Question 4509: Every Centered Variance Contribution

Original question and givens

S={0,…,2}, F=P(S), masses=[9, 2, 1], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-8, 2, 9] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

• G001 — S={0,…,2}, F=P(S), masses=[9, 2, 1], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-8, 2, 9] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. 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(9, 2, 1)

total mass numerator = 12

probability weight0 := div(9, 12)

probability weight0 = 3/4

probability weight1 := div(2, 12)

probability weight1 = 1/6

probability weight2 := div(1, 12)

probability weight2 = 1/12

values=[-8, 2, 9];weights=[3/4, 1/6, 1/12]

known finite probability law,not a sampled estimate

target variable weighted value0 := mul(3/4, -8)

target variable weighted value0 = -6

target variable weighted value1 := mul(1/6, 2)

target variable weighted value1 = 1/3

target variable weighted value2 := mul(1/12, 9)

target variable weighted value2 = 3/4

target variable mean := sum(-6, 1/3, 3/4)

target variable mean = -59/12

centered difference0 := sub(-8, -59/12)

centered difference0 = -37/12

centered square0 := square(-37/12)

centered square0 = 1369/144

centered contribution0 := mul(3/4, 1369/144)

centered contribution0 = 1369/192

centered difference1 := sub(2, -59/12)

centered difference1 = 83/12

centered square1 := square(83/12)

centered square1 = 6889/144

centered contribution1 := mul(1/6, 6889/144)

centered contribution1 = 6889/864

centered difference2 := sub(9, -59/12)

centered difference2 = 167/12

centered square2 := square(167/12)

centered square2 = 27889/144

centered contribution2 := mul(1/12, 27889/144)

centered contribution2 = 27889/1728

[1369/192, 6889/864, 27889/1728]

all indexed centered-square contributions retained

Reverse checks

• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Recompute centered contribution2 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute centered square2 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute centered difference2 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute centered contribution1 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute centered square1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute centered difference1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute centered contribution0 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute centered square0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute centered difference0 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

[1369/192, 6889/864, 27889/1728]

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.