Question 4145: Expectation Difference on a Shared Model

Original question and givens

S={0,…,2}, F=P(S), masses=[20, 2, 1], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-19, 2, 20] in index order and Y=[1, -1, -19] 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=[20, 2, 1], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-19, 2, 20] in index order and Y=[1, -1, -19] 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(20, 2, 1)

total mass numerator = 23

indexed masses=[20, 2, 1];D=23

probabilities=[20/23, 2/23, 1/23]

X weighted numerator0 := mul(20, -19)

X weighted numerator0 = -380

X contribution0 := div(-380, 23)

X contribution0 = -380/23

X weighted numerator1 := mul(2, 2)

X weighted numerator1 = 4

X contribution1 := div(4, 23)

X contribution1 = 4/23

X weighted numerator2 := mul(1, 20)

X weighted numerator2 = 20

X contribution2 := div(20, 23)

X contribution2 = 20/23

X expectation := sum(-380/23, 4/23, 20/23)

X expectation = -356/23

Y weighted numerator0 := mul(20, 1)

Y weighted numerator0 = 20

Y contribution0 := div(20, 23)

Y contribution0 = 20/23

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

Y weighted numerator1 = -2

Y contribution1 := div(-2, 23)

Y contribution1 = -2/23

Y weighted numerator2 := mul(1, -19)

Y weighted numerator2 = -19

Y contribution2 := div(-19, 23)

Y contribution2 = -19/23

Y expectation := sum(20/23, -2/23, -19/23)

Y expectation = -1/23

difference of expectations := sub(-356/23, -1/23)

difference of expectations = -355/23

E[X]=-356/23;E[Y]=-1/23

E[X]−E[Y]=E[X−Y]=-355/23

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

-355/23

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.