# 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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r04-finite-joint-distribution-0100/)

Finite joint distribution · qf-mod26-joint-mass-lookup · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations/validation-report.json)