# Question 4586: Second Moment and Squared Mean [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-0036) 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. R0008-04586 · q-20261008-advance-20261010-r06-variance-0036 ## Original question and givens S={0,…,2}, F=P(S), masses=[11, 2, 1], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-10, 2, 11] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers. - G001 — S={0,…,2}, F=P(S), masses=[11, 2, 1], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-10, 2, 11] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. 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(11, 2, 1) After total mass numerator = 14 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(11, 14) After probability weight0 = 11/14 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, 14) After probability weight1 = 1/7 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, 14) After probability weight2 = 1/14 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=[-10, 2, 11];weights=[11/14, 1/7, 1/14] 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 target variable weighted value0 := mul(11/14, -10) After target variable weighted value0 = -55/7 Conditions: none Multiply aligned probabilities and values,retaining negative values and zero weights exactly. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore target variable weighted value1 := mul(1/7, 2) After target variable weighted value1 = 2/7 Conditions: none Multiply aligned probabilities and values,retaining negative values and zero weights exactly. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore target variable weighted value2 := mul(1/14, 11) After target variable weighted value2 = 11/14 Conditions: none Multiply aligned probabilities and values,retaining negative values and zero weights exactly. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore target variable mean := sum(-55/7, 2/7, 11/14) After target variable mean = -95/14 Conditions: none Add every weighted term to obtain the exact mean. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore pointwise square0 := square(-10) After pointwise square0 = 100 Conditions: none The second moment squares each point value before weighting. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore pointwise square1 := square(2) After pointwise square1 = 4 Conditions: none The second moment squares each point value before weighting. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore pointwise square2 := square(11) After pointwise square2 = 121 Conditions: none The second moment squares each point value before weighting. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore squared variable weighted value0 := mul(11/14, 100) After squared variable weighted value0 = 550/7 Conditions: none Multiply aligned probabilities and values,retaining negative values and zero weights exactly. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore squared variable weighted value1 := mul(1/7, 4) After squared variable weighted value1 = 4/7 Conditions: none Multiply aligned probabilities and values,retaining negative values and zero weights exactly. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore squared variable weighted value2 := mul(1/14, 121) After squared variable weighted value2 = 121/14 Conditions: none Multiply aligned probabilities and values,retaining negative values and zero weights exactly. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore squared variable mean := sum(550/7, 4/7, 121/14) After squared variable mean = 1229/14 Conditions: none Add every weighted term to obtain the exact mean. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore mean squared := square(-95/14) After mean squared = 9025/196 Conditions: none The squared mean instead averages first and then squares;the order differs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore E[X²]=1229/14;(E[X])²=9025/196 After Var(X)=E[X²]−(E[X])² under this finite law Conditions: none These two values are distinct operations;their difference is the variance of the complete law.Report them in the requested order without a sample correction. Step identifiers and dependenciesRule: second_moment_vs_squared_mean · Uses: G001, S017 ## Reverse checks - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Recompute mean squared with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute squared variable mean with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute squared variable weighted value2 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute squared variable weighted value1 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute squared variable weighted value0 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute pointwise square2 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute pointwise square1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute pointwise square0 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute target variable mean with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute target variable weighted value2 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute target variable weighted value1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute target variable weighted value0 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 (1229/14, 9025/196) 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: 2538f3f2d84f456082a26681ce4663ce5f48fcbca41f8651efdb4f8fe4ee3734 - candidate record SHA-256: 927656aafb68dbc9332bbf4206c0b67c9c982197bf830e0da3a69622c9d122da - 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-0036.json)