Question 4177: Product Expectation versus Product of Means

Original question and givens

S={0,…,1}, F=P(S), masses=[1, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[3, 3] in index order and Y=[-1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

• G001 — S={0,…,1}, F=P(S), masses=[1, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[3, 3] in index order and Y=[-1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

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-expectation

Step-by-step derivation

total mass numerator := sum(1, 2)

total mass numerator = 3

indexed masses=[1, 2];D=3

probabilities=[1/3, 2/3]

X weighted numerator0 := mul(1, 3)

X weighted numerator0 = 3

X contribution0 := div(3, 3)

X contribution0 = 1

X weighted numerator1 := mul(2, 3)

X weighted numerator1 = 6

X contribution1 := div(6, 3)

X contribution1 = 2

X expectation := sum(1, 2)

X expectation = 3

Y weighted numerator0 := mul(1, -1)

Y weighted numerator0 = -1

Y contribution0 := div(-1, 3)

Y contribution0 = -1/3

Y weighted numerator1 := mul(2, 2)

Y weighted numerator1 = 4

Y contribution1 := div(4, 3)

Y contribution1 = 4/3

Y expectation := sum(-1/3, 4/3)

Y expectation = 1

pointwise XY0 := mul(3, -1)

pointwise XY0 = -3

pointwise XY1 := mul(3, 2)

pointwise XY1 = 6

XY weighted numerator0 := mul(1, -3)

XY weighted numerator0 = -3

XY contribution0 := div(-3, 3)

XY contribution0 = -1

XY weighted numerator1 := mul(2, 6)

XY weighted numerator1 = 12

XY contribution1 := div(12, 3)

XY contribution1 = 4

XY expectation := sum(-1, 4)

XY expectation = 3

product of means := mul(3, 1)

product of means = 3

E[XY]=3;E[X]E[Y]=3

report in this order;no independence inferred

Reverse checks

• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute product of means with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute XY expectation with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute XY contribution1 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute XY weighted numerator1 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute XY contribution0 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute XY weighted numerator0 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute pointwise XY1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute pointwise XY0 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Y expectation with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Y contribution1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Y weighted numerator1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Y contribution0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Y weighted numerator0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute X expectation with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute X contribution1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute X weighted numerator1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute X contribution0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute X weighted numerator0 with exact rational arithmetic, not rounded decimals.
• 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 total mass numerator with exact rational arithmetic, not rounded decimals.

Result

(3/1, 3/1)

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.