Question 4577: Second Moment and Squared Mean

Original question and givens

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

total mass numerator = 2

probability weight0 := div(1, 2)

probability weight0 = 1/2

probability weight1 := div(1, 2)

probability weight1 = 1/2

values=[-1, 1];weights=[1/2, 1/2]

known finite probability law,not a sampled estimate

target variable weighted value0 := mul(1/2, -1)

target variable weighted value0 = -1/2

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

target variable weighted value1 = 1/2

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

target variable mean = 0

pointwise square0 := square(-1)

pointwise square0 = 1

pointwise square1 := square(1)

pointwise square1 = 1

squared variable weighted value0 := mul(1/2, 1)

squared variable weighted value0 = 1/2

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

squared variable weighted value1 = 1/2

squared variable mean := sum(1/2, 1/2)

squared variable mean = 1

mean squared := square(0)

mean squared = 0

E[X²]=1;(E[X])²=0

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

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute mean squared with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute squared variable mean with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute squared variable weighted value1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute squared variable weighted value0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute pointwise square1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute pointwise square0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute target variable mean with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute target variable weighted value1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute target variable weighted value0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

(1/1, 0/1)

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.