One-Law Covariance, Symmetry, Affine Changes and Zero-Covariance Dependence
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Covariance · qf-mod26-covariance-zero-counterexample · candidate
Canonical question JSONL · English JSONL · Validation report