Question 4540: Variance of a Shared-Model Difference

Original question and givens

S={0,…,2}, F=P(S), masses=[15, 2, 1], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-14, 2, 15] in index order and Y=[1, -1, -14] 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=[15, 2, 1], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-14, 2, 15] in index order and Y=[1, -1, -14] 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(15, 2, 1)

total mass numerator = 18

probability weight0 := div(15, 18)

probability weight0 = 5/6

probability weight1 := div(2, 18)

probability weight1 = 1/9

probability weight2 := div(1, 18)

probability weight2 = 1/18

values=[-14, 2, 15];weights=[5/6, 1/9, 1/18]

known finite probability law,not a sampled estimate

pointwise difference0 := sub(-14, 1)

pointwise difference0 = -15

pointwise difference1 := sub(2, -1)

pointwise difference1 = 3

pointwise difference2 := sub(15, -14)

pointwise difference2 = 29

[-15, 3, 29]

compute the variance of this shared-model difference

target variable weighted value0 := mul(5/6, -15)

target variable weighted value0 = -25/2

target variable weighted value1 := mul(1/9, 3)

target variable weighted value1 = 1/3

target variable weighted value2 := mul(1/18, 29)

target variable weighted value2 = 29/18

target variable mean := sum(-25/2, 1/3, 29/18)

target variable mean = -95/9

centered difference0 := sub(-15, -95/9)

centered difference0 = -40/9

centered square0 := square(-40/9)

centered square0 = 1600/81

centered contribution0 := mul(5/6, 1600/81)

centered contribution0 = 4000/243

centered difference1 := sub(3, -95/9)

centered difference1 = 122/9

centered square1 := square(122/9)

centered square1 = 14884/81

centered contribution1 := mul(1/9, 14884/81)

centered contribution1 = 14884/729

centered difference2 := sub(29, -95/9)

centered difference2 = 356/9

centered square2 := square(356/9)

centered square2 = 126736/81

centered contribution2 := mul(1/18, 126736/81)

centered contribution2 = 63368/729

[4000/243, 14884/729, 63368/729]

all indexed centered-square contributions retained

variance of the difference := sum(4000/243, 14884/729, 63368/729)

variance of the difference = 10028/81

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

10028/81

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.