Question 35809: 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, 8), (3, 1007)}. In the displayed order of S, the nonnegative probability weights are [1/(14), 10/(14), 3/(14), 0/(14)], 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, -2), (1, -2), (2, 8), (3, 1007)}. In the displayed order of S, the nonnegative probability weights are [1/(14), 10/(14), 3/(14), 0/(14)], 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, 10, 3, 0)

Variance ORIGINAL nonnegative RAW weights sum = 14

source roles = [[0, 1, 2, 3], powerset(S), [-2, -2, 8, 1007], [1, 10, 3, 0], 14, 4, null]

derived = [Yes]

Variance transformed ORIGINAL value i=0 := mul(4, -2)

Variance transformed ORIGINAL value i=0 = -8

Variance weight times transformed value i=0 := mul(1, -8)

Variance weight times transformed value i=0 = -8

source roles = [0, 0, -2, 1, 4, null]

derived = [-8, 1/14]

Variance transformed ORIGINAL value i=1 := mul(4, -2)

Variance transformed ORIGINAL value i=1 = -8

Variance weight times transformed value i=1 := mul(10, -8)

Variance weight times transformed value i=1 = -80

source roles = [1, 1, -2, 10, 4, null]

derived = [-8, 5/7]

Variance transformed ORIGINAL value i=2 := mul(4, 8)

Variance transformed ORIGINAL value i=2 = 32

Variance weight times transformed value i=2 := mul(3, 32)

Variance weight times transformed value i=2 = 96

source roles = [2, 2, 8, 3, 4, null]

derived = [32, 3/14]

Variance transformed ORIGINAL value i=3 := mul(4, 1007)

Variance transformed ORIGINAL value i=3 = 4028

Variance weight times transformed value i=3 := mul(0, 4028)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1007, 0, 4, null]

derived = [4028, 0]

Variance mean numerator sum wY := sum(-8, -80, 96, 0)

Variance mean numerator sum wY = 8

Variance exact expectation M over q := div(8, 14)

Variance exact expectation M over q = 4/7

Variance i=0 Y times q := mul(-8, 14)

Variance i=0 Y times q = -112

Variance i=0 centered integer numerator := sub(-112, 8)

Variance i=0 centered integer numerator = -120

Variance i=0 centered numerator squared := mul(-120, -120)

Variance i=0 centered numerator squared = 14400

Variance i=0 weight times squared numerator := mul(1, 14400)

Variance i=0 weight times squared numerator = 14400

Variance i=1 Y times q := mul(-8, 14)

Variance i=1 Y times q = -112

Variance i=1 centered integer numerator := sub(-112, 8)

Variance i=1 centered integer numerator = -120

Variance i=1 centered numerator squared := mul(-120, -120)

Variance i=1 centered numerator squared = 14400

Variance i=1 weight times squared numerator := mul(10, 14400)

Variance i=1 weight times squared numerator = 144000

Variance i=2 Y times q := mul(32, 14)

Variance i=2 Y times q = 448

Variance i=2 centered integer numerator := sub(448, 8)

Variance i=2 centered integer numerator = 440

Variance i=2 centered numerator squared := mul(440, 440)

Variance i=2 centered numerator squared = 193600

Variance i=2 weight times squared numerator := mul(3, 193600)

Variance i=2 weight times squared numerator = 580800

Variance i=3 Y times q := mul(4028, 14)

Variance i=3 Y times q = 56392

Variance i=3 centered integer numerator := sub(56392, 8)

Variance i=3 centered integer numerator = 56384

Variance i=3 centered numerator squared := mul(56384, 56384)

Variance i=3 centered numerator squared = 3179155456

Variance i=3 weight times squared numerator := mul(0, 3179155456)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(14400, 144000, 580800, 0)

Variance sum ALL weighted squared numerators = 739200

Variance denominator q squared := mul(14, 14)

Variance denominator q squared = 196

Variance denominator q cubed binary second factor := mul(196, 14)

Variance denominator q cubed binary second factor = 2744

Variance probability population centered sum over q cubed := div(739200, 2744)

Variance probability population centered sum over q cubed = 13200/49

source roles = [[-8, -8, 32, 4028], [1, 10, 3, 0], 14, 4/7, 739200, 2744]

derived = [13200/49, [-8, 32], No]

source roles = [scale_variance, 4, null]

derived = [13200/49]

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

13200/49

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.