Question 4176: Product Expectation versus Product of Means

Original question and givens

S={0,…,1}, F=P(S), masses=[1, 1], D=2, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 1] in index order and Y=[0, 1] 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, 1], D=2, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 1] in index order and Y=[0, 1] 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, 1)

total mass numerator = 2

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

probabilities=[1/2, 1/2]

X weighted numerator0 := mul(1, 0)

X weighted numerator0 = 0

X contribution0 := div(0, 2)

X contribution0 = 0

X weighted numerator1 := mul(1, 1)

X weighted numerator1 = 1

X contribution1 := div(1, 2)

X contribution1 = 1/2

X expectation := sum(0, 1/2)

X expectation = 1/2

Y weighted numerator0 := mul(1, 0)

Y weighted numerator0 = 0

Y contribution0 := div(0, 2)

Y contribution0 = 0

Y weighted numerator1 := mul(1, 1)

Y weighted numerator1 = 1

Y contribution1 := div(1, 2)

Y contribution1 = 1/2

Y expectation := sum(0, 1/2)

Y expectation = 1/2

pointwise XY0 := mul(0, 0)

pointwise XY0 = 0

pointwise XY1 := mul(1, 1)

pointwise XY1 = 1

XY weighted numerator0 := mul(1, 0)

XY weighted numerator0 = 0

XY contribution0 := div(0, 2)

XY contribution0 = 0

XY weighted numerator1 := mul(1, 1)

XY weighted numerator1 = 1

XY contribution1 := div(1, 2)

XY contribution1 = 1/2

XY expectation := sum(0, 1/2)

XY expectation = 1/2

product of means := mul(1/2, 1/2)

product of means = 1/4

E[XY]=1/2;E[X]E[Y]=1/4

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

(1/2, 1/4)

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.