Question 4198: Product Expectation versus Product of Means

Original question and givens

S={0,…,2}, F=P(S), masses=[23, 2, 1], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-22, 2, 23] in index order and Y=[24, -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=[23, 2, 1], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-22, 2, 23] in index order and Y=[24, -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(23, 2, 1)

total mass numerator = 26

indexed masses=[23, 2, 1];D=26

probabilities=[23/26, 1/13, 1/26]

X weighted numerator0 := mul(23, -22)

X weighted numerator0 = -506

X contribution0 := div(-506, 26)

X contribution0 = -253/13

X weighted numerator1 := mul(2, 2)

X weighted numerator1 = 4

X contribution1 := div(4, 26)

X contribution1 = 2/13

X weighted numerator2 := mul(1, 23)

X weighted numerator2 = 23

X contribution2 := div(23, 26)

X contribution2 = 23/26

X expectation := sum(-253/13, 2/13, 23/26)

X expectation = -479/26

Y weighted numerator0 := mul(23, 24)

Y weighted numerator0 = 552

Y contribution0 := div(552, 26)

Y contribution0 = 276/13

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

Y weighted numerator1 = -2

Y contribution1 := div(-2, 26)

Y contribution1 = -1/13

Y weighted numerator2 := mul(1, 2)

Y weighted numerator2 = 2

Y contribution2 := div(2, 26)

Y contribution2 = 1/13

Y expectation := sum(276/13, -1/13, 1/13)

Y expectation = 276/13

pointwise XY0 := mul(-22, 24)

pointwise XY0 = -528

pointwise XY1 := mul(2, -1)

pointwise XY1 = -2

pointwise XY2 := mul(23, 2)

pointwise XY2 = 46

XY weighted numerator0 := mul(23, -528)

XY weighted numerator0 = -12144

XY contribution0 := div(-12144, 26)

XY contribution0 = -6072/13

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

XY weighted numerator1 = -4

XY contribution1 := div(-4, 26)

XY contribution1 = -2/13

XY weighted numerator2 := mul(1, 46)

XY weighted numerator2 = 46

XY contribution2 := div(46, 26)

XY contribution2 = 23/13

XY expectation := sum(-6072/13, -2/13, 23/13)

XY expectation = -6051/13

product of means := mul(-479/26, 276/13)

product of means = -66102/169

E[XY]=-6051/13;E[X]E[Y]=-66102/169

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

(-6051/13, -66102/169)

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.