Full Joint Factorization, Value Events and One-Way Covariance Consequences
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Independence of random variables · qf-mod28-rvind-zero-covariance · candidate
Canonical question JSONL · English JSONL · Validation report