Exact Joint Qualification, Marginals, Rectangle Events and Pair Masses
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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-r04-finite-joint-distribution-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. Qualify a Whole Exact Joint Probability Table
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, 3/8] 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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[-12, -1], [-11, 0], [-10, 1], [-9, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-12, -1], [-11, 0], [-10, 1], [-9, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
2. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[-11, -1], [-10, 0], [-9, 1], [-8, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-11, -1], [-10, 0], [-9, 1], [-8, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
3. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[-10, -1], [-9, 0], [-8, 1], [-7, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. 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], [-9, 0], [-8, 1], [-7, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
4. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[-9, -1], [-8, 0], [-7, 1], [-6, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-9, -1], [-8, 0], [-7, 1], [-6, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
5. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[-8, -1], [-7, 0], [-6, 1], [-5, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-8, -1], [-7, 0], [-6, 1], [-5, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
6. Qualify a Whole Exact Joint Probability Table
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, 3/8] 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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[-7, -1], [-6, 0], [-5, 1], [-4, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-7, -1], [-6, 0], [-5, 1], [-4, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
7. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[-6, -1], [-5, 0], [-4, 1], [-3, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-6, -1], [-5, 0], [-4, 1], [-3, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
8. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[-5, -1], [-4, 0], [-3, 1], [-2, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. 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], [-4, 0], [-3, 1], [-2, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
9. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[-4, -1], [-3, 0], [-2, 1], [-1, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-4, -1], [-3, 0], [-2, 1], [-1, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
10. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[-3, -1], [-2, 0], [-1, 1], [0, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. 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], [-2, 0], [-1, 1], [0, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
11. Qualify a Whole Exact Joint Probability Table
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, 3/8] 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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[-2, -1], [-1, 0], [0, 1], [1, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-2, -1], [-1, 0], [0, 1], [1, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
12. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[-1, -1], [0, 0], [1, 1], [2, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. 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], [0, 0], [1, 1], [2, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
13. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[0, -1], [1, 0], [2, 1], [3, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. 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], [1, 0], [2, 1], [3, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
14. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[1, -1], [2, 0], [3, 1], [4, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. 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], [2, 0], [3, 1], [4, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
15. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[2, -1], [3, 0], [4, 1], [5, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[2, -1], [3, 0], [4, 1], [5, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
16. Qualify a Whole Exact Joint Probability Table
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, 3/8] 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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[3, -1], [4, 0], [5, 1], [6, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. 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], [4, 0], [5, 1], [6, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
17. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[4, -1], [5, 0], [6, 1], [7, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[4, -1], [5, 0], [6, 1], [7, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
18. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[5, -1], [6, 0], [7, 1], [8, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. 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], [6, 0], [7, 1], [8, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
19. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[6, -1], [7, 0], [8, 1], [9, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[6, -1], [7, 0], [8, 1], [9, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
20. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[7, -1], [8, 0], [9, 1], [10, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[7, -1], [8, 0], [9, 1], [10, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
21. Qualify a Whole Exact Joint Probability Table
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, 3/8] 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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[8, -1], [9, 0], [10, 1], [11, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[8, -1], [9, 0], [10, 1], [11, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
22. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[9, -1], [10, 0], [11, 1], [12, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[9, -1], [10, 0], [11, 1], [12, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
23. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[10, -1], [11, 0], [12, 1], [13, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. 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], [11, 0], [12, 1], [13, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
24. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[11, -1], [12, 0], [13, 1], [14, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[11, -1], [12, 0], [13, 1], [14, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
25. Qualify a Whole Exact Joint Probability Table
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. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).
View answer and explanation
([[12, -1], [13, 0], [14, 1], [15, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)
Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[12, -1], [13, 0], [14, 1], [15, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-qualification · candidate
26. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[-12, 1/8], [-11, 1/4], [-10, 1/8], [-9, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-12, 1/8], [-11, 1/4], [-10, 1/8], [-9, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
27. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[-11, 1/8], [-10, 1/4], [-9, 1/8], [-8, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-11, 1/8], [-10, 1/4], [-9, 1/8], [-8, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
28. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[-10, 1/8], [-9, 1/4], [-8, 1/8], [-7, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-10, 1/8], [-9, 1/4], [-8, 1/8], [-7, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
29. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[-9, 1/8], [-8, 1/4], [-7, 1/8], [-6, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-9, 1/8], [-8, 1/4], [-7, 1/8], [-6, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
30. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[-8, 1/8], [-7, 1/4], [-6, 1/8], [-5, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-8, 1/8], [-7, 1/4], [-6, 1/8], [-5, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
31. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[-7, 1/8], [-6, 1/4], [-5, 1/8], [-4, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-7, 1/8], [-6, 1/4], [-5, 1/8], [-4, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
32. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[-6, 1/8], [-5, 1/4], [-4, 1/8], [-3, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-6, 1/8], [-5, 1/4], [-4, 1/8], [-3, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
33. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[-5, 1/8], [-4, 1/4], [-3, 1/8], [-2, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-5, 1/8], [-4, 1/4], [-3, 1/8], [-2, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
34. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[-4, 1/8], [-3, 1/4], [-2, 1/8], [-1, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-4, 1/8], [-3, 1/4], [-2, 1/8], [-1, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
35. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[-3, 1/8], [-2, 1/4], [-1, 1/8], [0, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-3, 1/8], [-2, 1/4], [-1, 1/8], [0, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
36. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[-2, 1/8], [-1, 1/4], [0, 1/8], [1, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-2, 1/8], [-1, 1/4], [0, 1/8], [1, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
37. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
38. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[0, 1/8], [1, 1/4], [2, 1/8], [3, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[0, 1/8], [1, 1/4], [2, 1/8], [3, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
39. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[1, 1/8], [2, 1/4], [3, 1/8], [4, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[1, 1/8], [2, 1/4], [3, 1/8], [4, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
40. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[2, 1/8], [3, 1/4], [4, 1/8], [5, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[2, 1/8], [3, 1/4], [4, 1/8], [5, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
41. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[3, 1/8], [4, 1/4], [5, 1/8], [6, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[3, 1/8], [4, 1/4], [5, 1/8], [6, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
42. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[4, 1/8], [5, 1/4], [6, 1/8], [7, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[4, 1/8], [5, 1/4], [6, 1/8], [7, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
43. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[5, 1/8], [6, 1/4], [7, 1/8], [8, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[5, 1/8], [6, 1/4], [7, 1/8], [8, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
44. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[6, 1/8], [7, 1/4], [8, 1/8], [9, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[6, 1/8], [7, 1/4], [8, 1/8], [9, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
45. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[7, 1/8], [8, 1/4], [9, 1/8], [10, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[7, 1/8], [8, 1/4], [9, 1/8], [10, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
46. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[8, 1/8], [9, 1/4], [10, 1/8], [11, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[8, 1/8], [9, 1/4], [10, 1/8], [11, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
47. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[9, 1/8], [10, 1/4], [11, 1/8], [12, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[9, 1/8], [10, 1/4], [11, 1/8], [12, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
48. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[10, 1/8], [11, 1/4], [12, 1/8], [13, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[10, 1/8], [11, 1/4], [12, 1/8], [13, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
49. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[11, 1/8], [12, 1/4], [13, 1/8], [14, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[11, 1/8], [12, 1/4], [13, 1/8], [14, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
50. Aggregate Both Marginals from the Same Qualified 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
View answer and explanation
([[12, 1/8], [13, 1/4], [14, 1/8], [15, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])
Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[12, 1/8], [13, 1/4], [14, 1/8], [15, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]]).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-marginals · candidate
51. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[-12, -10], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([-12, -10], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-12, -10], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
52. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[-11, -9], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([-11, -9], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-11, -9], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
53. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[-10, -8], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([-10, -8], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-10, -8], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
54. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[-9, -7], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([-9, -7], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-9, -7], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
55. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[-8, -6], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([-8, -6], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-8, -6], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
56. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[-7, -5], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([-7, -5], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. 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], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
57. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[-6, -4], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([-6, -4], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-6, -4], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
58. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[-5, -3], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([-5, -3], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-5, -3], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
59. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[-4, -2], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([-4, -2], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-4, -2], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
60. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[-3, -1], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([-3, -1], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. 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], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
61. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[-2, 0], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([-2, 0], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. 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, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
62. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[-1, 1], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([-1, 1], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. 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], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
63. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[0, 2], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([0, 2], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. 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], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
64. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[1, 3], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([1, 3], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. 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, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
65. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[2, 4], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([2, 4], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. 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, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
66. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[3, 5], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([3, 5], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. 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], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
67. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[4, 6], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([4, 6], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, 6], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
68. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[5, 7], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([5, 7], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, 7], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
69. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[6, 8], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([6, 8], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([6, 8], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
70. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[7, 9], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([7, 9], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([7, 9], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
71. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[8, 10], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([8, 10], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([8, 10], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
72. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[9, 11], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([9, 11], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([9, 11], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
73. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[10, 12], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([10, 12], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([10, 12], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
74. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[11, 13], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([11, 13], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([11, 13], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
75. Both Coordinates Must Belong to a Rectangle Event
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. The joint table must qualify first. x_event=[12, 14], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.
View answer and explanation
([12, 14], [-1, 1], 1/4)
Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([12, 14], [-1, 1], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-rectangle-event · candidate
76. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[-12, -1]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([-12, -1], 1/8)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-12, -1], 1/8).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
77. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[-10, 0]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([-10, 0], 1/4)
Use its mass in this exact joint table, not a product of marginal masses. 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], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
78. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[-8, 1]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([-8, 1], 1/8)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-8, 1], 1/8).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
79. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[3, 99]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([3, 99], 0)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 99], 0).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
80. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[-8, -1]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([-8, -1], 1/8)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-8, -1], 1/8).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
81. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[-6, 0]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([-6, 0], 1/4)
Use its mass in this exact joint table, not a product of marginal masses. 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], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
82. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[-4, 1]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([-4, 1], 1/8)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-4, 1], 1/8).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
83. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[7, 99]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([7, 99], 0)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([7, 99], 0).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
84. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[-4, -1]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([-4, -1], 1/8)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-4, -1], 1/8).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
85. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[-2, 0]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([-2, 0], 1/4)
Use its mass in this exact joint table, not a product of marginal masses. 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], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
86. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[0, 1]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([0, 1], 1/8)
Use its mass in this exact joint table, not a product of marginal masses. 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], 1/8).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
87. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[11, 99]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([11, 99], 0)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([11, 99], 0).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
88. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[0, -1]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([0, -1], 1/8)
Use its mass in this exact joint table, not a product of marginal masses. 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], 1/8).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
89. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[2, 0]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([2, 0], 1/4)
Use its mass in this exact joint table, not a product of marginal masses. 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], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
90. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[4, 1]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([4, 1], 1/8)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, 1], 1/8).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
91. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[15, 99]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([15, 99], 0)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([15, 99], 0).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
92. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[4, -1]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([4, -1], 1/8)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, -1], 1/8).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
93. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[6, 0]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([6, 0], 1/4)
Use its mass in this exact joint table, not a product of marginal masses. 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], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
94. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[8, 1]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([8, 1], 1/8)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([8, 1], 1/8).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
95. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[19, 99]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([19, 99], 0)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([19, 99], 0).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
96. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[8, -1]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([8, -1], 1/8)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([8, -1], 1/8).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
97. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[10, 0]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([10, 0], 1/4)
Use its mass in this exact joint table, not a product of marginal masses. 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], 1/4).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
98. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[12, 1]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([12, 1], 1/8)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([12, 1], 1/8).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
99. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[23, 99]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([23, 99], 0)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([23, 99], 0).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
100. Look Up One Pair without Assuming a Product 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. The joint table must qualify first; requested pair=[12, -1]. Give (pair,mass), using zero when the pair is unlisted.
View answer and explanation
([12, -1], 1/8)
Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([12, -1], 1/8).
Reviewed step-by-step derivation
Finite joint distribution · qf-mod26-joint-mass-lookup · candidate
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