Question 4147: Expectation Difference on a Shared Model

Original question and givens

S={0,…,2}, F=P(S), masses=[22, 2, 1], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-21, 2, 22] in index order and Y=[1, -1, -21] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

• G001 — S={0,…,2}, F=P(S), masses=[22, 2, 1], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-21, 2, 22] in index order and Y=[1, -1, -21] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−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(22, 2, 1)

total mass numerator = 25

indexed masses=[22, 2, 1];D=25

probabilities=[22/25, 2/25, 1/25]

X weighted numerator0 := mul(22, -21)

X weighted numerator0 = -462

X contribution0 := div(-462, 25)

X contribution0 = -462/25

X weighted numerator1 := mul(2, 2)

X weighted numerator1 = 4

X contribution1 := div(4, 25)

X contribution1 = 4/25

X weighted numerator2 := mul(1, 22)

X weighted numerator2 = 22

X contribution2 := div(22, 25)

X contribution2 = 22/25

X expectation := sum(-462/25, 4/25, 22/25)

X expectation = -436/25

Y weighted numerator0 := mul(22, 1)

Y weighted numerator0 = 22

Y contribution0 := div(22, 25)

Y contribution0 = 22/25

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

Y weighted numerator1 = -2

Y contribution1 := div(-2, 25)

Y contribution1 = -2/25

Y weighted numerator2 := mul(1, -21)

Y weighted numerator2 = -21

Y contribution2 := div(-21, 25)

Y contribution2 = -21/25

Y expectation := sum(22/25, -2/25, -21/25)

Y expectation = -1/25

difference of expectations := sub(-436/25, -1/25)

difference of expectations = -87/5

E[X]=-436/25;E[Y]=-1/25

E[X]−E[Y]=E[X−Y]=-87/5

Reverse checks

• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Recompute difference of expectations 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

-87/5

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.