Question 14065: Every Joint Grid Entry Must Factorize

Original question and givens

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?).

• G001 — 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?).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-random-variable-independence

Step-by-step derivation

joint grid model whole mass sum := sum(8/49, 6/49, 20/49, 15/49)

joint grid model whole mass sum = 1

joint grid model:raw aligned masses=[[8, 49], [6, 49], [20, 49], [15, 49]]

normalized masses=[8/49, 6/49, 20/49, 15/49];nonnegative=Yes;total=1

original x row labels=[2, 27];original y column labels=[-3, 34]

normalized aligned row-by-column joint grid=[[8/49, 6/49], [20/49, 15/49]]

X row0 marginal := sum(8/49, 6/49)

X row0 marginal = 2/7

X row1 marginal := sum(20/49, 15/49)

X row1 marginal = 5/7

Y column0 marginal := sum(8/49, 20/49)

Y column0 marginal = 4/7

Y column1 marginal := sum(6/49, 15/49)

Y column1 marginal = 3/7

joint cell row0 column0 marginal product := mul(2/7, 4/7)

joint cell row0 column0 marginal product = 8/49

original row=0;column=0;value labels=[2, -3];joint mass=8/49;marginal product=8/49

cell factorizes=Yes

joint cell row0 column1 marginal product := mul(2/7, 3/7)

joint cell row0 column1 marginal product = 6/49

original row=0;column=1;value labels=[2, 34];joint mass=6/49;marginal product=6/49

cell factorizes=Yes

joint cell row1 column0 marginal product := mul(5/7, 4/7)

joint cell row1 column0 marginal product = 20/49

original row=1;column=0;value labels=[27, -3];joint mass=20/49;marginal product=20/49

cell factorizes=Yes

joint cell row1 column1 marginal product := mul(5/7, 3/7)

joint cell row1 column1 marginal product = 15/49

original row=1;column=1;value labels=[27, 34];joint mass=15/49;marginal product=15/49

cell factorizes=Yes

all joint factorization failures=[]

random variables independent=Yes;all-cell factorization covers every value-event rectangle

Reverse checks

• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute joint cell row1 column1 marginal product with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute joint cell row1 column0 marginal product with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute joint cell row0 column1 marginal product with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute joint cell row0 column0 marginal product with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Y column1 marginal with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Y column0 marginal with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute X row1 marginal with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute X row0 marginal with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Recompute joint grid model whole mass sum with exact rational arithmetic, not rounded decimals.

Result

([2, 27], [-3, 34], [2/7, 5/7], [4/7, 3/7], [], Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.