Question 4580: Second Moment and Squared Mean

Original question and givens

S={0,…,2}, F=P(S), masses=[5, 2, 1], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-4, 2, 5] 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=[5, 2, 1], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-4, 2, 5] 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(5, 2, 1)

total mass numerator = 8

probability weight0 := div(5, 8)

probability weight0 = 5/8

probability weight1 := div(2, 8)

probability weight1 = 1/4

probability weight2 := div(1, 8)

probability weight2 = 1/8

values=[-4, 2, 5];weights=[5/8, 1/4, 1/8]

known finite probability law,not a sampled estimate

target variable weighted value0 := mul(5/8, -4)

target variable weighted value0 = -5/2

target variable weighted value1 := mul(1/4, 2)

target variable weighted value1 = 1/2

target variable weighted value2 := mul(1/8, 5)

target variable weighted value2 = 5/8

target variable mean := sum(-5/2, 1/2, 5/8)

target variable mean = -11/8

pointwise square0 := square(-4)

pointwise square0 = 16

pointwise square1 := square(2)

pointwise square1 = 4

pointwise square2 := square(5)

pointwise square2 = 25

squared variable weighted value0 := mul(5/8, 16)

squared variable weighted value0 = 10

squared variable weighted value1 := mul(1/4, 4)

squared variable weighted value1 = 1

squared variable weighted value2 := mul(1/8, 25)

squared variable weighted value2 = 25/8

squared variable mean := sum(10, 1, 25/8)

squared variable mean = 113/8

mean squared := square(-11/8)

mean squared = 121/64

E[X²]=113/8;(E[X])²=121/64

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

(113/8, 121/64)

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.