Question 4188: Product Expectation versus Product of Means

Original question and givens

S={0,…,2}, F=P(S), masses=[13, 2, 1], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-12, 2, 13] in index order and Y=[14, -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=[13, 2, 1], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-12, 2, 13] in index order and Y=[14, -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(13, 2, 1)

total mass numerator = 16

indexed masses=[13, 2, 1];D=16

probabilities=[13/16, 1/8, 1/16]

X weighted numerator0 := mul(13, -12)

X weighted numerator0 = -156

X contribution0 := div(-156, 16)

X contribution0 = -39/4

X weighted numerator1 := mul(2, 2)

X weighted numerator1 = 4

X contribution1 := div(4, 16)

X contribution1 = 1/4

X weighted numerator2 := mul(1, 13)

X weighted numerator2 = 13

X contribution2 := div(13, 16)

X contribution2 = 13/16

X expectation := sum(-39/4, 1/4, 13/16)

X expectation = -139/16

Y weighted numerator0 := mul(13, 14)

Y weighted numerator0 = 182

Y contribution0 := div(182, 16)

Y contribution0 = 91/8

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

Y weighted numerator1 = -2

Y contribution1 := div(-2, 16)

Y contribution1 = -1/8

Y weighted numerator2 := mul(1, 2)

Y weighted numerator2 = 2

Y contribution2 := div(2, 16)

Y contribution2 = 1/8

Y expectation := sum(91/8, -1/8, 1/8)

Y expectation = 91/8

pointwise XY0 := mul(-12, 14)

pointwise XY0 = -168

pointwise XY1 := mul(2, -1)

pointwise XY1 = -2

pointwise XY2 := mul(13, 2)

pointwise XY2 = 26

XY weighted numerator0 := mul(13, -168)

XY weighted numerator0 = -2184

XY contribution0 := div(-2184, 16)

XY contribution0 = -273/2

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

XY weighted numerator1 = -4

XY contribution1 := div(-4, 16)

XY contribution1 = -1/4

XY weighted numerator2 := mul(1, 26)

XY weighted numerator2 = 26

XY contribution2 := div(26, 16)

XY contribution2 = 13/8

XY expectation := sum(-273/2, -1/4, 13/8)

XY expectation = -1081/8

product of means := mul(-139/16, 91/8)

product of means = -12649/128

E[XY]=-1081/8;E[X]E[Y]=-12649/128

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

(-1081/8, -12649/128)

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.