Question 4599: Second Moment and Squared Mean

Original question and givens

S={0,…,2}, F=P(S), masses=[24, 2, 1], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-23, 2, 24] 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=[24, 2, 1], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-23, 2, 24] 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

Step-by-step derivation

total mass numerator := sum(24, 2, 1)

total mass numerator = 27

probability weight0 := div(24, 27)

probability weight0 = 8/9

probability weight1 := div(2, 27)

probability weight1 = 2/27

probability weight2 := div(1, 27)

probability weight2 = 1/27

values=[-23, 2, 24];weights=[8/9, 2/27, 1/27]

known finite probability law,not a sampled estimate

target variable weighted value0 := mul(8/9, -23)

target variable weighted value0 = -184/9

target variable weighted value1 := mul(2/27, 2)

target variable weighted value1 = 4/27

target variable weighted value2 := mul(1/27, 24)

target variable weighted value2 = 8/9

target variable mean := sum(-184/9, 4/27, 8/9)

target variable mean = -524/27

pointwise square0 := square(-23)

pointwise square0 = 529

pointwise square1 := square(2)

pointwise square1 = 4

pointwise square2 := square(24)

pointwise square2 = 576

squared variable weighted value0 := mul(8/9, 529)

squared variable weighted value0 = 4232/9

squared variable weighted value1 := mul(2/27, 4)

squared variable weighted value1 = 8/27

squared variable weighted value2 := mul(1/27, 576)

squared variable weighted value2 = 64/3

squared variable mean := sum(4232/9, 8/27, 64/3)

squared variable mean = 13280/27

mean squared := square(-524/27)

mean squared = 274576/729

E[X²]=13280/27;(E[X])²=274576/729

Var(X)=E[X²]−(E[X])² under this finite law

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

(13280/27, 274576/729)

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.