Question 35815: 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, 4), (1, 4), (2, 20), (3, 1013)}. In the displayed order of S, the nonnegative probability weights are [1/(22), 16/(22), 5/(22), 0/(22)], summing exactly to one. Put Z=(5)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, 4), (1, 4), (2, 20), (3, 1013)}. In the displayed order of S, the nonnegative probability weights are [1/(22), 16/(22), 5/(22), 0/(22)], summing exactly to one. Put Z=(5)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, 16, 5, 0)

Variance ORIGINAL nonnegative RAW weights sum = 22

source roles = [[0, 1, 2, 3], powerset(S), [4, 4, 20, 1013], [1, 16, 5, 0], 22, 5, null]

derived = [Yes]

Variance transformed ORIGINAL value i=0 := mul(5, 4)

Variance transformed ORIGINAL value i=0 = 20

Variance weight times transformed value i=0 := mul(1, 20)

Variance weight times transformed value i=0 = 20

source roles = [0, 0, 4, 1, 5, null]

derived = [20, 1/22]

Variance transformed ORIGINAL value i=1 := mul(5, 4)

Variance transformed ORIGINAL value i=1 = 20

Variance weight times transformed value i=1 := mul(16, 20)

Variance weight times transformed value i=1 = 320

source roles = [1, 1, 4, 16, 5, null]

derived = [20, 8/11]

Variance transformed ORIGINAL value i=2 := mul(5, 20)

Variance transformed ORIGINAL value i=2 = 100

Variance weight times transformed value i=2 := mul(5, 100)

Variance weight times transformed value i=2 = 500

source roles = [2, 2, 20, 5, 5, null]

derived = [100, 5/22]

Variance transformed ORIGINAL value i=3 := mul(5, 1013)

Variance transformed ORIGINAL value i=3 = 5065

Variance weight times transformed value i=3 := mul(0, 5065)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1013, 0, 5, null]

derived = [5065, 0]

Variance mean numerator sum wY := sum(20, 320, 500, 0)

Variance mean numerator sum wY = 840

Variance exact expectation M over q := div(840, 22)

Variance exact expectation M over q = 420/11

Variance i=0 Y times q := mul(20, 22)

Variance i=0 Y times q = 440

Variance i=0 centered integer numerator := sub(440, 840)

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(20, 22)

Variance i=1 Y times q = 440

Variance i=1 centered integer numerator := sub(440, 840)

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(16, 160000)

Variance i=1 weight times squared numerator = 2560000

Variance i=2 Y times q := mul(100, 22)

Variance i=2 Y times q = 2200

Variance i=2 centered integer numerator := sub(2200, 840)

Variance i=2 centered integer numerator = 1360

Variance i=2 centered numerator squared := mul(1360, 1360)

Variance i=2 centered numerator squared = 1849600

Variance i=2 weight times squared numerator := mul(5, 1849600)

Variance i=2 weight times squared numerator = 9248000

Variance i=3 Y times q := mul(5065, 22)

Variance i=3 Y times q = 111430

Variance i=3 centered integer numerator := sub(111430, 840)

Variance i=3 centered integer numerator = 110590

Variance i=3 centered numerator squared := mul(110590, 110590)

Variance i=3 centered numerator squared = 12230148100

Variance i=3 weight times squared numerator := mul(0, 12230148100)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(160000, 2560000, 9248000, 0)

Variance sum ALL weighted squared numerators = 11968000

Variance denominator q squared := mul(22, 22)

Variance denominator q squared = 484

Variance denominator q cubed binary second factor := mul(484, 22)

Variance denominator q cubed binary second factor = 10648

Variance probability population centered sum over q cubed := div(11968000, 10648)

Variance probability population centered sum over q cubed = 136000/121

source roles = [[20, 20, 100, 5065], [1, 16, 5, 0], 22, 420/11, 11968000, 10648]

derived = [136000/121, [20, 100], No]

source roles = [scale_variance, 5, null]

derived = [136000/121]

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

136000/121

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.