Question 35851: Exact Finite Probability Variance

Original question and givens

The nonempty finite sample space is S={0, 1, 2}, with event family F=𝒫(S). The total function X:S→R has table {(0, 5), (1, 5), (2, 999)}. In the displayed order of S, the nonnegative probability weights are [1/(3), 2/(3), 0/(3)], summing exactly to one. Put Z=X. Find Var(Z)=E[(Z−E[Z])²] exactly as a reduced positive-denominator fraction. This is probability-distribution variance, not sample variance with an n−1 denominator.

• G001 — The nonempty finite sample space is S={0, 1, 2}, with event family F=𝒫(S). The total function X:S→R has table {(0, 5), (1, 5), (2, 999)}. In the displayed order of S, the nonnegative probability weights are [1/(3), 2/(3), 0/(3)], summing exactly to one. Put Z=X. Find Var(Z)=E[(Z−E[Z])²] exactly as a reduced positive-denominator fraction. This is probability-distribution variance, not sample variance with an n−1 denominator.

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

source roles = [qf-variance-weighted, [F, S, denominator, operation, values, weights]]

derived = [variance, variance, null]

Variance ORIGINAL nonnegative RAW weights sum := sum(1, 2, 0)

Variance ORIGINAL nonnegative RAW weights sum = 3

source roles = [[0, 1, 2], powerset(S), [5, 5, 999], [1, 2, 0], 3, null, null]

derived = [Yes]

Variance weight times transformed value i=0 := mul(1, 5)

Variance weight times transformed value i=0 = 5

source roles = [0, 0, 5, 1, null, null]

derived = [5, 1/3]

Variance weight times transformed value i=1 := mul(2, 5)

Variance weight times transformed value i=1 = 10

source roles = [1, 1, 5, 2, null, null]

derived = [5, 2/3]

Variance weight times transformed value i=2 := mul(0, 999)

Variance weight times transformed value i=2 = 0

source roles = [2, 2, 999, 0, null, null]

derived = [999, 0]

Variance mean numerator sum wY := sum(5, 10, 0)

Variance mean numerator sum wY = 15

Variance exact expectation M over q := div(15, 3)

Variance exact expectation M over q = 5

Variance i=0 Y times q := mul(5, 3)

Variance i=0 Y times q = 15

Variance i=0 centered integer numerator := sub(15, 15)

Variance i=0 centered integer numerator = 0

Variance i=0 centered numerator squared := mul(0, 0)

Variance i=0 centered numerator squared = 0

Variance i=0 weight times squared numerator := mul(1, 0)

Variance i=0 weight times squared numerator = 0

Variance i=1 Y times q := mul(5, 3)

Variance i=1 Y times q = 15

Variance i=1 centered integer numerator := sub(15, 15)

Variance i=1 centered integer numerator = 0

Variance i=1 centered numerator squared := mul(0, 0)

Variance i=1 centered numerator squared = 0

Variance i=1 weight times squared numerator := mul(2, 0)

Variance i=1 weight times squared numerator = 0

Variance i=2 Y times q := mul(999, 3)

Variance i=2 Y times q = 2997

Variance i=2 centered integer numerator := sub(2997, 15)

Variance i=2 centered integer numerator = 2982

Variance i=2 centered numerator squared := mul(2982, 2982)

Variance i=2 centered numerator squared = 8892324

Variance i=2 weight times squared numerator := mul(0, 8892324)

Variance i=2 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(0, 0, 0)

Variance sum ALL weighted squared numerators = 0

Variance denominator q squared := mul(3, 3)

Variance denominator q squared = 9

Variance denominator q cubed binary second factor := mul(9, 3)

Variance denominator q cubed binary second factor = 27

Variance probability population centered sum over q cubed := div(0, 27)

Variance probability population centered sum over q cubed = 0

source roles = [[5, 5, 999], [1, 2, 0], 3, 5, 0, 27]

derived = [0, [5], Yes]

source roles = [variance, null, null]

derived = [0]

Reverse checks

• S029 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
• S028 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
• S027 · premise_and_result_audit — Recompute Variance probability population centered sum over q cubed with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Variance denominator q cubed binary second factor with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Variance denominator q squared with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Variance sum ALL weighted squared numerators with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Variance i=2 weight times squared numerator with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Variance i=2 centered numerator squared with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Variance i=2 centered integer numerator with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Variance i=2 Y times q with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Variance i=1 weight times squared numerator with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Variance i=1 centered numerator squared with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Variance i=1 centered integer numerator with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Variance i=1 Y times q with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Variance i=0 weight times squared numerator with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Variance i=0 centered numerator squared with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Variance i=0 centered integer numerator with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Variance i=0 Y times q with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Variance exact expectation M over q with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Variance mean numerator sum wY with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
• S008 · premise_and_result_audit — Recompute Variance weight times transformed value i=2 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
• S006 · premise_and_result_audit — Recompute Variance weight times transformed value i=1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
• S004 · premise_and_result_audit — Recompute Variance weight times transformed value i=0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
• S002 · premise_and_result_audit — Recompute Variance ORIGINAL nonnegative RAW weights sum with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.

Result

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.