# Question 12220: Affine Shifts Drop Out of Covariance [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-0070) 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. R0023-12220 · q-20261013-advance-20261026-r05-covariance-0070 ## Original question and givens Use the complete list of distinct integer pairs=[[7, -1], [8, 0], [9, 1], [10, 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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[2, 7, 2, -7]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)). - G001 — Use the complete list of distinct integer pairs=[[7, -1], [8, 0], [9, 1], [10, 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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[2, 7, 2, -7]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)). ## 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=[[7, -1], [8, 0], [9, 1], [10, 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 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, 7) After original atom0 weighted X = 7/8 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, 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(7, -1) After original atom0 XY = -7 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, -7) After original atom0 weighted XY = -7/8 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, S005 S007 · calculationBefore original atom1 weighted X := mul(1/4, 8) After original atom1 weighted X = 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, 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(8, 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, 9) After original atom2 weighted X = 9/8 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, 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(9, 1) After original atom2 XY = 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, S012 S014 · calculationBefore original atom2 weighted XY := mul(1/8, 9) After original atom2 weighted XY = 9/8 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, S013 S015 · calculationBefore original atom3 weighted X := mul(1/2, 10) After original atom3 weighted X = 5 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, 6) After original atom3 weighted Y = 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, S015 S017 · calculationBefore original atom3 XY := mul(10, 6) After original atom3 XY = 60 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, 60) After original atom3 weighted XY = 30 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(7/8, 2, 9/8, 5) After original E_X = 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, S018 S020 · calculationBefore original E_Y := sum(-1/8, 0, 1/8, 3) After original E_Y = 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, S019 S021 · calculationBefore original E_XY := sum(-7/8, 0, 9/8, 30) After original E_XY = 121/4 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(9, 3) After original product of means = 27 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(121/4, 27) After original covariance = 13/4 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=9;E_Y=3;E_XY=121/4 After Cov=E_XY-E_X*E_Y=13/4 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 affine atom0 scaled X := mul(2, 7) After affine atom0 scaled X = 14 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 affine atom0 U := sum(14, 7) After affine atom0 U = 21 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 affine atom0 scaled Y := mul(2, -1) After affine atom0 scaled Y = -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 affine atom0 V := sum(-2, -7) After affine atom0 V = -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, S027 S029 · calculationBefore affine atom1 scaled X := mul(2, 8) After affine atom1 scaled X = 16 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 affine atom1 U := sum(16, 7) After affine atom1 U = 23 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 affine atom1 scaled Y := mul(2, 0) After affine atom1 scaled 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, S030 S032 · calculationBefore affine atom1 V := sum(0, -7) After affine atom1 V = -7 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 affine atom2 scaled X := mul(2, 9) After affine atom2 scaled X = 18 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 affine atom2 U := sum(18, 7) After affine atom2 U = 25 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 affine atom2 scaled Y := mul(2, 1) After affine atom2 scaled Y = 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, S034 S036 · calculationBefore affine atom2 V := sum(2, -7) After affine atom2 V = -5 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 affine atom3 scaled X := mul(2, 10) After affine atom3 scaled X = 20 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 affine atom3 U := sum(20, 7) After affine atom3 U = 27 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 affine atom3 scaled Y := mul(2, 6) After affine atom3 scaled Y = 12 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 affine atom3 V := sum(12, -7) After affine atom3 V = 5 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 affine transformed atom0 weighted X := mul(1/8, 21) After affine transformed atom0 weighted X = 21/8 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, S040 S042 · calculationBefore affine transformed atom0 weighted Y := mul(1/8, -9) After affine transformed atom0 weighted Y = -9/8 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, S041 S043 · calculationBefore affine transformed atom0 XY := mul(21, -9) After affine transformed atom0 XY = -189 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 affine transformed atom0 weighted XY := mul(1/8, -189) After affine transformed atom0 weighted XY = -189/8 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, S043 S045 · calculationBefore affine transformed atom1 weighted X := mul(1/4, 23) After affine transformed atom1 weighted X = 23/4 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 · calculationBefore affine transformed atom1 weighted Y := mul(1/4, -7) After affine transformed atom1 weighted Y = -7/4 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, S045 S047 · calculationBefore affine transformed atom1 XY := mul(23, -7) After affine transformed atom1 XY = -161 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, S046 S048 · calculationBefore affine transformed atom1 weighted XY := mul(1/4, -161) After affine transformed atom1 weighted XY = -161/4 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, S047 S049 · calculationBefore affine transformed atom2 weighted X := mul(1/8, 25) After affine transformed atom2 weighted X = 25/8 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, S048 S050 · calculationBefore affine transformed atom2 weighted Y := mul(1/8, -5) After affine transformed atom2 weighted Y = -5/8 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, S049 S051 · calculationBefore affine transformed atom2 XY := mul(25, -5) After affine transformed atom2 XY = -125 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, S050 S052 · calculationBefore affine transformed atom2 weighted XY := mul(1/8, -125) After affine transformed atom2 weighted XY = -125/8 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, S051 S053 · calculationBefore affine transformed atom3 weighted X := mul(1/2, 27) After affine transformed atom3 weighted X = 27/2 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, S052 S054 · calculationBefore affine transformed atom3 weighted Y := mul(1/2, 5) After affine transformed atom3 weighted Y = 5/2 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, S053 S055 · calculationBefore affine transformed atom3 XY := mul(27, 5) After affine transformed atom3 XY = 135 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, S054 S056 · calculationBefore affine transformed atom3 weighted XY := mul(1/2, 135) After affine transformed atom3 weighted XY = 135/2 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, S055 S057 · calculationBefore affine transformed E_X := sum(21/8, 23/4, 25/8, 27/2) After affine transformed E_X = 25 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, S056 S058 · calculationBefore affine transformed E_Y := sum(-9/8, -7/4, -5/8, 5/2) After affine transformed 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, S057 S059 · calculationBefore affine transformed E_XY := sum(-189/8, -161/4, -125/8, 135/2) After affine transformed E_XY = -12 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, S058 S060 · calculationBefore affine transformed product of means := mul(25, -1) After affine transformed product of means = -25 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, S059 S061 · calculationBefore affine transformed covariance := sub(-12, -25) After affine transformed covariance = 13 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, S060 S062 · impliesBefore affine transformed:E_X=25;E_Y=-1;E_XY=-12 After Cov=E_XY-E_X*E_Y=13 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, S061 S063 · calculationBefore affine two scale product := mul(2, 2) After affine two scale product = 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, S062 S064 · calculationBefore affine scaled original covariance := mul(4, 13/4) After affine scaled original covariance = 13 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, S063 S065 · impliesBefore U=(2)*X+(7);V=(2)*Y+(-7);same indexed atoms and weights After E_UV-E_U*E_V=(2)*(2)*(E_XY-E_X*E_Y)=13 Conditions: none Expand E[UV]=acE[XY]+adE[X]+bcE[Y]+bd and subtract(aE[X]+b)(cE[Y]+d);all shift terms cancel.Even zero scaling may duplicate image points:retain original probability atoms rather than treating images as a new distinct support. Step identifiers and dependenciesRule: affine_covariance_shift_cancellation_not_new_joint_support · Uses: G001, S064 ## Reverse checks - S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S064 · premise_and_result_audit — Recompute affine scaled original covariance with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Recompute affine two scale product with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S061 · premise_and_result_audit — Recompute affine transformed covariance with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute affine transformed product of means with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute affine transformed E_XY with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute affine transformed E_Y with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute affine transformed E_X with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute affine transformed atom3 weighted XY with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute affine transformed atom3 XY with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute affine transformed atom3 weighted Y with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute affine transformed atom3 weighted X with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute affine transformed atom2 weighted XY with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute affine transformed atom2 XY with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute affine transformed atom2 weighted Y with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute affine transformed atom2 weighted X with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute affine transformed atom1 weighted XY with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute affine transformed atom1 XY with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute affine transformed atom1 weighted Y with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute affine transformed atom1 weighted X with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute affine transformed atom0 weighted XY with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute affine transformed atom0 XY with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute affine transformed atom0 weighted Y with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute affine transformed atom0 weighted X with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute affine atom3 V with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute affine atom3 scaled Y with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute affine atom3 U with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute affine atom3 scaled X with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute affine atom2 V with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute affine atom2 scaled Y with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute affine atom2 U with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute affine atom2 scaled X with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute affine atom1 V with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute affine atom1 scaled Y with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute affine atom1 U with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute affine atom1 scaled X with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute affine atom0 V with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute affine atom0 scaled Y with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute affine atom0 U with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute affine atom0 scaled 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 (2, 7, 2, -7, 13/4, 13, Cov(aX+b,cY+d)=ac Cov(X,Y)) 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: fd96534c3fc61e518e1caba4733ecf7a4738044422a8fe57a0c10b14b3da95c4 - candidate record SHA-256: 9c40eb46a2e2dcc9ab3b9607437ffbac723a5356ae88d76f5e2cdf5321d40ff3 - 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-0070.json)