Explore related knowledge and questions →
← Question derivation index · Original question batch
R0023-12236 · q-20261013-advance-20261026-r05-covariance-0011
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 (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].
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 (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].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=[[-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.
Rule: qualify_whole_joint_law_preserve_alignment · Uses: G001, S001
Before
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.
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(-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.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
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.
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(-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.
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, 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.
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(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.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
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.
Rule: covariance_same_joint_three_moments · Uses: G001, S023
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.(0, 1, 5/4, 5/4)
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: 5809d0666210cb444e4b8ef1b1461e31b8d8f8635fadf18fb6ae4a2cf72a811ccandidate record SHA-256: 1b33712e43d1965587641b3550a980402f0274ed70dff3f006e52ae02df60c5bsemantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ffquestion-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18Lean: NOT_RUN · human review: NOT_PERFORMED