Question 35824: 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, -29), (1, -29), (2, -4), (3, 1022)}. In the displayed order of S, the nonnegative probability weights are [1/(32), 25/(32), 6/(32), 0/(32)], 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, -29), (1, -29), (2, -4), (3, 1022)}. In the displayed order of S, the nonnegative probability weights are [1/(32), 25/(32), 6/(32), 0/(32)], 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, 25, 6, 0)

Variance ORIGINAL nonnegative RAW weights sum = 32

source roles = [[0, 1, 2, 3], powerset(S), [-29, -29, -4, 1022], [1, 25, 6, 0], 32, -4, null]

derived = [Yes]

Variance transformed ORIGINAL value i=0 := mul(-4, -29)

Variance transformed ORIGINAL value i=0 = 116

Variance weight times transformed value i=0 := mul(1, 116)

Variance weight times transformed value i=0 = 116

source roles = [0, 0, -29, 1, -4, null]

derived = [116, 1/32]

Variance transformed ORIGINAL value i=1 := mul(-4, -29)

Variance transformed ORIGINAL value i=1 = 116

Variance weight times transformed value i=1 := mul(25, 116)

Variance weight times transformed value i=1 = 2900

source roles = [1, 1, -29, 25, -4, null]

derived = [116, 25/32]

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

Variance transformed ORIGINAL value i=2 = 16

Variance weight times transformed value i=2 := mul(6, 16)

Variance weight times transformed value i=2 = 96

source roles = [2, 2, -4, 6, -4, null]

derived = [16, 3/16]

Variance transformed ORIGINAL value i=3 := mul(-4, 1022)

Variance transformed ORIGINAL value i=3 = -4088

Variance weight times transformed value i=3 := mul(0, -4088)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1022, 0, -4, null]

derived = [-4088, 0]

Variance mean numerator sum wY := sum(116, 2900, 96, 0)

Variance mean numerator sum wY = 3112

Variance exact expectation M over q := div(3112, 32)

Variance exact expectation M over q = 389/4

Variance i=0 Y times q := mul(116, 32)

Variance i=0 Y times q = 3712

Variance i=0 centered integer numerator := sub(3712, 3112)

Variance i=0 centered integer numerator = 600

Variance i=0 centered numerator squared := mul(600, 600)

Variance i=0 centered numerator squared = 360000

Variance i=0 weight times squared numerator := mul(1, 360000)

Variance i=0 weight times squared numerator = 360000

Variance i=1 Y times q := mul(116, 32)

Variance i=1 Y times q = 3712

Variance i=1 centered integer numerator := sub(3712, 3112)

Variance i=1 centered integer numerator = 600

Variance i=1 centered numerator squared := mul(600, 600)

Variance i=1 centered numerator squared = 360000

Variance i=1 weight times squared numerator := mul(25, 360000)

Variance i=1 weight times squared numerator = 9000000

Variance i=2 Y times q := mul(16, 32)

Variance i=2 Y times q = 512

Variance i=2 centered integer numerator := sub(512, 3112)

Variance i=2 centered integer numerator = -2600

Variance i=2 centered numerator squared := mul(-2600, -2600)

Variance i=2 centered numerator squared = 6760000

Variance i=2 weight times squared numerator := mul(6, 6760000)

Variance i=2 weight times squared numerator = 40560000

Variance i=3 Y times q := mul(-4088, 32)

Variance i=3 Y times q = -130816

Variance i=3 centered integer numerator := sub(-130816, 3112)

Variance i=3 centered integer numerator = -133928

Variance i=3 centered numerator squared := mul(-133928, -133928)

Variance i=3 centered numerator squared = 17936709184

Variance i=3 weight times squared numerator := mul(0, 17936709184)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(360000, 9000000, 40560000, 0)

Variance sum ALL weighted squared numerators = 49920000

Variance denominator q squared := mul(32, 32)

Variance denominator q squared = 1024

Variance denominator q cubed binary second factor := mul(1024, 32)

Variance denominator q cubed binary second factor = 32768

Variance probability population centered sum over q cubed := div(49920000, 32768)

Variance probability population centered sum over q cubed = 24375/16

source roles = [[116, 116, 16, -4088], [1, 25, 6, 0], 32, 389/4, 49920000, 32768]

derived = [24375/16, [16, 116], No]

source roles = [scale_variance, -4, null]

derived = [24375/16]

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

24375/16

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.