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R0023-12273 · q-20261013-advance-20261026-r05-covariance-0048
Use the complete list of distinct integer pairs=[[10, -1], [11, 0], [12, 1], [13, 4]] 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=[[10, -1], [11, 0], [12, 1], [13, 4]] 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?).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.
Before
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.
Rule: exact_rational_arithmetic · Uses: G001
Before
original distinct joint pairs=[[10, -1], [11, 0], [12, 1], [13, 4]];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.
Rule: qualify_whole_joint_law_preserve_alignment · Uses: G001, S001
Before
original atom0 weighted X := mul(1/8, 10)
After
original atom0 weighted X = 5/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
original atom0 XY := mul(10, -1)
After
original atom0 XY = -10
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
original atom0 weighted XY := mul(1/8, -10)
After
original atom0 weighted XY = -5/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
original atom1 weighted X := mul(1/4, 11)
After
original atom1 weighted X = 11/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
original atom1 XY := mul(11, 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.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
original atom2 weighted X := mul(1/8, 12)
After
original atom2 weighted X = 3/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
original atom2 XY := mul(12, 1)
After
original atom2 XY = 12
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
original atom2 weighted XY := mul(1/8, 12)
After
original atom2 weighted XY = 3/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
original atom3 weighted X := mul(1/2, 13)
After
original atom3 weighted X = 13/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
original atom3 weighted Y := mul(1/2, 4)
After
original atom3 weighted Y = 2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
original atom3 XY := mul(13, 4)
After
original atom3 XY = 52
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
original atom3 weighted XY := mul(1/2, 52)
After
original atom3 weighted XY = 26
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
original E_X := sum(5/4, 11/4, 3/2, 13/2)
After
original E_X = 12
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
original E_Y := sum(-1/8, 0, 1/8, 2)
After
original E_Y = 2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
original E_XY := sum(-5/4, 0, 3/2, 26)
After
original E_XY = 105/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
original product of means := mul(12, 2)
After
original product of means = 24
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
original covariance := sub(105/4, 24)
After
original covariance = 9/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
original:E_X=12;E_Y=2;E_XY=105/4
After
Cov=E_XY-E_X*E_Y=9/4
Conditions: none
All three expectations use the same atoms and masses.Do not mix marginal models or assume independence.
Rule: covariance_same_joint_three_moments · Uses: G001, S023
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
swapped atom0 weighted Y := mul(1/8, 10)
After
swapped atom0 weighted Y = 5/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
swapped atom0 XY := mul(-1, 10)
After
swapped atom0 XY = -10
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
swapped atom0 weighted XY := mul(1/8, -10)
After
swapped atom0 weighted XY = -5/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
swapped atom1 weighted Y := mul(1/4, 11)
After
swapped atom1 weighted Y = 11/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
swapped atom1 XY := mul(0, 11)
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.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
swapped atom2 weighted Y := mul(1/8, 12)
After
swapped atom2 weighted Y = 3/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
swapped atom2 XY := mul(1, 12)
After
swapped atom2 XY = 12
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
swapped atom2 weighted XY := mul(1/8, 12)
After
swapped atom2 weighted XY = 3/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
swapped atom3 weighted X := mul(1/2, 4)
After
swapped atom3 weighted X = 2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
swapped atom3 weighted Y := mul(1/2, 13)
After
swapped atom3 weighted Y = 13/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
swapped atom3 XY := mul(4, 13)
After
swapped atom3 XY = 52
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
swapped atom3 weighted XY := mul(1/2, 52)
After
swapped atom3 weighted XY = 26
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
swapped E_X := sum(-1/8, 0, 1/8, 2)
After
swapped E_X = 2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
swapped E_Y := sum(5/4, 11/4, 3/2, 13/2)
After
swapped E_Y = 12
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
swapped E_XY := sum(-5/4, 0, 3/2, 26)
After
swapped E_XY = 105/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
swapped product of means := mul(2, 12)
After
swapped product of means = 24
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
swapped covariance := sub(105/4, 24)
After
swapped covariance = 9/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S044
Before
swapped:E_X=2;E_Y=12;E_XY=105/4
After
Cov=E_XY-E_X*E_Y=9/4
Conditions: none
All three expectations use the same atoms and masses.Do not mix marginal models or assume independence.
Rule: covariance_same_joint_three_moments · Uses: G001, S045
Before
original covariance=9/4;swapped covariance=9/4
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.
Rule: covariance_swap_both_roles_symmetry · Uses: G001, S046
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.(9/4, 9/4, Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: e52703e3e81cb1852b7003a0a43cd2cdd0aed9bc36c0b297395c3c63798ede9ecandidate record SHA-256: afecfabf13dc0889588990f6db56dcc9cd789e978d988b9a7d7b2c953cfbb223semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ffquestion-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18Lean: NOT_RUN · human review: NOT_PERFORMED