Question 35806: 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, -11), (1, -11), (2, -4), (3, 1004)}. In the displayed order of S, the nonnegative probability weights are [1/(12), 7/(12), 4/(12), 0/(12)], summing exactly to one. Put Z=(-1)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, -11), (1, -11), (2, -4), (3, 1004)}. In the displayed order of S, the nonnegative probability weights are [1/(12), 7/(12), 4/(12), 0/(12)], summing exactly to one. Put Z=(-1)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, 7, 4, 0)

Variance ORIGINAL nonnegative RAW weights sum = 12

source roles = [[0, 1, 2, 3], powerset(S), [-11, -11, -4, 1004], [1, 7, 4, 0], 12, -1, null]

derived = [Yes]

Variance transformed ORIGINAL value i=0 := mul(-1, -11)

Variance transformed ORIGINAL value i=0 = 11

Variance weight times transformed value i=0 := mul(1, 11)

Variance weight times transformed value i=0 = 11

source roles = [0, 0, -11, 1, -1, null]

derived = [11, 1/12]

Variance transformed ORIGINAL value i=1 := mul(-1, -11)

Variance transformed ORIGINAL value i=1 = 11

Variance weight times transformed value i=1 := mul(7, 11)

Variance weight times transformed value i=1 = 77

source roles = [1, 1, -11, 7, -1, null]

derived = [11, 7/12]

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

Variance transformed ORIGINAL value i=2 = 4

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

Variance weight times transformed value i=2 = 16

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

derived = [4, 1/3]

Variance transformed ORIGINAL value i=3 := mul(-1, 1004)

Variance transformed ORIGINAL value i=3 = -1004

Variance weight times transformed value i=3 := mul(0, -1004)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1004, 0, -1, null]

derived = [-1004, 0]

Variance mean numerator sum wY := sum(11, 77, 16, 0)

Variance mean numerator sum wY = 104

Variance exact expectation M over q := div(104, 12)

Variance exact expectation M over q = 26/3

Variance i=0 Y times q := mul(11, 12)

Variance i=0 Y times q = 132

Variance i=0 centered integer numerator := sub(132, 104)

Variance i=0 centered integer numerator = 28

Variance i=0 centered numerator squared := mul(28, 28)

Variance i=0 centered numerator squared = 784

Variance i=0 weight times squared numerator := mul(1, 784)

Variance i=0 weight times squared numerator = 784

Variance i=1 Y times q := mul(11, 12)

Variance i=1 Y times q = 132

Variance i=1 centered integer numerator := sub(132, 104)

Variance i=1 centered integer numerator = 28

Variance i=1 centered numerator squared := mul(28, 28)

Variance i=1 centered numerator squared = 784

Variance i=1 weight times squared numerator := mul(7, 784)

Variance i=1 weight times squared numerator = 5488

Variance i=2 Y times q := mul(4, 12)

Variance i=2 Y times q = 48

Variance i=2 centered integer numerator := sub(48, 104)

Variance i=2 centered integer numerator = -56

Variance i=2 centered numerator squared := mul(-56, -56)

Variance i=2 centered numerator squared = 3136

Variance i=2 weight times squared numerator := mul(4, 3136)

Variance i=2 weight times squared numerator = 12544

Variance i=3 Y times q := mul(-1004, 12)

Variance i=3 Y times q = -12048

Variance i=3 centered integer numerator := sub(-12048, 104)

Variance i=3 centered integer numerator = -12152

Variance i=3 centered numerator squared := mul(-12152, -12152)

Variance i=3 centered numerator squared = 147671104

Variance i=3 weight times squared numerator := mul(0, 147671104)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(784, 5488, 12544, 0)

Variance sum ALL weighted squared numerators = 18816

Variance denominator q squared := mul(12, 12)

Variance denominator q squared = 144

Variance denominator q cubed binary second factor := mul(144, 12)

Variance denominator q cubed binary second factor = 1728

Variance probability population centered sum over q cubed := div(18816, 1728)

Variance probability population centered sum over q cubed = 98/9

source roles = [[11, 11, 4, -1004], [1, 7, 4, 0], 12, 26/3, 18816, 1728]

derived = [98/9, [4, 11], No]

source roles = [scale_variance, -1, null]

derived = [98/9]

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

98/9

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.