Question 35819: 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, 8), (1, 8), (2, 28), (3, 1017)}. In the displayed order of S, the nonnegative probability weights are [1/(26), 20/(26), 5/(26), 0/(26)], summing exactly to one. Put Z=(4)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, 8), (1, 8), (2, 28), (3, 1017)}. In the displayed order of S, the nonnegative probability weights are [1/(26), 20/(26), 5/(26), 0/(26)], summing exactly to one. Put Z=(4)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, 20, 5, 0)

Variance ORIGINAL nonnegative RAW weights sum = 26

source roles = [[0, 1, 2, 3], powerset(S), [8, 8, 28, 1017], [1, 20, 5, 0], 26, 4, null]

derived = [Yes]

Variance transformed ORIGINAL value i=0 := mul(4, 8)

Variance transformed ORIGINAL value i=0 = 32

Variance weight times transformed value i=0 := mul(1, 32)

Variance weight times transformed value i=0 = 32

source roles = [0, 0, 8, 1, 4, null]

derived = [32, 1/26]

Variance transformed ORIGINAL value i=1 := mul(4, 8)

Variance transformed ORIGINAL value i=1 = 32

Variance weight times transformed value i=1 := mul(20, 32)

Variance weight times transformed value i=1 = 640

source roles = [1, 1, 8, 20, 4, null]

derived = [32, 10/13]

Variance transformed ORIGINAL value i=2 := mul(4, 28)

Variance transformed ORIGINAL value i=2 = 112

Variance weight times transformed value i=2 := mul(5, 112)

Variance weight times transformed value i=2 = 560

source roles = [2, 2, 28, 5, 4, null]

derived = [112, 5/26]

Variance transformed ORIGINAL value i=3 := mul(4, 1017)

Variance transformed ORIGINAL value i=3 = 4068

Variance weight times transformed value i=3 := mul(0, 4068)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1017, 0, 4, null]

derived = [4068, 0]

Variance mean numerator sum wY := sum(32, 640, 560, 0)

Variance mean numerator sum wY = 1232

Variance exact expectation M over q := div(1232, 26)

Variance exact expectation M over q = 616/13

Variance i=0 Y times q := mul(32, 26)

Variance i=0 Y times q = 832

Variance i=0 centered integer numerator := sub(832, 1232)

Variance i=0 centered integer numerator = -400

Variance i=0 centered numerator squared := mul(-400, -400)

Variance i=0 centered numerator squared = 160000

Variance i=0 weight times squared numerator := mul(1, 160000)

Variance i=0 weight times squared numerator = 160000

Variance i=1 Y times q := mul(32, 26)

Variance i=1 Y times q = 832

Variance i=1 centered integer numerator := sub(832, 1232)

Variance i=1 centered integer numerator = -400

Variance i=1 centered numerator squared := mul(-400, -400)

Variance i=1 centered numerator squared = 160000

Variance i=1 weight times squared numerator := mul(20, 160000)

Variance i=1 weight times squared numerator = 3200000

Variance i=2 Y times q := mul(112, 26)

Variance i=2 Y times q = 2912

Variance i=2 centered integer numerator := sub(2912, 1232)

Variance i=2 centered integer numerator = 1680

Variance i=2 centered numerator squared := mul(1680, 1680)

Variance i=2 centered numerator squared = 2822400

Variance i=2 weight times squared numerator := mul(5, 2822400)

Variance i=2 weight times squared numerator = 14112000

Variance i=3 Y times q := mul(4068, 26)

Variance i=3 Y times q = 105768

Variance i=3 centered integer numerator := sub(105768, 1232)

Variance i=3 centered integer numerator = 104536

Variance i=3 centered numerator squared := mul(104536, 104536)

Variance i=3 centered numerator squared = 10927775296

Variance i=3 weight times squared numerator := mul(0, 10927775296)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(160000, 3200000, 14112000, 0)

Variance sum ALL weighted squared numerators = 17472000

Variance denominator q squared := mul(26, 26)

Variance denominator q squared = 676

Variance denominator q cubed binary second factor := mul(676, 26)

Variance denominator q cubed binary second factor = 17576

Variance probability population centered sum over q cubed := div(17472000, 17576)

Variance probability population centered sum over q cubed = 168000/169

source roles = [[32, 32, 112, 4068], [1, 20, 5, 0], 26, 616/13, 17472000, 17576]

derived = [168000/169, [32, 112], No]

source roles = [scale_variance, 4, null]

derived = [168000/169]

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

168000/169

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.