Question 4545: Variance of a Shared-Model Difference

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 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=[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 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(20, 2, 1)

total mass numerator = 23

probability weight0 := div(20, 23)

probability weight0 = 20/23

probability weight1 := div(2, 23)

probability weight1 = 2/23

probability weight2 := div(1, 23)

probability weight2 = 1/23

values=[-19, 2, 20];weights=[20/23, 2/23, 1/23]

known finite probability law,not a sampled estimate

pointwise difference0 := sub(-19, 1)

pointwise difference0 = -20

pointwise difference1 := sub(2, -1)

pointwise difference1 = 3

pointwise difference2 := sub(20, -19)

pointwise difference2 = 39

[-20, 3, 39]

compute the variance of this shared-model difference

target variable weighted value0 := mul(20/23, -20)

target variable weighted value0 = -400/23

target variable weighted value1 := mul(2/23, 3)

target variable weighted value1 = 6/23

target variable weighted value2 := mul(1/23, 39)

target variable weighted value2 = 39/23

target variable mean := sum(-400/23, 6/23, 39/23)

target variable mean = -355/23

centered difference0 := sub(-20, -355/23)

centered difference0 = -105/23

centered square0 := square(-105/23)

centered square0 = 11025/529

centered contribution0 := mul(20/23, 11025/529)

centered contribution0 = 220500/12167

centered difference1 := sub(3, -355/23)

centered difference1 = 424/23

centered square1 := square(424/23)

centered square1 = 179776/529

centered contribution1 := mul(2/23, 179776/529)

centered contribution1 = 359552/12167

centered difference2 := sub(39, -355/23)

centered difference2 = 1252/23

centered square2 := square(1252/23)

centered square2 = 1567504/529

centered contribution2 := mul(1/23, 1567504/529)

centered contribution2 = 1567504/12167

[220500/12167, 359552/12167, 1567504/12167]

all indexed centered-square contributions retained

variance of the difference := sum(220500/12167, 359552/12167, 1567504/12167)

variance of the difference = 93372/529

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

93372/529

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.