Question 4196: Product Expectation versus Product of Means

Original question and givens

S={0,…,2}, F=P(S), masses=[21, 2, 1], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-20, 2, 21] in index order and Y=[22, -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=[21, 2, 1], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-20, 2, 21] in index order and Y=[22, -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(21, 2, 1)

total mass numerator = 24

indexed masses=[21, 2, 1];D=24

probabilities=[7/8, 1/12, 1/24]

X weighted numerator0 := mul(21, -20)

X weighted numerator0 = -420

X contribution0 := div(-420, 24)

X contribution0 = -35/2

X weighted numerator1 := mul(2, 2)

X weighted numerator1 = 4

X contribution1 := div(4, 24)

X contribution1 = 1/6

X weighted numerator2 := mul(1, 21)

X weighted numerator2 = 21

X contribution2 := div(21, 24)

X contribution2 = 7/8

X expectation := sum(-35/2, 1/6, 7/8)

X expectation = -395/24

Y weighted numerator0 := mul(21, 22)

Y weighted numerator0 = 462

Y contribution0 := div(462, 24)

Y contribution0 = 77/4

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

Y weighted numerator1 = -2

Y contribution1 := div(-2, 24)

Y contribution1 = -1/12

Y weighted numerator2 := mul(1, 2)

Y weighted numerator2 = 2

Y contribution2 := div(2, 24)

Y contribution2 = 1/12

Y expectation := sum(77/4, -1/12, 1/12)

Y expectation = 77/4

pointwise XY0 := mul(-20, 22)

pointwise XY0 = -440

pointwise XY1 := mul(2, -1)

pointwise XY1 = -2

pointwise XY2 := mul(21, 2)

pointwise XY2 = 42

XY weighted numerator0 := mul(21, -440)

XY weighted numerator0 = -9240

XY contribution0 := div(-9240, 24)

XY contribution0 = -385

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

XY weighted numerator1 = -4

XY contribution1 := div(-4, 24)

XY contribution1 = -1/6

XY weighted numerator2 := mul(1, 42)

XY weighted numerator2 = 42

XY contribution2 := div(42, 24)

XY contribution2 = 7/4

XY expectation := sum(-385, -1/6, 7/4)

XY expectation = -4601/12

product of means := mul(-395/24, 77/4)

product of means = -30415/96

E[XY]=-4601/12;E[X]E[Y]=-30415/96

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

(-4601/12, -30415/96)

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.