Question 4183: Product Expectation versus Product of Means

Original question and givens

S={0,…,2}, F=P(S), masses=[8, 2, 1], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-7, 2, 8] in index order and Y=[9, -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,…,2}, F=P(S), masses=[8, 2, 1], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-7, 2, 8] in index order and Y=[9, -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(8, 2, 1)

total mass numerator = 11

indexed masses=[8, 2, 1];D=11

probabilities=[8/11, 2/11, 1/11]

X weighted numerator0 := mul(8, -7)

X weighted numerator0 = -56

X contribution0 := div(-56, 11)

X contribution0 = -56/11

X weighted numerator1 := mul(2, 2)

X weighted numerator1 = 4

X contribution1 := div(4, 11)

X contribution1 = 4/11

X weighted numerator2 := mul(1, 8)

X weighted numerator2 = 8

X contribution2 := div(8, 11)

X contribution2 = 8/11

X expectation := sum(-56/11, 4/11, 8/11)

X expectation = -4

Y weighted numerator0 := mul(8, 9)

Y weighted numerator0 = 72

Y contribution0 := div(72, 11)

Y contribution0 = 72/11

Y weighted numerator1 := mul(2, -1)

Y weighted numerator1 = -2

Y contribution1 := div(-2, 11)

Y contribution1 = -2/11

Y weighted numerator2 := mul(1, 2)

Y weighted numerator2 = 2

Y contribution2 := div(2, 11)

Y contribution2 = 2/11

Y expectation := sum(72/11, -2/11, 2/11)

Y expectation = 72/11

pointwise XY0 := mul(-7, 9)

pointwise XY0 = -63

pointwise XY1 := mul(2, -1)

pointwise XY1 = -2

pointwise XY2 := mul(8, 2)

pointwise XY2 = 16

XY weighted numerator0 := mul(8, -63)

XY weighted numerator0 = -504

XY contribution0 := div(-504, 11)

XY contribution0 = -504/11

XY weighted numerator1 := mul(2, -2)

XY weighted numerator1 = -4

XY contribution1 := div(-4, 11)

XY contribution1 = -4/11

XY weighted numerator2 := mul(1, 16)

XY weighted numerator2 = 16

XY contribution2 := div(16, 11)

XY contribution2 = 16/11

XY expectation := sum(-504/11, -4/11, 16/11)

XY expectation = -492/11

product of means := mul(-4, 72/11)

product of means = -288/11

E[XY]=-492/11;E[X]E[Y]=-288/11

report in this order;no independence inferred

Reverse checks

• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Recompute product of means with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute XY expectation with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute XY contribution2 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute XY weighted numerator2 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute XY contribution1 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute XY weighted numerator1 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute XY contribution0 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute XY weighted numerator0 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute pointwise XY2 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute pointwise XY1 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute pointwise XY0 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Y expectation with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Y contribution2 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Y weighted numerator2 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Y contribution1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Y weighted numerator1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Y contribution0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Y weighted numerator0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute X expectation with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute X contribution2 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute X weighted numerator2 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

(-492/11, -288/11)

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.