# One-Law Covariance, Symmetry, Affine Changes and Zero-Covariance Dependence

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261013-advance-20261026-r05-covariance-foundations · 2026-10-13 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[-12, -1], [-11, 0], [-10, 1], [-9, 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].

View answer and explanation(-10, 1, -35/4, 5/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-10, 1, -35/4, 5/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0001/)

Covariance · qf-mod26-covariance-compute · candidate





## 2. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[-11, -1], [-10, 0], [-9, 1], [-8, 3]] 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].

View answer and explanation(-9, 3/2, -47/4, 7/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-9, 3/2, -47/4, 7/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0002/)

Covariance · qf-mod26-covariance-compute · candidate





## 3. Compute Three Moments from One Joint Law

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 (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].

View answer and explanation(-8, 2, -55/4, 9/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-8, 2, -55/4, 9/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0003/)

Covariance · qf-mod26-covariance-compute · candidate





## 4. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[-9, -1], [-8, 0], [-7, 1], [-6, 5]] 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].

View answer and explanation(-7, 5/2, -59/4, 11/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-7, 5/2, -59/4, 11/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0004/)

Covariance · qf-mod26-covariance-compute · candidate





## 5. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[-8, -1], [-7, 0], [-6, 1], [-5, 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. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].

View answer and explanation(-6, 3, -59/4, 13/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-6, 3, -59/4, 13/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0005/)

Covariance · qf-mod26-covariance-compute · candidate





## 6. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[-7, -1], [-6, 0], [-5, 1], [-4, 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].

View answer and explanation(-5, 1, -15/4, 5/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-5, 1, -15/4, 5/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0006/)

Covariance · qf-mod26-covariance-compute · candidate





## 7. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[-6, -1], [-5, 0], [-4, 1], [-3, 3]] 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].

View answer and explanation(-4, 3/2, -17/4, 7/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-4, 3/2, -17/4, 7/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0007/)

Covariance · qf-mod26-covariance-compute · candidate





## 8. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[-5, -1], [-4, 0], [-3, 1], [-2, 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 (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].

View answer and explanation(-3, 2, -15/4, 9/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-3, 2, -15/4, 9/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0008/)

Covariance · qf-mod26-covariance-compute · candidate





## 9. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[-4, -1], [-3, 0], [-2, 1], [-1, 5]] 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].

View answer and explanation(-2, 5/2, -9/4, 11/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 5/2, -9/4, 11/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0009/)

Covariance · qf-mod26-covariance-compute · candidate





## 10. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[-3, -1], [-2, 0], [-1, 1], [0, 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. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].

View answer and explanation(-1, 3, 1/4, 13/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, 3, 1/4, 13/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0010/)

Covariance · qf-mod26-covariance-compute · candidate





## 11. Compute Three Moments from One Joint Law

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].

View answer and explanation(0, 1, 5/4, 5/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, 5/4, 5/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0011/)

Covariance · qf-mod26-covariance-compute · candidate





## 12. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[-1, -1], [0, 0], [1, 1], [2, 3]] 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].

View answer and explanation(1, 3/2, 13/4, 7/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 3/2, 13/4, 7/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0012/)

Covariance · qf-mod26-covariance-compute · candidate





## 13. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[0, -1], [1, 0], [2, 1], [3, 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 (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].

View answer and explanation(2, 2, 25/4, 9/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 2, 25/4, 9/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0013/)

Covariance · qf-mod26-covariance-compute · candidate





## 14. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[1, -1], [2, 0], [3, 1], [4, 5]] 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].

View answer and explanation(3, 5/2, 41/4, 11/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 5/2, 41/4, 11/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0014/)

Covariance · qf-mod26-covariance-compute · candidate





## 15. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[2, -1], [3, 0], [4, 1], [5, 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. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].

View answer and explanation(4, 3, 61/4, 13/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 3, 61/4, 13/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0015/)

Covariance · qf-mod26-covariance-compute · candidate





## 16. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[3, -1], [4, 0], [5, 1], [6, 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].

View answer and explanation(5, 1, 25/4, 5/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 1, 25/4, 5/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0016/)

Covariance · qf-mod26-covariance-compute · candidate





## 17. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[4, -1], [5, 0], [6, 1], [7, 3]] 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].

View answer and explanation(6, 3/2, 43/4, 7/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 3/2, 43/4, 7/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0017/)

Covariance · qf-mod26-covariance-compute · candidate





## 18. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[5, -1], [6, 0], [7, 1], [8, 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 (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].

View answer and explanation(7, 2, 65/4, 9/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7, 2, 65/4, 9/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0018/)

Covariance · qf-mod26-covariance-compute · candidate





## 19. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[6, -1], [7, 0], [8, 1], [9, 5]] 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].

View answer and explanation(8, 5/2, 91/4, 11/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (8, 5/2, 91/4, 11/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0019/)

Covariance · qf-mod26-covariance-compute · candidate





## 20. Compute Three Moments from One Joint Law

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. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].

View answer and explanation(9, 3, 121/4, 13/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9, 3, 121/4, 13/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0020/)

Covariance · qf-mod26-covariance-compute · candidate





## 21. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[8, -1], [9, 0], [10, 1], [11, 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].

View answer and explanation(10, 1, 45/4, 5/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (10, 1, 45/4, 5/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0021/)

Covariance · qf-mod26-covariance-compute · candidate





## 22. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[9, -1], [10, 0], [11, 1], [12, 3]] 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].

View answer and explanation(11, 3/2, 73/4, 7/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11, 3/2, 73/4, 7/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0022/)

Covariance · qf-mod26-covariance-compute · candidate





## 23. Compute Three Moments from One Joint Law

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 (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].

View answer and explanation(12, 2, 105/4, 9/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (12, 2, 105/4, 9/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0023/)

Covariance · qf-mod26-covariance-compute · candidate





## 24. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[11, -1], [12, 0], [13, 1], [14, 5]] 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].

View answer and explanation(13, 5/2, 141/4, 11/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13, 5/2, 141/4, 11/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0024/)

Covariance · qf-mod26-covariance-compute · candidate





## 25. Compute Three Moments from One Joint Law

Use the complete list of distinct integer pairs=[[12, -1], [13, 0], [14, 1], [15, 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. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].

View answer and explanation(14, 3, 181/4, 13/4)

Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (14, 3, 181/4, 13/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0025/)

Covariance · qf-mod26-covariance-compute · candidate





## 26. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[-12, -1], [-11, 0], [-10, 1], [-9, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).

View answer and explanation(5/4, 5/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/4, 5/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0026/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 27. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[-11, -1], [-10, 0], [-9, 1], [-8, 3]] 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?).

View answer and explanation(7/4, 7/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7/4, 7/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0027/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 28. Symmetry of Covariance under Swapping Coordinates

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?).

View answer and explanation(9/4, 9/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9/4, 9/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0028/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 29. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[-9, -1], [-8, 0], [-7, 1], [-6, 5]] 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?).

View answer and explanation(11/4, 11/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11/4, 11/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0029/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 30. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[-8, -1], [-7, 0], [-6, 1], [-5, 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. Give (Cov_XY,Cov_YX,equal?).

View answer and explanation(13/4, 13/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13/4, 13/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0030/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 31. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[-7, -1], [-6, 0], [-5, 1], [-4, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).

View answer and explanation(5/4, 5/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/4, 5/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0031/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 32. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[-6, -1], [-5, 0], [-4, 1], [-3, 3]] 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?).

View answer and explanation(7/4, 7/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7/4, 7/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0032/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 33. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[-5, -1], [-4, 0], [-3, 1], [-2, 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?).

View answer and explanation(9/4, 9/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9/4, 9/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0033/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 34. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[-4, -1], [-3, 0], [-2, 1], [-1, 5]] 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?).

View answer and explanation(11/4, 11/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11/4, 11/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0034/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 35. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[-3, -1], [-2, 0], [-1, 1], [0, 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. Give (Cov_XY,Cov_YX,equal?).

View answer and explanation(13/4, 13/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13/4, 13/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0035/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 36. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[-2, -1], [-1, 0], [0, 1], [1, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).

View answer and explanation(5/4, 5/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/4, 5/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0036/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 37. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[-1, -1], [0, 0], [1, 1], [2, 3]] 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?).

View answer and explanation(7/4, 7/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7/4, 7/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0037/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 38. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[0, -1], [1, 0], [2, 1], [3, 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?).

View answer and explanation(9/4, 9/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9/4, 9/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0038/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 39. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[1, -1], [2, 0], [3, 1], [4, 5]] 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?).

View answer and explanation(11/4, 11/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11/4, 11/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0039/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 40. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[2, -1], [3, 0], [4, 1], [5, 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. Give (Cov_XY,Cov_YX,equal?).

View answer and explanation(13/4, 13/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13/4, 13/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0040/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 41. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[3, -1], [4, 0], [5, 1], [6, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).

View answer and explanation(5/4, 5/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/4, 5/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0041/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 42. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[4, -1], [5, 0], [6, 1], [7, 3]] 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?).

View answer and explanation(7/4, 7/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7/4, 7/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0042/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 43. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[5, -1], [6, 0], [7, 1], [8, 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?).

View answer and explanation(9/4, 9/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9/4, 9/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0043/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 44. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[6, -1], [7, 0], [8, 1], [9, 5]] 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?).

View answer and explanation(11/4, 11/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11/4, 11/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0044/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 45. Symmetry of Covariance under Swapping Coordinates

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. Give (Cov_XY,Cov_YX,equal?).

View answer and explanation(13/4, 13/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13/4, 13/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0045/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 46. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[8, -1], [9, 0], [10, 1], [11, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).

View answer and explanation(5/4, 5/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/4, 5/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0046/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 47. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[9, -1], [10, 0], [11, 1], [12, 3]] 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?).

View answer and explanation(7/4, 7/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7/4, 7/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0047/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 48. Symmetry of Covariance under Swapping Coordinates

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?).

View answer and explanation(9/4, 9/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9/4, 9/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0048/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 49. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[11, -1], [12, 0], [13, 1], [14, 5]] 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?).

View answer and explanation(11/4, 11/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11/4, 11/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0049/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 50. Symmetry of Covariance under Swapping Coordinates

Use the complete list of distinct integer pairs=[[12, -1], [13, 0], [14, 1], [15, 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. Give (Cov_XY,Cov_YX,equal?).

View answer and explanation(13/4, 13/4, Yes)

Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13/4, 13/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0050/)

Covariance · qf-mod26-covariance-symmetry · candidate





## 51. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[-12, -1], [-11, 0], [-10, 1], [-9, 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. Transform to aX+b and cY+d with [a,b,c,d]=[-3, -12, -2, 12]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(-3, -12, -2, 12, 5/4, 15/2, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-3, -12, -2, 12, 5/4, 15/2, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0051/)

Covariance · qf-mod26-covariance-affine · candidate





## 52. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[-11, -1], [-10, 0], [-9, 1], [-8, 3]] 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, -11, -1, 11]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(-2, -11, -1, 11, 7/4, 7/2, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, -11, -1, 11, 7/4, 7/2, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0052/)

Covariance · qf-mod26-covariance-affine · candidate





## 53. Affine Shifts Drop Out of Covariance

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. Transform to aX+b and cY+d with [a,b,c,d]=[-1, -10, 0, 10]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(-1, -10, 0, 10, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -10, 0, 10, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0053/)

Covariance · qf-mod26-covariance-affine · candidate





## 54. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[-9, -1], [-8, 0], [-7, 1], [-6, 5]] 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]=[0, -9, 1, 9]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(0, -9, 1, 9, 11/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -9, 1, 9, 11/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0054/)

Covariance · qf-mod26-covariance-affine · candidate





## 55. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[-8, -1], [-7, 0], [-6, 1], [-5, 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]=[1, -8, 2, 8]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(1, -8, 2, 8, 13/4, 13/2, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, -8, 2, 8, 13/4, 13/2, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0055/)

Covariance · qf-mod26-covariance-affine · candidate





## 56. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[-7, -1], [-6, 0], [-5, 1], [-4, 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. 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)).

View answer and explanation(2, -7, -2, 7, 5/4, -5, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, -7, -2, 7, 5/4, -5, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0056/)

Covariance · qf-mod26-covariance-affine · candidate





## 57. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[-6, -1], [-5, 0], [-4, 1], [-3, 3]] 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]=[3, -6, -1, 6]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(3, -6, -1, 6, 7/4, -21/4, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, -6, -1, 6, 7/4, -21/4, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0057/)

Covariance · qf-mod26-covariance-affine · candidate





## 58. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[-5, -1], [-4, 0], [-3, 1], [-2, 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. Transform to aX+b and cY+d with [a,b,c,d]=[-3, -5, 0, 5]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(-3, -5, 0, 5, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-3, -5, 0, 5, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0058/)

Covariance · qf-mod26-covariance-affine · candidate





## 59. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[-4, -1], [-3, 0], [-2, 1], [-1, 5]] 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, -4, 1, 4]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(-2, -4, 1, 4, 11/4, -11/2, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, -4, 1, 4, 11/4, -11/2, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0059/)

Covariance · qf-mod26-covariance-affine · candidate





## 60. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[-3, -1], [-2, 0], [-1, 1], [0, 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]=[-1, -3, 2, 3]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(-1, -3, 2, 3, 13/4, -13/2, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -3, 2, 3, 13/4, -13/2, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0060/)

Covariance · qf-mod26-covariance-affine · candidate





## 61. Affine Shifts Drop Out of Covariance

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. Transform to aX+b and cY+d with [a,b,c,d]=[0, -2, -2, 2]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(0, -2, -2, 2, 5/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -2, -2, 2, 5/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0061/)

Covariance · qf-mod26-covariance-affine · candidate





## 62. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[-1, -1], [0, 0], [1, 1], [2, 3]] 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]=[1, -1, -1, 1]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(1, -1, -1, 1, 7/4, -7/4, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, -1, -1, 1, 7/4, -7/4, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0062/)

Covariance · qf-mod26-covariance-affine · candidate





## 63. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[0, -1], [1, 0], [2, 1], [3, 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. Transform to aX+b and cY+d with [a,b,c,d]=[2, 0, 0, 0]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(2, 0, 0, 0, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, 0, 0, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0063/)

Covariance · qf-mod26-covariance-affine · candidate





## 64. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[1, -1], [2, 0], [3, 1], [4, 5]] 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]=[3, 1, 1, -1]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(3, 1, 1, -1, 11/4, 33/4, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 1, 1, -1, 11/4, 33/4, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0064/)

Covariance · qf-mod26-covariance-affine · candidate





## 65. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[2, -1], [3, 0], [4, 1], [5, 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]=[-3, 2, 2, -2]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(-3, 2, 2, -2, 13/4, -39/2, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-3, 2, 2, -2, 13/4, -39/2, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0065/)

Covariance · qf-mod26-covariance-affine · candidate





## 66. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[3, -1], [4, 0], [5, 1], [6, 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. Transform to aX+b and cY+d with [a,b,c,d]=[-2, 3, -2, -3]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(-2, 3, -2, -3, 5/4, 5, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 3, -2, -3, 5/4, 5, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0066/)

Covariance · qf-mod26-covariance-affine · candidate





## 67. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[4, -1], [5, 0], [6, 1], [7, 3]] 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]=[-1, 4, -1, -4]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(-1, 4, -1, -4, 7/4, 7/4, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, 4, -1, -4, 7/4, 7/4, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0067/)

Covariance · qf-mod26-covariance-affine · candidate





## 68. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[5, -1], [6, 0], [7, 1], [8, 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. Transform to aX+b and cY+d with [a,b,c,d]=[0, 5, 0, -5]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(0, 5, 0, -5, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 5, 0, -5, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0068/)

Covariance · qf-mod26-covariance-affine · candidate





## 69. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[6, -1], [7, 0], [8, 1], [9, 5]] 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]=[1, 6, 1, -6]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(1, 6, 1, -6, 11/4, 11/4, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 6, 1, -6, 11/4, 11/4, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0069/)

Covariance · qf-mod26-covariance-affine · candidate





## 70. Affine Shifts Drop Out of Covariance

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)).

View answer and explanation(2, 7, 2, -7, 13/4, 13, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 7, 2, -7, 13/4, 13, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0070/)

Covariance · qf-mod26-covariance-affine · candidate





## 71. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[8, -1], [9, 0], [10, 1], [11, 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. Transform to aX+b and cY+d with [a,b,c,d]=[3, 8, -2, -8]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(3, 8, -2, -8, 5/4, -15/2, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 8, -2, -8, 5/4, -15/2, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0071/)

Covariance · qf-mod26-covariance-affine · candidate





## 72. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[9, -1], [10, 0], [11, 1], [12, 3]] 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]=[-3, 9, -1, -9]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(-3, 9, -1, -9, 7/4, 21/4, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-3, 9, -1, -9, 7/4, 21/4, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0072/)

Covariance · qf-mod26-covariance-affine · candidate





## 73. Affine Shifts Drop Out of Covariance

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. Transform to aX+b and cY+d with [a,b,c,d]=[-2, 10, 0, -10]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(-2, 10, 0, -10, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 10, 0, -10, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0073/)

Covariance · qf-mod26-covariance-affine · candidate





## 74. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[11, -1], [12, 0], [13, 1], [14, 5]] 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]=[-1, 11, 1, -11]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(-1, 11, 1, -11, 11/4, -11/4, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, 11, 1, -11, 11/4, -11/4, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0074/)

Covariance · qf-mod26-covariance-affine · candidate





## 75. Affine Shifts Drop Out of Covariance

Use the complete list of distinct integer pairs=[[12, -1], [13, 0], [14, 1], [15, 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]=[0, 12, 2, -12]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).

View answer and explanation(0, 12, 2, -12, 13/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))

The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 12, 2, -12, 13/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y)).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0075/)

Covariance · qf-mod26-covariance-affine · candidate





## 76. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=1, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(1, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0076/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 77. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=2, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(2, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0077/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 78. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=3, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(3, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0078/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 79. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=4, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(4, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0079/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 80. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=5, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(5, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0080/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 81. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=6, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(6, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0081/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 82. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=7, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(7, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0082/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 83. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=8, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(8, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (8, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0083/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 84. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=9, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(9, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0084/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 85. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=10, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(10, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (10, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0085/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 86. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=11, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(11, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0086/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 87. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=12, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(12, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (12, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0087/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 88. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=13, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(13, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0088/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 89. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=14, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(14, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (14, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0089/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 90. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=15, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(15, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (15, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0090/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 91. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=16, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(16, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (16, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0091/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 92. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=17, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(17, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (17, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0092/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 93. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=18, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(18, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (18, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0093/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 94. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=19, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(19, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (19, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0094/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 95. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=20, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(20, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (20, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0095/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 96. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=21, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(21, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (21, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0096/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 97. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=22, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(22, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (22, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0097/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 98. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=23, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(23, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (23, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0098/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 99. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=24, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(24, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (24, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0099/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate





## 100. Zero Covariance Can Coexist with Dependence

Take X uniformly on [−scale,0,scale] for nonzero positive scale=25, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

View answer and explanation(25, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)

The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (25, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r05-covariance-0100/)

Covariance · qf-mod26-covariance-zero-counterexample · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261026-r05-covariance-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261026-r05-covariance-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261026-r05-covariance-foundations/validation-report.json)