Question 4549: Variance of a Shared-Model Difference

Original question and givens

S={0,…,2}, F=P(S), masses=[24, 2, 1], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-23, 2, 24] in index order and Y=[1, -1, -23] on the same S and weights, with no independence premise. Find Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

• G001 — S={0,…,2}, F=P(S), masses=[24, 2, 1], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-23, 2, 24] in index order and Y=[1, -1, -23] on the same S and weights, with no independence premise. Find Var(X−Y), not Var(X)+Var(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-variance

Step-by-step derivation

total mass numerator := sum(24, 2, 1)

total mass numerator = 27

probability weight0 := div(24, 27)

probability weight0 = 8/9

probability weight1 := div(2, 27)

probability weight1 = 2/27

probability weight2 := div(1, 27)

probability weight2 = 1/27

values=[-23, 2, 24];weights=[8/9, 2/27, 1/27]

known finite probability law,not a sampled estimate

pointwise difference0 := sub(-23, 1)

pointwise difference0 = -24

pointwise difference1 := sub(2, -1)

pointwise difference1 = 3

pointwise difference2 := sub(24, -23)

pointwise difference2 = 47

[-24, 3, 47]

compute the variance of this shared-model difference

target variable weighted value0 := mul(8/9, -24)

target variable weighted value0 = -64/3

target variable weighted value1 := mul(2/27, 3)

target variable weighted value1 = 2/9

target variable weighted value2 := mul(1/27, 47)

target variable weighted value2 = 47/27

target variable mean := sum(-64/3, 2/9, 47/27)

target variable mean = -523/27

centered difference0 := sub(-24, -523/27)

centered difference0 = -125/27

centered square0 := square(-125/27)

centered square0 = 15625/729

centered contribution0 := mul(8/9, 15625/729)

centered contribution0 = 125000/6561

centered difference1 := sub(3, -523/27)

centered difference1 = 604/27

centered square1 := square(604/27)

centered square1 = 364816/729

centered contribution1 := mul(2/27, 364816/729)

centered contribution1 = 729632/19683

centered difference2 := sub(47, -523/27)

centered difference2 = 1792/27

centered square2 := square(1792/27)

centered square2 = 3211264/729

centered contribution2 := mul(1/27, 3211264/729)

centered contribution2 = 3211264/19683

[125000/6561, 729632/19683, 3211264/19683]

all indexed centered-square contributions retained

variance of the difference := sum(125000/6561, 729632/19683, 3211264/19683)

variance of the difference = 159848/729

Reverse checks

• S024 · premise_and_result_audit — Recompute variance of the difference with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Recompute centered contribution2 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute centered square2 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute centered difference2 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute centered contribution1 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute centered square1 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute centered difference1 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute centered contribution0 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute centered square0 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute centered difference0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute target variable mean with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute target variable weighted value2 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute target variable weighted value1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute target variable weighted value0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute pointwise difference2 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute pointwise difference1 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute pointwise difference0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute probability weight2 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute probability weight1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute probability weight0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Recompute total mass numerator with exact rational arithmetic, not rounded decimals.

Result

159848/729

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.