# Question 12261: Symmetry of Covariance under Swapping Coordinates [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-0036) 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. R0023-12261 · q-20261013-advance-20261026-r05-covariance-0036 ## Original question and givens Use the complete list of distinct integer pairs=[[-2, -1], [-1, 0], [0, 1], [1, 2]] 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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?). - G001 — Use the complete list of distinct integer pairs=[[-2, -1], [-1, 0], [0, 1], [1, 2]] 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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?). ## 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 joint law total mass := sum(1/8, 1/4, 1/8, 1/2) 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 S002 · impliesBefore original distinct joint pairs=[[-2, -1], [-1, 0], [0, 1], [1, 2]];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 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, S001 S003 · calculationBefore original atom0 weighted X := mul(1/8, -2) After original atom0 weighted X = -1/4 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 S004 · calculationBefore original atom0 weighted Y := mul(1/8, -1) After original atom0 weighted Y = -1/8 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 XY := mul(-2, -1) After original atom0 XY = 2 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 weighted XY := mul(1/8, 2) After original atom0 weighted XY = 1/4 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 atom1 weighted X := mul(1/4, -1) After original atom1 weighted X = -1/4 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 Y := mul(1/4, 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, S007 S009 · calculationBefore original atom1 XY := mul(-1, 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, S008 S010 · calculationBefore original atom1 weighted XY := mul(1/4, 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, S009 S011 · calculationBefore original atom2 weighted X := mul(1/8, 0) After original atom2 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, S010 S012 · calculationBefore original atom2 weighted Y := mul(1/8, 1) After original atom2 weighted Y = 1/8 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 XY := mul(0, 1) After original atom2 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, S012 S014 · calculationBefore original atom2 weighted XY := mul(1/8, 0) After original atom2 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, S013 S015 · calculationBefore original atom3 weighted X := mul(1/2, 1) After original atom3 weighted X = 1/2 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 atom3 weighted Y := mul(1/2, 2) After original atom3 weighted Y = 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, S015 S017 · calculationBefore original atom3 XY := mul(1, 2) After original atom3 XY = 2 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 atom3 weighted XY := mul(1/2, 2) After original atom3 weighted XY = 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, S017 S019 · calculationBefore original E_X := sum(-1/4, -1/4, 0, 1/2) 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, S018 S020 · calculationBefore original E_Y := sum(-1/8, 0, 1/8, 1) After original E_Y = 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, S019 S021 · calculationBefore original E_XY := sum(1/4, 0, 0, 1) After original E_XY = 5/4 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, S020 S022 · calculationBefore original product of means := mul(0, 1) 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, S021 S023 · calculationBefore original covariance := sub(5/4, 0) After original covariance = 5/4 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 · impliesBefore original:E_X=0;E_Y=1;E_XY=5/4 After Cov=E_XY-E_X*E_Y=5/4 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, S023 S025 · calculationBefore swapped atom0 weighted X := mul(1/8, -1) After swapped atom0 weighted X = -1/8 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 · calculationBefore swapped atom0 weighted Y := mul(1/8, -2) After swapped atom0 weighted Y = -1/4 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, S025 S027 · calculationBefore swapped atom0 XY := mul(-1, -2) After swapped atom0 XY = 2 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, S026 S028 · calculationBefore swapped atom0 weighted XY := mul(1/8, 2) After swapped atom0 weighted XY = 1/4 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, S027 S029 · calculationBefore swapped atom1 weighted X := mul(1/4, 0) After swapped 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, S028 S030 · calculationBefore swapped atom1 weighted Y := mul(1/4, -1) After swapped atom1 weighted Y = -1/4 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, S029 S031 · calculationBefore swapped atom1 XY := mul(0, -1) After swapped 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, S030 S032 · calculationBefore swapped atom1 weighted XY := mul(1/4, 0) After swapped 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, S031 S033 · calculationBefore swapped atom2 weighted X := mul(1/8, 1) After swapped atom2 weighted X = 1/8 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, S032 S034 · calculationBefore swapped atom2 weighted Y := mul(1/8, 0) After swapped atom2 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, S033 S035 · calculationBefore swapped atom2 XY := mul(1, 0) After swapped atom2 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, S034 S036 · calculationBefore swapped atom2 weighted XY := mul(1/8, 0) After swapped atom2 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, S035 S037 · calculationBefore swapped atom3 weighted X := mul(1/2, 2) After swapped atom3 weighted X = 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, S036 S038 · calculationBefore swapped atom3 weighted Y := mul(1/2, 1) After swapped atom3 weighted Y = 1/2 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, S037 S039 · calculationBefore swapped atom3 XY := mul(2, 1) After swapped atom3 XY = 2 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, S038 S040 · calculationBefore swapped atom3 weighted XY := mul(1/2, 2) After swapped atom3 weighted XY = 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, S039 S041 · calculationBefore swapped E_X := sum(-1/8, 0, 1/8, 1) After swapped E_X = 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, S040 S042 · calculationBefore swapped E_Y := sum(-1/4, -1/4, 0, 1/2) After swapped E_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, S041 S043 · calculationBefore swapped E_XY := sum(1/4, 0, 0, 1) After swapped E_XY = 5/4 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, S042 S044 · calculationBefore swapped product of means := mul(1, 0) After swapped 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, S043 S045 · calculationBefore swapped covariance := sub(5/4, 0) After swapped covariance = 5/4 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, S044 S046 · impliesBefore swapped:E_X=1;E_Y=0;E_XY=5/4 After Cov=E_XY-E_X*E_Y=5/4 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, S045 S047 · impliesBefore original covariance=5/4;swapped covariance=5/4 After equal by XY=YX and E_X*E_Y=E_Y*E_X Conditions: none Swap both variable and mean roles.Commutativity leaves both the product expectation and product of means unchanged. Step identifiers and dependenciesRule: covariance_swap_both_roles_symmetry · Uses: G001, S046 ## Reverse checks - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S045 · premise_and_result_audit — Recompute swapped covariance with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute swapped product of means with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute swapped E_XY with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute swapped E_Y with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute swapped E_X with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute swapped atom3 weighted XY with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute swapped atom3 XY with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute swapped atom3 weighted Y with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute swapped atom3 weighted X with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute swapped atom2 weighted XY with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute swapped atom2 XY with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute swapped atom2 weighted Y with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute swapped atom2 weighted X with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute swapped atom1 weighted XY with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute swapped atom1 XY with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute swapped atom1 weighted Y with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute swapped atom1 weighted X with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute swapped atom0 weighted XY with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute swapped atom0 XY with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute swapped atom0 weighted Y with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute swapped atom0 weighted X with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Recompute original covariance with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute original product of means with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute original E_XY with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute original E_Y with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute original E_X with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute original atom3 weighted XY with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute original atom3 XY with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute original atom3 weighted Y with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute original atom3 weighted X with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute original atom2 weighted XY with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute original atom2 XY with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute original atom2 weighted Y with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute original atom2 weighted X with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute original atom1 weighted XY with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute original atom1 XY with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute original atom1 weighted Y with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute original atom1 weighted X with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute original atom0 weighted XY with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute original atom0 XY with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute original atom0 weighted Y with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute original atom0 weighted X 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 total mass with exact rational arithmetic, not rounded decimals. ## Result (5/4, 5/4, 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: 15ffc3b428f006efc6917be6660becf0a03ce04c79ea4d570a671e35ea1de510 - candidate record SHA-256: 4fe35fc38610b09c18722dbe19d89ce0f429e01dedfafdc76510cea93f155369 - 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-0036.json)