Question 35802: 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, -7), (1, -7), (2, -4), (3, 1000)}. In the displayed order of S, the nonnegative probability weights are [1/(8), 3/(8), 4/(8), 0/(8)], summing exactly to one. Put Z=(-2)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, -7), (1, -7), (2, -4), (3, 1000)}. In the displayed order of S, the nonnegative probability weights are [1/(8), 3/(8), 4/(8), 0/(8)], summing exactly to one. Put Z=(-2)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, 3, 4, 0)

Variance ORIGINAL nonnegative RAW weights sum = 8

source roles = [[0, 1, 2, 3], powerset(S), [-7, -7, -4, 1000], [1, 3, 4, 0], 8, -2, null]

derived = [Yes]

Variance transformed ORIGINAL value i=0 := mul(-2, -7)

Variance transformed ORIGINAL value i=0 = 14

Variance weight times transformed value i=0 := mul(1, 14)

Variance weight times transformed value i=0 = 14

source roles = [0, 0, -7, 1, -2, null]

derived = [14, 1/8]

Variance transformed ORIGINAL value i=1 := mul(-2, -7)

Variance transformed ORIGINAL value i=1 = 14

Variance weight times transformed value i=1 := mul(3, 14)

Variance weight times transformed value i=1 = 42

source roles = [1, 1, -7, 3, -2, null]

derived = [14, 3/8]

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

Variance transformed ORIGINAL value i=2 = 8

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

Variance weight times transformed value i=2 = 32

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

derived = [8, 1/2]

Variance transformed ORIGINAL value i=3 := mul(-2, 1000)

Variance transformed ORIGINAL value i=3 = -2000

Variance weight times transformed value i=3 := mul(0, -2000)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1000, 0, -2, null]

derived = [-2000, 0]

Variance mean numerator sum wY := sum(14, 42, 32, 0)

Variance mean numerator sum wY = 88

Variance exact expectation M over q := div(88, 8)

Variance exact expectation M over q = 11

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

Variance i=0 Y times q = 112

Variance i=0 centered integer numerator := sub(112, 88)

Variance i=0 centered integer numerator = 24

Variance i=0 centered numerator squared := mul(24, 24)

Variance i=0 centered numerator squared = 576

Variance i=0 weight times squared numerator := mul(1, 576)

Variance i=0 weight times squared numerator = 576

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

Variance i=1 Y times q = 112

Variance i=1 centered integer numerator := sub(112, 88)

Variance i=1 centered integer numerator = 24

Variance i=1 centered numerator squared := mul(24, 24)

Variance i=1 centered numerator squared = 576

Variance i=1 weight times squared numerator := mul(3, 576)

Variance i=1 weight times squared numerator = 1728

Variance i=2 Y times q := mul(8, 8)

Variance i=2 Y times q = 64

Variance i=2 centered integer numerator := sub(64, 88)

Variance i=2 centered integer numerator = -24

Variance i=2 centered numerator squared := mul(-24, -24)

Variance i=2 centered numerator squared = 576

Variance i=2 weight times squared numerator := mul(4, 576)

Variance i=2 weight times squared numerator = 2304

Variance i=3 Y times q := mul(-2000, 8)

Variance i=3 Y times q = -16000

Variance i=3 centered integer numerator := sub(-16000, 88)

Variance i=3 centered integer numerator = -16088

Variance i=3 centered numerator squared := mul(-16088, -16088)

Variance i=3 centered numerator squared = 258823744

Variance i=3 weight times squared numerator := mul(0, 258823744)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(576, 1728, 2304, 0)

Variance sum ALL weighted squared numerators = 4608

Variance denominator q squared := mul(8, 8)

Variance denominator q squared = 64

Variance denominator q cubed binary second factor := mul(64, 8)

Variance denominator q cubed binary second factor = 512

Variance probability population centered sum over q cubed := div(4608, 512)

Variance probability population centered sum over q cubed = 9

source roles = [[14, 14, 8, -2000], [1, 3, 4, 0], 8, 11, 4608, 512]

derived = [9, [8, 14], No]

source roles = [scale_variance, -2, null]

derived = [9]

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

9/1

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.