# Full Joint Factorization, Value Events and One-Way Covariance Consequences

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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-20261014-advance-20261028-r06-random-variable-independence-foundations · 2026-10-14 · 100 questions

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



## 1. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[-12, 13], y_values=[-3, 20], and aligned row-by-column exact masses=[[1/2, 0/1], [0/1, 1/2]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([-12, 13], [-3, 20], [1/2, 1/2], [1/2, 1/2], [[-12, -3, 1/2, 1/4], [-12, 20, 0, 1/4], [13, -3, 0, 1/4], [13, 20, 1/2, 1/4]], No)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-12, 13], [-3, 20], [1/2, 1/2], [1/2, 1/2], [[-12, -3, 1/2, 1/4], [-12, 20, 0, 1/4], [13, -3, 0, 1/4], [13, 20, 1/2, 1/4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0001/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 2. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[-11, 14], y_values=[-2, 21], and aligned row-by-column exact masses=[[30/49, 5/49], [12/49, 2/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([-11, 14], [-2, 21], [5/7, 2/7], [6/7, 1/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-11, 14], [-2, 21], [5/7, 2/7], [6/7, 1/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0002/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 3. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[-10, 15], y_values=[-1, 22], and aligned row-by-column exact masses=[[24/49, 4/49], [18/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([-10, 15], [-1, 22], [4/7, 3/7], [6/7, 1/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-10, 15], [-1, 22], [4/7, 3/7], [6/7, 1/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0003/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 4. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[-9, 16], y_values=[0, 23], and aligned row-by-column exact masses=[[18/49, 3/49], [24/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([-9, 16], [0, 23], [3/7, 4/7], [6/7, 1/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-9, 16], [0, 23], [3/7, 4/7], [6/7, 1/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0004/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 5. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[-8, 17], y_values=[1, 24], and aligned row-by-column exact masses=[[12/49, 2/49], [30/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([-8, 17], [1, 24], [2/7, 5/7], [6/7, 1/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-8, 17], [1, 24], [2/7, 5/7], [6/7, 1/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0005/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 6. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[-7, 18], y_values=[2, 25], and aligned row-by-column exact masses=[[1/2, 0/1], [0/1, 1/2]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([-7, 18], [2, 25], [1/2, 1/2], [1/2, 1/2], [[-7, 2, 1/2, 1/4], [-7, 25, 0, 1/4], [18, 2, 0, 1/4], [18, 25, 1/2, 1/4]], No)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-7, 18], [2, 25], [1/2, 1/2], [1/2, 1/2], [[-7, 2, 1/2, 1/4], [-7, 25, 0, 1/4], [18, 2, 0, 1/4], [18, 25, 1/2, 1/4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0006/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 7. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[-6, 19], y_values=[3, 26], and aligned row-by-column exact masses=[[25/49, 10/49], [10/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([-6, 19], [3, 26], [5/7, 2/7], [5/7, 2/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-6, 19], [3, 26], [5/7, 2/7], [5/7, 2/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0007/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 8. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[-5, 20], y_values=[-3, 27], and aligned row-by-column exact masses=[[20/49, 8/49], [15/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([-5, 20], [-3, 27], [4/7, 3/7], [5/7, 2/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-5, 20], [-3, 27], [4/7, 3/7], [5/7, 2/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0008/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 9. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[-4, 21], y_values=[-2, 28], and aligned row-by-column exact masses=[[15/49, 6/49], [20/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([-4, 21], [-2, 28], [3/7, 4/7], [5/7, 2/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-4, 21], [-2, 28], [3/7, 4/7], [5/7, 2/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0009/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 10. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[-3, 22], y_values=[-1, 29], and aligned row-by-column exact masses=[[10/49, 4/49], [25/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([-3, 22], [-1, 29], [2/7, 5/7], [5/7, 2/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-3, 22], [-1, 29], [2/7, 5/7], [5/7, 2/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0010/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 11. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[-2, 23], y_values=[0, 30], and aligned row-by-column exact masses=[[1/2, 0/1], [0/1, 1/2]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([-2, 23], [0, 30], [1/2, 1/2], [1/2, 1/2], [[-2, 0, 1/2, 1/4], [-2, 30, 0, 1/4], [23, 0, 0, 1/4], [23, 30, 1/2, 1/4]], No)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-2, 23], [0, 30], [1/2, 1/2], [1/2, 1/2], [[-2, 0, 1/2, 1/4], [-2, 30, 0, 1/4], [23, 0, 0, 1/4], [23, 30, 1/2, 1/4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0011/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 12. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[-1, 24], y_values=[1, 31], and aligned row-by-column exact masses=[[20/49, 15/49], [8/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([-1, 24], [1, 31], [5/7, 2/7], [4/7, 3/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-1, 24], [1, 31], [5/7, 2/7], [4/7, 3/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0012/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 13. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[0, 25], y_values=[2, 32], and aligned row-by-column exact masses=[[16/49, 12/49], [12/49, 9/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([0, 25], [2, 32], [4/7, 3/7], [4/7, 3/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 25], [2, 32], [4/7, 3/7], [4/7, 3/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0013/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 14. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[1, 26], y_values=[3, 33], and aligned row-by-column exact masses=[[12/49, 9/49], [16/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([1, 26], [3, 33], [3/7, 4/7], [4/7, 3/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([1, 26], [3, 33], [3/7, 4/7], [4/7, 3/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0014/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 15. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[2, 27], y_values=[-3, 34], and aligned row-by-column exact masses=[[8/49, 6/49], [20/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([2, 27], [-3, 34], [2/7, 5/7], [4/7, 3/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([2, 27], [-3, 34], [2/7, 5/7], [4/7, 3/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0015/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 16. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[3, 28], y_values=[-2, 35], and aligned row-by-column exact masses=[[1/2, 0/1], [0/1, 1/2]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([3, 28], [-2, 35], [1/2, 1/2], [1/2, 1/2], [[3, -2, 1/2, 1/4], [3, 35, 0, 1/4], [28, -2, 0, 1/4], [28, 35, 1/2, 1/4]], No)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([3, 28], [-2, 35], [1/2, 1/2], [1/2, 1/2], [[3, -2, 1/2, 1/4], [3, 35, 0, 1/4], [28, -2, 0, 1/4], [28, 35, 1/2, 1/4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0016/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 17. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[4, 29], y_values=[-1, 36], and aligned row-by-column exact masses=[[15/49, 20/49], [6/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([4, 29], [-1, 36], [5/7, 2/7], [3/7, 4/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([4, 29], [-1, 36], [5/7, 2/7], [3/7, 4/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0017/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 18. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[5, 30], y_values=[0, 37], and aligned row-by-column exact masses=[[12/49, 16/49], [9/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([5, 30], [0, 37], [4/7, 3/7], [3/7, 4/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([5, 30], [0, 37], [4/7, 3/7], [3/7, 4/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0018/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 19. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[6, 31], y_values=[1, 38], and aligned row-by-column exact masses=[[9/49, 12/49], [12/49, 16/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([6, 31], [1, 38], [3/7, 4/7], [3/7, 4/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([6, 31], [1, 38], [3/7, 4/7], [3/7, 4/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0019/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 20. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[7, 32], y_values=[2, 39], and aligned row-by-column exact masses=[[6/49, 8/49], [15/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([7, 32], [2, 39], [2/7, 5/7], [3/7, 4/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([7, 32], [2, 39], [2/7, 5/7], [3/7, 4/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0020/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 21. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[8, 33], y_values=[3, 40], and aligned row-by-column exact masses=[[1/2, 0/1], [0/1, 1/2]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([8, 33], [3, 40], [1/2, 1/2], [1/2, 1/2], [[8, 3, 1/2, 1/4], [8, 40, 0, 1/4], [33, 3, 0, 1/4], [33, 40, 1/2, 1/4]], No)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([8, 33], [3, 40], [1/2, 1/2], [1/2, 1/2], [[8, 3, 1/2, 1/4], [8, 40, 0, 1/4], [33, 3, 0, 1/4], [33, 40, 1/2, 1/4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0021/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 22. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[9, 34], y_values=[-3, 41], and aligned row-by-column exact masses=[[10/49, 25/49], [4/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([9, 34], [-3, 41], [5/7, 2/7], [2/7, 5/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([9, 34], [-3, 41], [5/7, 2/7], [2/7, 5/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0022/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 23. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[10, 35], y_values=[-2, 42], and aligned row-by-column exact masses=[[8/49, 20/49], [6/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([10, 35], [-2, 42], [4/7, 3/7], [2/7, 5/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([10, 35], [-2, 42], [4/7, 3/7], [2/7, 5/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0023/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 24. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[11, 36], y_values=[-1, 43], and aligned row-by-column exact masses=[[6/49, 15/49], [8/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([11, 36], [-1, 43], [3/7, 4/7], [2/7, 5/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([11, 36], [-1, 43], [3/7, 4/7], [2/7, 5/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0024/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 25. Every Joint Grid Entry Must Factorize

Use the complete joint grid with original x_values=[12, 37], y_values=[0, 44], and aligned row-by-column exact masses=[[4/49, 10/49], [10/49, 25/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).

View answer and explanation([12, 37], [0, 44], [2/7, 5/7], [2/7, 5/7], [], Yes)

Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([12, 37], [0, 44], [2/7, 5/7], [2/7, 5/7], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0025/)

Independence of random variables · qf-mod28-rvind-qualification · candidate





## 26. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[-12, 13], y_values=[-3, 20], and aligned row-by-column exact masses=[[36/49, 6/49], [6/49, 1/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=-12, y=20 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(-12, 20, 6/49, 6/7, 1/7, 6/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (-12, 20, 6/49, 6/7, 1/7, 6/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0026/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 27. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[-11, 14], y_values=[-2, 21], and aligned row-by-column exact masses=[[30/49, 5/49], [12/49, 2/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=14, y=-2 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(14, -2, 12/49, 2/7, 6/7, 12/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (14, -2, 12/49, 2/7, 6/7, 12/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0027/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 28. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[-10, 15], y_values=[-1, 22], and aligned row-by-column exact masses=[[24/49, 4/49], [18/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=-10, y=22 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(-10, 22, 4/49, 4/7, 1/7, 4/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (-10, 22, 4/49, 4/7, 1/7, 4/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0028/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 29. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[-9, 16], y_values=[0, 23], and aligned row-by-column exact masses=[[18/49, 3/49], [24/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=16, y=0 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(16, 0, 24/49, 4/7, 6/7, 24/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (16, 0, 24/49, 4/7, 6/7, 24/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0029/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 30. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[-8, 17], y_values=[1, 24], and aligned row-by-column exact masses=[[12/49, 2/49], [30/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=-8, y=24 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(-8, 24, 2/49, 2/7, 1/7, 2/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (-8, 24, 2/49, 2/7, 1/7, 2/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0030/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 31. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[-7, 18], y_values=[2, 25], and aligned row-by-column exact masses=[[30/49, 12/49], [5/49, 2/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=18, y=2 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(18, 2, 5/49, 1/7, 5/7, 5/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (18, 2, 5/49, 1/7, 5/7, 5/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0031/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 32. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[-6, 19], y_values=[3, 26], and aligned row-by-column exact masses=[[25/49, 10/49], [10/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=-6, y=26 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(-6, 26, 10/49, 5/7, 2/7, 10/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (-6, 26, 10/49, 5/7, 2/7, 10/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0032/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 33. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[-5, 20], y_values=[-3, 27], and aligned row-by-column exact masses=[[20/49, 8/49], [15/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=20, y=-3 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(20, -3, 15/49, 3/7, 5/7, 15/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (20, -3, 15/49, 3/7, 5/7, 15/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0033/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 34. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[-4, 21], y_values=[-2, 28], and aligned row-by-column exact masses=[[15/49, 6/49], [20/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=-4, y=28 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(-4, 28, 6/49, 3/7, 2/7, 6/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (-4, 28, 6/49, 3/7, 2/7, 6/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0034/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 35. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[-3, 22], y_values=[-1, 29], and aligned row-by-column exact masses=[[10/49, 4/49], [25/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=22, y=-1 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(22, -1, 25/49, 5/7, 5/7, 25/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (22, -1, 25/49, 5/7, 5/7, 25/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0035/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 36. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[-2, 23], y_values=[0, 30], and aligned row-by-column exact masses=[[24/49, 18/49], [4/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=-2, y=30 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(-2, 30, 18/49, 6/7, 3/7, 18/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (-2, 30, 18/49, 6/7, 3/7, 18/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0036/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 37. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[-1, 24], y_values=[1, 31], and aligned row-by-column exact masses=[[20/49, 15/49], [8/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=24, y=1 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(24, 1, 8/49, 2/7, 4/7, 8/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (24, 1, 8/49, 2/7, 4/7, 8/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0037/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 38. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[0, 25], y_values=[2, 32], and aligned row-by-column exact masses=[[16/49, 12/49], [12/49, 9/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=0, y=32 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(0, 32, 12/49, 4/7, 3/7, 12/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, 32, 12/49, 4/7, 3/7, 12/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0038/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 39. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[1, 26], y_values=[3, 33], and aligned row-by-column exact masses=[[12/49, 9/49], [16/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=26, y=3 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(26, 3, 16/49, 4/7, 4/7, 16/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (26, 3, 16/49, 4/7, 4/7, 16/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0039/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 40. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[2, 27], y_values=[-3, 34], and aligned row-by-column exact masses=[[8/49, 6/49], [20/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=2, y=34 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(2, 34, 6/49, 2/7, 3/7, 6/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, 34, 6/49, 2/7, 3/7, 6/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0040/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 41. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[3, 28], y_values=[-2, 35], and aligned row-by-column exact masses=[[18/49, 24/49], [3/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=28, y=-2 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(28, -2, 3/49, 1/7, 3/7, 3/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (28, -2, 3/49, 1/7, 3/7, 3/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0041/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 42. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[4, 29], y_values=[-1, 36], and aligned row-by-column exact masses=[[15/49, 20/49], [6/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=4, y=36 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(4, 36, 20/49, 5/7, 4/7, 20/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, 36, 20/49, 5/7, 4/7, 20/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0042/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 43. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[5, 30], y_values=[0, 37], and aligned row-by-column exact masses=[[12/49, 16/49], [9/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=30, y=0 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(30, 0, 9/49, 3/7, 3/7, 9/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (30, 0, 9/49, 3/7, 3/7, 9/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0043/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 44. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[6, 31], y_values=[1, 38], and aligned row-by-column exact masses=[[9/49, 12/49], [12/49, 16/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=6, y=38 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(6, 38, 12/49, 3/7, 4/7, 12/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, 38, 12/49, 3/7, 4/7, 12/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0044/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 45. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[7, 32], y_values=[2, 39], and aligned row-by-column exact masses=[[6/49, 8/49], [15/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=32, y=2 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(32, 2, 15/49, 5/7, 3/7, 15/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (32, 2, 15/49, 5/7, 3/7, 15/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0045/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 46. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[8, 33], y_values=[3, 40], and aligned row-by-column exact masses=[[12/49, 30/49], [2/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=8, y=40 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(8, 40, 30/49, 6/7, 5/7, 30/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (8, 40, 30/49, 6/7, 5/7, 30/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0046/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 47. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[9, 34], y_values=[-3, 41], and aligned row-by-column exact masses=[[10/49, 25/49], [4/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=34, y=-3 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(34, -3, 4/49, 2/7, 2/7, 4/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (34, -3, 4/49, 2/7, 2/7, 4/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0047/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 48. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[10, 35], y_values=[-2, 42], and aligned row-by-column exact masses=[[8/49, 20/49], [6/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=10, y=42 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(10, 42, 20/49, 4/7, 5/7, 20/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (10, 42, 20/49, 4/7, 5/7, 20/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0048/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 49. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[11, 36], y_values=[-1, 43], and aligned row-by-column exact masses=[[6/49, 15/49], [8/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=36, y=-1 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(36, -1, 8/49, 4/7, 2/7, 8/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (36, -1, 8/49, 4/7, 2/7, 8/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0049/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 50. One Cell Equality Is Not Whole-Law Independence

Use the complete joint grid with original x_values=[12, 37], y_values=[0, 44], and aligned row-by-column exact masses=[[4/49, 10/49], [10/49, 25/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=12, y=44 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).

View answer and explanation(12, 44, 10/49, 2/7, 5/7, 10/49, Yes)

Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (12, 44, 10/49, 2/7, 5/7, 10/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0050/)

Independence of random variables · qf-mod28-rvind-marginal-product · candidate





## 51. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[-12, 13], y_values=[-3, 20], and aligned row-by-column exact masses=[[36/49, 6/49], [6/49, 1/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[-12], y_event=[20] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([-12], [20], 6/7, 1/7, 6/49, 6/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-12], [20], 6/7, 1/7, 6/49, 6/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0051/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 52. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[-11, 14], y_values=[-2, 21], and aligned row-by-column exact masses=[[30/49, 5/49], [12/49, 2/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[14], y_event=[-2] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([14], [-2], 2/7, 6/7, 12/49, 12/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([14], [-2], 2/7, 6/7, 12/49, 12/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0052/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 53. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[-10, 15], y_values=[-1, 22], and aligned row-by-column exact masses=[[24/49, 4/49], [18/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[-10], y_event=[22] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([-10], [22], 4/7, 1/7, 4/49, 4/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-10], [22], 4/7, 1/7, 4/49, 4/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0053/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 54. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[-9, 16], y_values=[0, 23], and aligned row-by-column exact masses=[[18/49, 3/49], [24/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[16], y_event=[0] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([16], [0], 4/7, 6/7, 24/49, 24/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([16], [0], 4/7, 6/7, 24/49, 24/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0054/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 55. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[-8, 17], y_values=[1, 24], and aligned row-by-column exact masses=[[12/49, 2/49], [30/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[-8], y_event=[24] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([-8], [24], 2/7, 1/7, 2/49, 2/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-8], [24], 2/7, 1/7, 2/49, 2/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0055/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 56. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[-7, 18], y_values=[2, 25], and aligned row-by-column exact masses=[[30/49, 12/49], [5/49, 2/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[18], y_event=[2] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([18], [2], 1/7, 5/7, 5/49, 5/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([18], [2], 1/7, 5/7, 5/49, 5/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0056/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 57. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[-6, 19], y_values=[3, 26], and aligned row-by-column exact masses=[[25/49, 10/49], [10/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[-6], y_event=[26] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([-6], [26], 5/7, 2/7, 10/49, 10/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-6], [26], 5/7, 2/7, 10/49, 10/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0057/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 58. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[-5, 20], y_values=[-3, 27], and aligned row-by-column exact masses=[[20/49, 8/49], [15/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[20], y_event=[-3] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([20], [-3], 3/7, 5/7, 15/49, 15/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([20], [-3], 3/7, 5/7, 15/49, 15/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0058/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 59. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[-4, 21], y_values=[-2, 28], and aligned row-by-column exact masses=[[15/49, 6/49], [20/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[-4], y_event=[28] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([-4], [28], 3/7, 2/7, 6/49, 6/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-4], [28], 3/7, 2/7, 6/49, 6/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0059/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 60. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[-3, 22], y_values=[-1, 29], and aligned row-by-column exact masses=[[10/49, 4/49], [25/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[22], y_event=[-1] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([22], [-1], 5/7, 5/7, 25/49, 25/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([22], [-1], 5/7, 5/7, 25/49, 25/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0060/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 61. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[-2, 23], y_values=[0, 30], and aligned row-by-column exact masses=[[24/49, 18/49], [4/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[-2], y_event=[30] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([-2], [30], 6/7, 3/7, 18/49, 18/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-2], [30], 6/7, 3/7, 18/49, 18/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0061/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 62. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[-1, 24], y_values=[1, 31], and aligned row-by-column exact masses=[[20/49, 15/49], [8/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[24], y_event=[1] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([24], [1], 2/7, 4/7, 8/49, 8/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([24], [1], 2/7, 4/7, 8/49, 8/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0062/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 63. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[0, 25], y_values=[2, 32], and aligned row-by-column exact masses=[[16/49, 12/49], [12/49, 9/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[0], y_event=[32] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([0], [32], 4/7, 3/7, 12/49, 12/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], [32], 4/7, 3/7, 12/49, 12/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0063/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 64. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[1, 26], y_values=[3, 33], and aligned row-by-column exact masses=[[12/49, 9/49], [16/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[26], y_event=[3] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([26], [3], 4/7, 4/7, 16/49, 16/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([26], [3], 4/7, 4/7, 16/49, 16/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0064/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 65. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[2, 27], y_values=[-3, 34], and aligned row-by-column exact masses=[[8/49, 6/49], [20/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[2], y_event=[34] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([2], [34], 2/7, 3/7, 6/49, 6/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([2], [34], 2/7, 3/7, 6/49, 6/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0065/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 66. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[3, 28], y_values=[-2, 35], and aligned row-by-column exact masses=[[18/49, 24/49], [3/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[28], y_event=[-2] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([28], [-2], 1/7, 3/7, 3/49, 3/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([28], [-2], 1/7, 3/7, 3/49, 3/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0066/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 67. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[4, 29], y_values=[-1, 36], and aligned row-by-column exact masses=[[15/49, 20/49], [6/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[4], y_event=[36] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([4], [36], 5/7, 4/7, 20/49, 20/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([4], [36], 5/7, 4/7, 20/49, 20/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0067/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 68. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[5, 30], y_values=[0, 37], and aligned row-by-column exact masses=[[12/49, 16/49], [9/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[30], y_event=[0] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([30], [0], 3/7, 3/7, 9/49, 9/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([30], [0], 3/7, 3/7, 9/49, 9/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0068/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 69. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[6, 31], y_values=[1, 38], and aligned row-by-column exact masses=[[9/49, 12/49], [12/49, 16/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[6], y_event=[38] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([6], [38], 3/7, 4/7, 12/49, 12/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([6], [38], 3/7, 4/7, 12/49, 12/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0069/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 70. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[7, 32], y_values=[2, 39], and aligned row-by-column exact masses=[[6/49, 8/49], [15/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[32], y_event=[2] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([32], [2], 5/7, 3/7, 15/49, 15/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([32], [2], 5/7, 3/7, 15/49, 15/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0070/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 71. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[8, 33], y_values=[3, 40], and aligned row-by-column exact masses=[[12/49, 30/49], [2/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[8], y_event=[40] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([8], [40], 6/7, 5/7, 30/49, 30/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([8], [40], 6/7, 5/7, 30/49, 30/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0071/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 72. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[9, 34], y_values=[-3, 41], and aligned row-by-column exact masses=[[10/49, 25/49], [4/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[34], y_event=[-3] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([34], [-3], 2/7, 2/7, 4/49, 4/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([34], [-3], 2/7, 2/7, 4/49, 4/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0072/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 73. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[10, 35], y_values=[-2, 42], and aligned row-by-column exact masses=[[8/49, 20/49], [6/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[10], y_event=[42] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([10], [42], 4/7, 5/7, 20/49, 20/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([10], [42], 4/7, 5/7, 20/49, 20/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0073/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 74. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[11, 36], y_values=[-1, 43], and aligned row-by-column exact masses=[[6/49, 15/49], [8/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[36], y_event=[-1] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([36], [-1], 4/7, 2/7, 8/49, 8/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([36], [-1], 4/7, 2/7, 8/49, 8/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0074/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 75. A Selected Pair of Value Events under the Same Joint Law

Use the complete joint grid with original x_values=[12, 37], y_values=[0, 44], and aligned row-by-column exact masses=[[4/49, 10/49], [10/49, 25/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[12], y_event=[44] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

View answer and explanation([12], [44], 2/7, 5/7, 10/49, 10/49, Yes)

Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([12], [44], 2/7, 5/7, 10/49, 10/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0075/)

Independence of random variables · qf-mod28-rvind-event-preimage · candidate





## 76. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[-12, 13], y_values=[-3, 20], and aligned row-by-column exact masses=[[36/49, 6/49], [6/49, 1/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, -59/7, 2/7, -118/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, -59/7, 2/7, -118/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0076/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 77. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[-11, 14], y_values=[-2, 21], and aligned row-by-column exact masses=[[30/49, 5/49], [12/49, 2/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, -27/7, 9/7, -243/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, -27/7, 9/7, -243/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0077/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 78. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[-10, 15], y_values=[-1, 22], and aligned row-by-column exact masses=[[24/49, 4/49], [18/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 5/7, 16/7, 80/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 5/7, 16/7, 80/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0078/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 79. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[-9, 16], y_values=[0, 23], and aligned row-by-column exact masses=[[18/49, 3/49], [24/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 37/7, 23/7, 851/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 37/7, 23/7, 851/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0079/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 80. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[-8, 17], y_values=[1, 24], and aligned row-by-column exact masses=[[12/49, 2/49], [30/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 69/7, 30/7, 2070/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 69/7, 30/7, 2070/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0080/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 81. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[-7, 18], y_values=[2, 25], and aligned row-by-column exact masses=[[30/49, 12/49], [5/49, 2/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, -24/7, 60/7, -1440/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, -24/7, 60/7, -1440/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0081/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 82. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[-6, 19], y_values=[3, 26], and aligned row-by-column exact masses=[[25/49, 10/49], [10/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 8/7, 67/7, 536/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 8/7, 67/7, 536/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0082/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 83. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[-5, 20], y_values=[-3, 27], and aligned row-by-column exact masses=[[20/49, 8/49], [15/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 40/7, 39/7, 1560/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 40/7, 39/7, 1560/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0083/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 84. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[-4, 21], y_values=[-2, 28], and aligned row-by-column exact masses=[[15/49, 6/49], [20/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 72/7, 46/7, 3312/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 72/7, 46/7, 3312/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0084/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 85. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[-3, 22], y_values=[-1, 29], and aligned row-by-column exact masses=[[10/49, 4/49], [25/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 104/7, 53/7, 5512/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 104/7, 53/7, 5512/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0085/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 86. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[-2, 23], y_values=[0, 30], and aligned row-by-column exact masses=[[24/49, 18/49], [4/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 11/7, 90/7, 990/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 11/7, 90/7, 990/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0086/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 87. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[-1, 24], y_values=[1, 31], and aligned row-by-column exact masses=[[20/49, 15/49], [8/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 43/7, 97/7, 4171/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 43/7, 97/7, 4171/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0087/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 88. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[0, 25], y_values=[2, 32], and aligned row-by-column exact masses=[[16/49, 12/49], [12/49, 9/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 75/7, 104/7, 7800/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 75/7, 104/7, 7800/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0088/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 89. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[1, 26], y_values=[3, 33], and aligned row-by-column exact masses=[[12/49, 9/49], [16/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 107/7, 111/7, 11877/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 107/7, 111/7, 11877/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0089/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 90. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[2, 27], y_values=[-3, 34], and aligned row-by-column exact masses=[[8/49, 6/49], [20/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 139/7, 90/7, 12510/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 139/7, 90/7, 12510/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0090/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 91. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[3, 28], y_values=[-2, 35], and aligned row-by-column exact masses=[[18/49, 24/49], [3/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 46/7, 134/7, 6164/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 46/7, 134/7, 6164/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0091/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 92. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[4, 29], y_values=[-1, 36], and aligned row-by-column exact masses=[[15/49, 20/49], [6/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 78/7, 141/7, 10998/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 78/7, 141/7, 10998/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0092/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 93. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[5, 30], y_values=[0, 37], and aligned row-by-column exact masses=[[12/49, 16/49], [9/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 110/7, 148/7, 16280/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 110/7, 148/7, 16280/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0093/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 94. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[6, 31], y_values=[1, 38], and aligned row-by-column exact masses=[[9/49, 12/49], [12/49, 16/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 142/7, 155/7, 22010/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 142/7, 155/7, 22010/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0094/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 95. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[7, 32], y_values=[2, 39], and aligned row-by-column exact masses=[[6/49, 8/49], [15/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 174/7, 162/7, 28188/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 174/7, 162/7, 28188/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0095/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 96. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[8, 33], y_values=[3, 40], and aligned row-by-column exact masses=[[12/49, 30/49], [2/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 81/7, 206/7, 16686/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 81/7, 206/7, 16686/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0096/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 97. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[9, 34], y_values=[-3, 41], and aligned row-by-column exact masses=[[10/49, 25/49], [4/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 113/7, 199/7, 22487/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 113/7, 199/7, 22487/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0097/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 98. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[10, 35], y_values=[-2, 42], and aligned row-by-column exact masses=[[8/49, 20/49], [6/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 145/7, 206/7, 29870/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 145/7, 206/7, 29870/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0098/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 99. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[11, 36], y_values=[-1, 43], and aligned row-by-column exact masses=[[6/49, 15/49], [8/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 177/7, 213/7, 37701/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 177/7, 213/7, 37701/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0099/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate





## 100. Keep Verified Independence and Zero Covariance Distinct

Use the complete joint grid with original x_values=[12, 37], y_values=[0, 44], and aligned row-by-column exact masses=[[4/49, 10/49], [10/49, 25/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.

View answer and explanation(Yes, 209/7, 220/7, 45980/49, 0, Yes, zero_covariance_converse_not_claimed)

Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 209/7, 220/7, 45980/49, 0, Yes, zero_covariance_converse_not_claimed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r06-random-variable-independence-0100/)

Independence of random variables · qf-mod28-rvind-zero-covariance · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261028-r06-random-variable-independence-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261028-r06-random-variable-independence-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261028-r06-random-variable-independence-foundations/validation-report.json)