# Question 4550: Variance of a Shared-Model Difference [Return to catalogue page 23](https://ocme.evemisslab.com/en/questions/page/23/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261010-r06-variance-foundations/#q-20261008-advance-20261010-r06-variance-0075) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0008-04550 · q-20261008-advance-20261010-r06-variance-0075 ## Original question and givens S={0,…,2}, F=P(S), masses=[25, 2, 1], D=28, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-24, 2, 25] in index order and Y=[1, -1, -24] 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=[25, 2, 1], D=28, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-24, 2, 25] in index order and Y=[1, -1, -24] 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](https://ocme.evemisslab.com/en/mko/mko-variance/) ## Step-by-step derivation S001 · calculationBefore total mass numerator := sum(25, 2, 1) After total mass numerator = 28 Conditions: none Nonnegative masses must sum to D to give total probability1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore probability weight0 := div(25, 28) After probability weight0 = 25/28 Conditions: none Divide the indexed mass by the common positive denominator D. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore probability weight1 := div(2, 28) After probability weight1 = 1/14 Conditions: none Divide the indexed mass by the common positive denominator D. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore probability weight2 := div(1, 28) After probability weight2 = 1/28 Conditions: none Divide the indexed mass by the common positive denominator D. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore values=[-24, 2, 25];weights=[25/28, 1/14, 1/28] After known finite probability law,not a sampled estimate Conditions: none This is the complete normalized probability law,so variance uses its probability weights,not an n−1 sample estimate.Retain original indices of zero-mass outcomes. Step identifiers and dependenciesRule: probability_variance_scope · Uses: G001, S004 S006 · calculationBefore pointwise difference0 := sub(-24, 1) After pointwise difference0 = -25 Conditions: none First take X_i−Y_i at the shared outcome,without re-pairing two models. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore pointwise difference1 := sub(2, -1) After pointwise difference1 = 3 Conditions: none First take X_i−Y_i at the shared outcome,without re-pairing two models. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore pointwise difference2 := sub(25, -24) After pointwise difference2 = 49 Conditions: none First take X_i−Y_i at the shared outcome,without re-pairing two models. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore [-25, 3, 49] After compute the variance of this shared-model difference Conditions: none Without an independence premise,do not replace Var(X−Y) by Var(X)+Var(Y).Form the difference variable first and retain the original weights. Step identifiers and dependenciesRule: difference_variance_not_sum · Uses: G001, S008 S010 · calculationBefore target variable weighted value0 := mul(25/28, -25) After target variable weighted value0 = -625/28 Conditions: none Multiply aligned probabilities and values,retaining negative values and zero weights exactly. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore target variable weighted value1 := mul(1/14, 3) After target variable weighted value1 = 3/14 Conditions: none Multiply aligned probabilities and values,retaining negative values and zero weights exactly. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore target variable weighted value2 := mul(1/28, 49) After target variable weighted value2 = 7/4 Conditions: none Multiply aligned probabilities and values,retaining negative values and zero weights exactly. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore target variable mean := sum(-625/28, 3/14, 7/4) After target variable mean = -285/14 Conditions: none Add every weighted term to obtain the exact mean. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore centered difference0 := sub(-25, -285/14) After centered difference0 = -65/14 Conditions: none Subtract this variable's own exact mean. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore centered square0 := square(-65/14) After centered square0 = 4225/196 Conditions: none The squared deviation is nonnegative;do not use a signed deviation itself as a variance term. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore centered contribution0 := mul(25/28, 4225/196) After centered contribution0 = 105625/5488 Conditions: none Multiply by the aligned probability;a zero weight contributes0 even for a large deviation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore centered difference1 := sub(3, -285/14) After centered difference1 = 327/14 Conditions: none Subtract this variable's own exact mean. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore centered square1 := square(327/14) After centered square1 = 106929/196 Conditions: none The squared deviation is nonnegative;do not use a signed deviation itself as a variance term. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore centered contribution1 := mul(1/14, 106929/196) After centered contribution1 = 106929/2744 Conditions: none Multiply by the aligned probability;a zero weight contributes0 even for a large deviation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore centered difference2 := sub(49, -285/14) After centered difference2 = 971/14 Conditions: none Subtract this variable's own exact mean. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore centered square2 := square(971/14) After centered square2 = 942841/196 Conditions: none The squared deviation is nonnegative;do not use a signed deviation itself as a variance term. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore centered contribution2 := mul(1/28, 942841/196) After centered contribution2 = 942841/5488 Conditions: none Multiply by the aligned probability;a zero weight contributes0 even for a large deviation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · impliesBefore [105625/5488, 106929/2744, 942841/5488] After all indexed centered-square contributions retained Conditions: none Retain one term for every outcome,including zeros.These are quantities of the complete probability law,not sample estimates. Step identifiers and dependenciesRule: complete_centered_variance_terms · Uses: G001, S022 S024 · calculationBefore variance of the difference := sum(105625/5488, 106929/2744, 942841/5488) After variance of the difference = 45083/196 Conditions: none Sum every centered-square contribution of the difference variable. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 ## 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 45083/196 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 46bebd2c625a38d9d7336e62f0ab7cc84aa04de28effd32661d2b0b0ee131deb - candidate record SHA-256: a6bf5782afc67c31051349e2e790be4d53b992d134315dc105ec90858bd44a56 - semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289 - question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261010-r06-variance-0075.json)