atom0 weighted Y = 400/3 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore original atom0 XY := mul(-20, 400) After original atom0 XY = -8000 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore original atom0 weighted XY := mul(1/3, -8000) After original atom0 weighted XY = -8000/3 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore original atom1 weighted X := mul(1/3, 0) After original atom1 weighted X = 0 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore original atom1 weighted Y := mul(1/3, 0) After original atom1 weighted Y = 0 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore original atom1 XY := mul(0, 0) After original atom1 XY = 0 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore original atom1 weighted XY := mul(1/3, 0) After original atom1 weighted XY = 0 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore original atom2 weighted X := mul(1/3, 20) After original atom2 weighted X = 20/3 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore original atom2 weighted Y := mul(1/3, 400) After original atom2 weighted Y = 400/3 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore original atom2 XY := mul(20, 400) After original atom2 XY = 8000 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore original atom2 weighted XY := mul(1/3, 8000) After original atom2 weighted XY = 8000/3 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore original E_X := sum(-20/3, 0, 20/3) After original E_X = 0 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore original E_Y := sum(400/3, 0, 400/3) After original E_Y = 800/3 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore original E_XY := sum(-8000/3, 0, 8000/3) After original E_XY = 0 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore original product of means := mul(0, 800/3) After original product of means = 0 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore original covariance := sub(0, 0) After original covariance = 0 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore original:E_X=0;E_Y=800/3;E_XY=0 After Cov=E_XY-E_X*E_Y=0 Conditions: none All three expectations use the same atoms and masses.Do not mix marginal models or assume independence. Step identifiers and dependenciesRule: covariance_same_joint_three_moments · Uses: G001, S020 S022 · calculationBefore joint zero event mass := sum(1/3) After joint zero event mass = 1/3 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore X zero marginal mass := sum(1/3) After X zero marginal mass = 1/3 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Y zero marginal mass := sum(1/3) After Y zero marginal mass = 1/3 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore zero event product of marginal masses := mul(1/3, 1/3) After zero event product of marginal masses = 1/9 Conditions: none Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore X uniform on [-20, 0, 20];Y=X^2;Cov=0;P(X=0,Y=0)=1/3 After P(X=0)*P(Y=0)=1/9;different=Yes Conditions: none Symmetry makes E[X] and E[XY] zero,hence zero covariance.Independence requires factorization for every event pair;the zero events with1/3≠1/9 give one exact refutation. Step identifiers and dependenciesRule: zero_covariance_explicit_independence_counterevent · Uses: G001, S025 ## Reverse checks - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Recompute zero event product of marginal masses with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Y zero marginal mass with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute X zero marginal mass with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute joint zero event mass with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Recompute original covariance with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute original product of means with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute original E_XY with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute original E_Y with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute original E_X with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute original atom2 weighted XY with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute original atom2 XY with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute original atom2 weighted Y with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute original atom2 weighted X with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute original atom1 weighted XY with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute original atom1 XY with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute original atom1 weighted Y with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute original atom1 weighted X with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute original atom0 weighted XY with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute original atom0 XY with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute original atom0 weighted Y with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute original atom0 weighted X 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 — Recompute joint law total mass with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Recompute zero covariance scale squared with exact rational arithmetic, not rounded decimals. ## Result (20, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent) 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: 97195207c3396d453df5b4530b4f6bc81253878f1e6da420e853949a5d9f369c - candidate record SHA-256: 3ee5effc29df424007ca37a0798200750b87300e43b5cf8a1e87b56c42175e83 - semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff - question-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261026-r05-covariance-0095.json)