Question 14038: One Cell Equality Is Not Whole-Law Independence

Original question and givens

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?).

• G001 — 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?).

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(16/49, 12/49, 12/49, 9/49)

joint grid model whole mass sum = 1

joint grid model:raw aligned masses=[[16, 49], [12, 49], [12, 49], [9, 49]]

normalized masses=[16/49, 12/49, 12/49, 9/49];nonnegative=Yes;total=1

original x row labels=[0, 25];original y column labels=[2, 32]

normalized aligned row-by-column joint grid=[[16/49, 12/49], [12/49, 9/49]]

X row0 marginal := sum(16/49, 12/49)

X row0 marginal = 4/7

X row1 marginal := sum(12/49, 9/49)

X row1 marginal = 3/7

Y column0 marginal := sum(16/49, 12/49)

Y column0 marginal = 4/7

Y column1 marginal := sum(12/49, 9/49)

Y column1 marginal = 3/7

selected cell marginal product := mul(4/7, 3/7)

selected cell marginal product = 12/49

requested value labels=[0, 32];original indices=[0, 1];joint=12/49

selected product=12/49;selected cell equality=Yes

Reverse checks

• 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 selected cell 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

(0, 32, 12/49, 4/7, 3/7, 12/49, 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.