Question 35810: 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, -15), (1, -15), (2, -4), (3, 1008)}. In the displayed order of S, the nonnegative probability weights are [1/(16), 11/(16), 4/(16), 0/(16)], 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, -15), (1, -15), (2, -4), (3, 1008)}. In the displayed order of S, the nonnegative probability weights are [1/(16), 11/(16), 4/(16), 0/(16)], 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, 11, 4, 0)

Variance ORIGINAL nonnegative RAW weights sum = 16

source roles = [[0, 1, 2, 3], powerset(S), [-15, -15, -4, 1008], [1, 11, 4, 0], 16, -5, null]

derived = [Yes]

Variance transformed ORIGINAL value i=0 := mul(-5, -15)

Variance transformed ORIGINAL value i=0 = 75

Variance weight times transformed value i=0 := mul(1, 75)

Variance weight times transformed value i=0 = 75

source roles = [0, 0, -15, 1, -5, null]

derived = [75, 1/16]

Variance transformed ORIGINAL value i=1 := mul(-5, -15)

Variance transformed ORIGINAL value i=1 = 75

Variance weight times transformed value i=1 := mul(11, 75)

Variance weight times transformed value i=1 = 825

source roles = [1, 1, -15, 11, -5, null]

derived = [75, 11/16]

Variance transformed ORIGINAL value i=2 := mul(-5, -4)

Variance transformed ORIGINAL value i=2 = 20

Variance weight times transformed value i=2 := mul(4, 20)

Variance weight times transformed value i=2 = 80

source roles = [2, 2, -4, 4, -5, null]

derived = [20, 1/4]

Variance transformed ORIGINAL value i=3 := mul(-5, 1008)

Variance transformed ORIGINAL value i=3 = -5040

Variance weight times transformed value i=3 := mul(0, -5040)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1008, 0, -5, null]

derived = [-5040, 0]

Variance mean numerator sum wY := sum(75, 825, 80, 0)

Variance mean numerator sum wY = 980

Variance exact expectation M over q := div(980, 16)

Variance exact expectation M over q = 245/4

Variance i=0 Y times q := mul(75, 16)

Variance i=0 Y times q = 1200

Variance i=0 centered integer numerator := sub(1200, 980)

Variance i=0 centered integer numerator = 220

Variance i=0 centered numerator squared := mul(220, 220)

Variance i=0 centered numerator squared = 48400

Variance i=0 weight times squared numerator := mul(1, 48400)

Variance i=0 weight times squared numerator = 48400

Variance i=1 Y times q := mul(75, 16)

Variance i=1 Y times q = 1200

Variance i=1 centered integer numerator := sub(1200, 980)

Variance i=1 centered integer numerator = 220

Variance i=1 centered numerator squared := mul(220, 220)

Variance i=1 centered numerator squared = 48400

Variance i=1 weight times squared numerator := mul(11, 48400)

Variance i=1 weight times squared numerator = 532400

Variance i=2 Y times q := mul(20, 16)

Variance i=2 Y times q = 320

Variance i=2 centered integer numerator := sub(320, 980)

Variance i=2 centered integer numerator = -660

Variance i=2 centered numerator squared := mul(-660, -660)

Variance i=2 centered numerator squared = 435600

Variance i=2 weight times squared numerator := mul(4, 435600)

Variance i=2 weight times squared numerator = 1742400

Variance i=3 Y times q := mul(-5040, 16)

Variance i=3 Y times q = -80640

Variance i=3 centered integer numerator := sub(-80640, 980)

Variance i=3 centered integer numerator = -81620

Variance i=3 centered numerator squared := mul(-81620, -81620)

Variance i=3 centered numerator squared = 6661824400

Variance i=3 weight times squared numerator := mul(0, 6661824400)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(48400, 532400, 1742400, 0)

Variance sum ALL weighted squared numerators = 2323200

Variance denominator q squared := mul(16, 16)

Variance denominator q squared = 256

Variance denominator q cubed binary second factor := mul(256, 16)

Variance denominator q cubed binary second factor = 4096

Variance probability population centered sum over q cubed := div(2323200, 4096)

Variance probability population centered sum over q cubed = 9075/16

source roles = [[75, 75, 20, -5040], [1, 11, 4, 0], 16, 245/4, 2323200, 4096]

derived = [9075/16, [20, 75], No]

source roles = [scale_variance, -5, null]

derived = [9075/16]

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

9075/16

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.