Question 35813: Probability Variance after Scaling

Original question and givens

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, 2), (1, 2), (2, 16), (3, 1011)}. In the displayed order of S, the nonnegative probability weights are [1/(18), 14/(18), 3/(18), 0/(18)], summing exactly to one. Put Z=(3)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, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, 2), (1, 2), (2, 16), (3, 1011)}. In the displayed order of S, the nonnegative probability weights are [1/(18), 14/(18), 3/(18), 0/(18)], summing exactly to one. Put Z=(3)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-scale, [F, S, denominator, k, operation, values, weights]]

derived = [variance, scale_variance, null]

Variance ORIGINAL nonnegative RAW weights sum := sum(1, 14, 3, 0)

Variance ORIGINAL nonnegative RAW weights sum = 18

source roles = [[0, 1, 2, 3], powerset(S), [2, 2, 16, 1011], [1, 14, 3, 0], 18, 3, null]

derived = [Yes]

Variance transformed ORIGINAL value i=0 := mul(3, 2)

Variance transformed ORIGINAL value i=0 = 6

Variance weight times transformed value i=0 := mul(1, 6)

Variance weight times transformed value i=0 = 6

source roles = [0, 0, 2, 1, 3, null]

derived = [6, 1/18]

Variance transformed ORIGINAL value i=1 := mul(3, 2)

Variance transformed ORIGINAL value i=1 = 6

Variance weight times transformed value i=1 := mul(14, 6)

Variance weight times transformed value i=1 = 84

source roles = [1, 1, 2, 14, 3, null]

derived = [6, 7/9]

Variance transformed ORIGINAL value i=2 := mul(3, 16)

Variance transformed ORIGINAL value i=2 = 48

Variance weight times transformed value i=2 := mul(3, 48)

Variance weight times transformed value i=2 = 144

source roles = [2, 2, 16, 3, 3, null]

derived = [48, 1/6]

Variance transformed ORIGINAL value i=3 := mul(3, 1011)

Variance transformed ORIGINAL value i=3 = 3033

Variance weight times transformed value i=3 := mul(0, 3033)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1011, 0, 3, null]

derived = [3033, 0]

Variance mean numerator sum wY := sum(6, 84, 144, 0)

Variance mean numerator sum wY = 234

Variance exact expectation M over q := div(234, 18)

Variance exact expectation M over q = 13

Variance i=0 Y times q := mul(6, 18)

Variance i=0 Y times q = 108

Variance i=0 centered integer numerator := sub(108, 234)

Variance i=0 centered integer numerator = -126

Variance i=0 centered numerator squared := mul(-126, -126)

Variance i=0 centered numerator squared = 15876

Variance i=0 weight times squared numerator := mul(1, 15876)

Variance i=0 weight times squared numerator = 15876

Variance i=1 Y times q := mul(6, 18)

Variance i=1 Y times q = 108

Variance i=1 centered integer numerator := sub(108, 234)

Variance i=1 centered integer numerator = -126

Variance i=1 centered numerator squared := mul(-126, -126)

Variance i=1 centered numerator squared = 15876

Variance i=1 weight times squared numerator := mul(14, 15876)

Variance i=1 weight times squared numerator = 222264

Variance i=2 Y times q := mul(48, 18)

Variance i=2 Y times q = 864

Variance i=2 centered integer numerator := sub(864, 234)

Variance i=2 centered integer numerator = 630

Variance i=2 centered numerator squared := mul(630, 630)

Variance i=2 centered numerator squared = 396900

Variance i=2 weight times squared numerator := mul(3, 396900)

Variance i=2 weight times squared numerator = 1190700

Variance i=3 Y times q := mul(3033, 18)

Variance i=3 Y times q = 54594

Variance i=3 centered integer numerator := sub(54594, 234)

Variance i=3 centered integer numerator = 54360

Variance i=3 centered numerator squared := mul(54360, 54360)

Variance i=3 centered numerator squared = 2955009600

Variance i=3 weight times squared numerator := mul(0, 2955009600)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(15876, 222264, 1190700, 0)

Variance sum ALL weighted squared numerators = 1428840

Variance denominator q squared := mul(18, 18)

Variance denominator q squared = 324

Variance denominator q cubed binary second factor := mul(324, 18)

Variance denominator q cubed binary second factor = 5832

Variance probability population centered sum over q cubed := div(1428840, 5832)

Variance probability population centered sum over q cubed = 245

source roles = [[6, 6, 48, 3033], [1, 14, 3, 0], 18, 13, 1428840, 5832]

derived = [245, [6, 48], No]

source roles = [scale_variance, 3, null]

derived = [245]

Reverse checks

• S039 · 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.
• S038 · 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.
• S037 · premise_and_result_audit — Recompute Variance probability population centered sum over q cubed with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Variance denominator q cubed binary second factor with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Variance denominator q squared with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute Variance sum ALL weighted squared numerators with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Variance i=3 weight times squared numerator with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Variance i=3 centered numerator squared with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Variance i=3 centered integer numerator with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute Variance i=3 Y times q with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Variance i=2 weight times squared numerator with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Variance i=2 centered numerator squared with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Variance i=2 centered integer numerator with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Variance i=2 Y times q with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Variance i=1 weight times squared numerator with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Variance i=1 centered numerator squared with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Variance i=1 centered integer numerator with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Variance i=1 Y times q with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Variance i=0 weight times squared numerator with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Variance i=0 centered numerator squared with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Variance i=0 centered integer numerator with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Variance i=0 Y times q with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Variance exact expectation M over q with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Variance mean numerator sum wY with exact rational arithmetic, not rounded decimals.
• S015 · 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.
• S014 · premise_and_result_audit — Recompute Variance weight times transformed value i=3 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Variance transformed ORIGINAL value i=3 with exact rational arithmetic, not rounded decimals.
• S012 · 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.
• S011 · premise_and_result_audit — Recompute Variance weight times transformed value i=2 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Variance transformed ORIGINAL value i=2 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=1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Variance transformed ORIGINAL value i=1 with exact rational arithmetic, not rounded decimals.
• S006 · 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.
• S005 · premise_and_result_audit — Recompute Variance weight times transformed value i=0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Variance transformed ORIGINAL 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

245/1

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.