Question 4538: Variance of a Shared-Model Difference

Original question and givens

S={0,…,2}, F=P(S), masses=[13, 2, 1], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-12, 2, 13] in index order and Y=[1, -1, -12] 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=[13, 2, 1], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-12, 2, 13] in index order and Y=[1, -1, -12] 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(13, 2, 1)

total mass numerator = 16

probability weight0 := div(13, 16)

probability weight0 = 13/16

probability weight1 := div(2, 16)

probability weight1 = 1/8

probability weight2 := div(1, 16)

probability weight2 = 1/16

values=[-12, 2, 13];weights=[13/16, 1/8, 1/16]

known finite probability law,not a sampled estimate

pointwise difference0 := sub(-12, 1)

pointwise difference0 = -13

pointwise difference1 := sub(2, -1)

pointwise difference1 = 3

pointwise difference2 := sub(13, -12)

pointwise difference2 = 25

[-13, 3, 25]

compute the variance of this shared-model difference

target variable weighted value0 := mul(13/16, -13)

target variable weighted value0 = -169/16

target variable weighted value1 := mul(1/8, 3)

target variable weighted value1 = 3/8

target variable weighted value2 := mul(1/16, 25)

target variable weighted value2 = 25/16

target variable mean := sum(-169/16, 3/8, 25/16)

target variable mean = -69/8

centered difference0 := sub(-13, -69/8)

centered difference0 = -35/8

centered square0 := square(-35/8)

centered square0 = 1225/64

centered contribution0 := mul(13/16, 1225/64)

centered contribution0 = 15925/1024

centered difference1 := sub(3, -69/8)

centered difference1 = 93/8

centered square1 := square(93/8)

centered square1 = 8649/64

centered contribution1 := mul(1/8, 8649/64)

centered contribution1 = 8649/512

centered difference2 := sub(25, -69/8)

centered difference2 = 269/8

centered square2 := square(269/8)

centered square2 = 72361/64

centered contribution2 := mul(1/16, 72361/64)

centered contribution2 = 72361/1024

[15925/1024, 8649/512, 72361/1024]

all indexed centered-square contributions retained

variance of the difference := sum(15925/1024, 8649/512, 72361/1024)

variance of the difference = 6599/64

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

6599/64

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.