Question 4581: Second Moment and Squared Mean

Original question and givens

S={0,…,2}, F=P(S), masses=[6, 2, 1], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-5, 2, 6] 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=[6, 2, 1], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-5, 2, 6] 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(6, 2, 1)

total mass numerator = 9

probability weight0 := div(6, 9)

probability weight0 = 2/3

probability weight1 := div(2, 9)

probability weight1 = 2/9

probability weight2 := div(1, 9)

probability weight2 = 1/9

values=[-5, 2, 6];weights=[2/3, 2/9, 1/9]

known finite probability law,not a sampled estimate

target variable weighted value0 := mul(2/3, -5)

target variable weighted value0 = -10/3

target variable weighted value1 := mul(2/9, 2)

target variable weighted value1 = 4/9

target variable weighted value2 := mul(1/9, 6)

target variable weighted value2 = 2/3

target variable mean := sum(-10/3, 4/9, 2/3)

target variable mean = -20/9

pointwise square0 := square(-5)

pointwise square0 = 25

pointwise square1 := square(2)

pointwise square1 = 4

pointwise square2 := square(6)

pointwise square2 = 36

squared variable weighted value0 := mul(2/3, 25)

squared variable weighted value0 = 50/3

squared variable weighted value1 := mul(2/9, 4)

squared variable weighted value1 = 8/9

squared variable weighted value2 := mul(1/9, 36)

squared variable weighted value2 = 4

squared variable mean := sum(50/3, 8/9, 4)

squared variable mean = 194/9

mean squared := square(-20/9)

mean squared = 400/81

E[X²]=194/9;(E[X])²=400/81

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

(194/9, 400/81)

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.