Question 4547: Variance of a Shared-Model Difference

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 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=[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 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(22, 2, 1)

total mass numerator = 25

probability weight0 := div(22, 25)

probability weight0 = 22/25

probability weight1 := div(2, 25)

probability weight1 = 2/25

probability weight2 := div(1, 25)

probability weight2 = 1/25

values=[-21, 2, 22];weights=[22/25, 2/25, 1/25]

known finite probability law,not a sampled estimate

pointwise difference0 := sub(-21, 1)

pointwise difference0 = -22

pointwise difference1 := sub(2, -1)

pointwise difference1 = 3

pointwise difference2 := sub(22, -21)

pointwise difference2 = 43

[-22, 3, 43]

compute the variance of this shared-model difference

target variable weighted value0 := mul(22/25, -22)

target variable weighted value0 = -484/25

target variable weighted value1 := mul(2/25, 3)

target variable weighted value1 = 6/25

target variable weighted value2 := mul(1/25, 43)

target variable weighted value2 = 43/25

target variable mean := sum(-484/25, 6/25, 43/25)

target variable mean = -87/5

centered difference0 := sub(-22, -87/5)

centered difference0 = -23/5

centered square0 := square(-23/5)

centered square0 = 529/25

centered contribution0 := mul(22/25, 529/25)

centered contribution0 = 11638/625

centered difference1 := sub(3, -87/5)

centered difference1 = 102/5

centered square1 := square(102/5)

centered square1 = 10404/25

centered contribution1 := mul(2/25, 10404/25)

centered contribution1 = 20808/625

centered difference2 := sub(43, -87/5)

centered difference2 = 302/5

centered square2 := square(302/5)

centered square2 = 91204/25

centered contribution2 := mul(1/25, 91204/25)

centered contribution2 = 91204/625

[11638/625, 20808/625, 91204/625]

all indexed centered-square contributions retained

variance of the difference := sum(11638/625, 20808/625, 91204/625)

variance of the difference = 4946/25

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

4946/25

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.