OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 12253: Symmetry of Covariance under Swapping Coordinates

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

Return to catalogue page 62

← Question derivation index · Original question batch

R0023-12253 · q-20261013-advance-20261026-r05-covariance-0028

Original question and givens

Use the complete list of distinct integer pairs=[[-10, -1], [-9, 0], [-8, 1], [-7, 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], [-9, 0], [-8, 1], [-7, 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?).

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.

Step-by-step derivation

S001 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001

S002 · implies

Before

original distinct joint pairs=[[-10, -1], [-9, 0], [-8, 1], [-7, 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.

Step identifiers and dependencies

Rule: qualify_whole_joint_law_preserve_alignment · Uses: G001, S001

S003 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

original atom1 weighted X := mul(1/4, -9)

After

original atom1 weighted X = -9/4

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

original atom1 XY := mul(-9, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

original atom2 weighted X := mul(1/8, -8)

After

original atom2 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

original atom2 XY := mul(-8, 1)

After

original atom2 XY = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

original atom2 weighted XY := mul(1/8, -8)

After

original atom2 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

original atom3 weighted X := mul(1/2, -7)

After

original atom3 weighted X = -7/2

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

original atom3 XY := mul(-7, 4)

After

original atom3 XY = -28

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

original atom3 weighted XY := mul(1/2, -28)

After

original atom3 weighted XY = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

original E_X := sum(-5/4, -9/4, -1, -7/2)

After

original E_X = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

original E_XY := sum(5/4, 0, -1, -14)

After

original E_XY = -55/4

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

original product of means := mul(-8, 2)

After

original product of means = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

original covariance := sub(-55/4, -16)

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

original:E_X=-8;E_Y=2;E_XY=-55/4

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.

Step identifiers and dependencies

Rule: covariance_same_joint_three_moments · Uses: G001, S023

S025 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

swapped atom1 weighted Y := mul(1/4, -9)

After

swapped atom1 weighted Y = -9/4

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

swapped atom1 XY := mul(0, -9)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

swapped atom2 weighted Y := mul(1/8, -8)

After

swapped atom2 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

swapped atom2 XY := mul(1, -8)

After

swapped atom2 XY = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

swapped atom2 weighted XY := mul(1/8, -8)

After

swapped atom2 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

swapped atom3 weighted Y := mul(1/2, -7)

After

swapped atom3 weighted Y = -7/2

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

swapped atom3 XY := mul(4, -7)

After

swapped atom3 XY = -28

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

swapped atom3 weighted XY := mul(1/2, -28)

After

swapped atom3 weighted XY = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

swapped E_Y := sum(-5/4, -9/4, -1, -7/2)

After

swapped E_Y = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

swapped E_XY := sum(5/4, 0, -1, -14)

After

swapped E_XY = -55/4

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

swapped product of means := mul(2, -8)

After

swapped product of means = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

swapped covariance := sub(-55/4, -16)

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · implies

Before

swapped:E_X=2;E_Y=-8;E_XY=-55/4

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.

Step identifiers and dependencies

Rule: covariance_same_joint_three_moments · Uses: G001, S045

S047 · implies

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.

Step identifiers and dependencies

Rule: 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

(9/4, 9/4, 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: 2131db0fde27100ef72a8cdafc7d78af4e717eae6d1e7d73580aaf80e3c04e5a
  • candidate record SHA-256: acbea6946681a70a49e38e72eaca0096ea4d23031720e2c20b967bce26db55ed
  • semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff
  • question-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON