Question 35866: Exact Finite Probability Variance

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, -21), (1, -21), (2, -4), (3, 1014)}. In the displayed order of S, the nonnegative probability weights are [1/(24), 17/(24), 6/(24), 0/(24)], summing exactly to one. Put Z=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, -21), (1, -21), (2, -4), (3, 1014)}. In the displayed order of S, the nonnegative probability weights are [1/(24), 17/(24), 6/(24), 0/(24)], summing exactly to one. Put Z=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-weighted, [F, S, denominator, operation, values, weights]]

derived = [variance, variance, null]

Variance ORIGINAL nonnegative RAW weights sum := sum(1, 17, 6, 0)

Variance ORIGINAL nonnegative RAW weights sum = 24

source roles = [[0, 1, 2, 3], powerset(S), [-21, -21, -4, 1014], [1, 17, 6, 0], 24, null, null]

derived = [Yes]

Variance weight times transformed value i=0 := mul(1, -21)

Variance weight times transformed value i=0 = -21

source roles = [0, 0, -21, 1, null, null]

derived = [-21, 1/24]

Variance weight times transformed value i=1 := mul(17, -21)

Variance weight times transformed value i=1 = -357

source roles = [1, 1, -21, 17, null, null]

derived = [-21, 17/24]

Variance weight times transformed value i=2 := mul(6, -4)

Variance weight times transformed value i=2 = -24

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

derived = [-4, 1/4]

Variance weight times transformed value i=3 := mul(0, 1014)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1014, 0, null, null]

derived = [1014, 0]

Variance mean numerator sum wY := sum(-21, -357, -24, 0)

Variance mean numerator sum wY = -402

Variance exact expectation M over q := div(-402, 24)

Variance exact expectation M over q = -67/4

Variance i=0 Y times q := mul(-21, 24)

Variance i=0 Y times q = -504

Variance i=0 centered integer numerator := sub(-504, -402)

Variance i=0 centered integer numerator = -102

Variance i=0 centered numerator squared := mul(-102, -102)

Variance i=0 centered numerator squared = 10404

Variance i=0 weight times squared numerator := mul(1, 10404)

Variance i=0 weight times squared numerator = 10404

Variance i=1 Y times q := mul(-21, 24)

Variance i=1 Y times q = -504

Variance i=1 centered integer numerator := sub(-504, -402)

Variance i=1 centered integer numerator = -102

Variance i=1 centered numerator squared := mul(-102, -102)

Variance i=1 centered numerator squared = 10404

Variance i=1 weight times squared numerator := mul(17, 10404)

Variance i=1 weight times squared numerator = 176868

Variance i=2 Y times q := mul(-4, 24)

Variance i=2 Y times q = -96

Variance i=2 centered integer numerator := sub(-96, -402)

Variance i=2 centered integer numerator = 306

Variance i=2 centered numerator squared := mul(306, 306)

Variance i=2 centered numerator squared = 93636

Variance i=2 weight times squared numerator := mul(6, 93636)

Variance i=2 weight times squared numerator = 561816

Variance i=3 Y times q := mul(1014, 24)

Variance i=3 Y times q = 24336

Variance i=3 centered integer numerator := sub(24336, -402)

Variance i=3 centered integer numerator = 24738

Variance i=3 centered numerator squared := mul(24738, 24738)

Variance i=3 centered numerator squared = 611968644

Variance i=3 weight times squared numerator := mul(0, 611968644)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(10404, 176868, 561816, 0)

Variance sum ALL weighted squared numerators = 749088

Variance denominator q squared := mul(24, 24)

Variance denominator q squared = 576

Variance denominator q cubed binary second factor := mul(576, 24)

Variance denominator q cubed binary second factor = 13824

Variance probability population centered sum over q cubed := div(749088, 13824)

Variance probability population centered sum over q cubed = 867/16

source roles = [[-21, -21, -4, 1014], [1, 17, 6, 0], 24, -67/4, 749088, 13824]

derived = [867/16, [-21, -4], No]

source roles = [variance, null, null]

derived = [867/16]

Reverse checks

• S035 · 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.
• S034 · 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.
• S033 · premise_and_result_audit — Recompute Variance probability population centered sum over q cubed with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Variance denominator q cubed binary second factor with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Variance denominator q squared with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute Variance sum ALL weighted squared numerators with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Variance i=3 weight times squared numerator with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Variance i=3 centered numerator squared with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Variance i=3 centered integer numerator with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Variance i=3 Y times q with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Variance i=2 weight times squared numerator with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Variance i=2 centered numerator squared with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Variance i=2 centered integer numerator with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Variance i=2 Y times q with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Variance i=1 weight times squared numerator with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Variance i=1 centered numerator squared with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Variance i=1 centered integer numerator with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Variance i=1 Y times q with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Variance i=0 weight times squared numerator with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Variance i=0 centered numerator squared with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Variance i=0 centered integer numerator with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Variance i=0 Y times q with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Variance exact expectation M over q with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Variance mean numerator sum wY with exact rational arithmetic, not rounded decimals.
• S011 · 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.
• S010 · premise_and_result_audit — Recompute Variance weight times transformed value i=3 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=2 with exact rational arithmetic, not rounded decimals.
• S007 · 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.
• S006 · premise_and_result_audit — Recompute Variance weight times transformed value i=1 with exact rational arithmetic, not rounded decimals.
• S005 · 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.
• S004 · premise_and_result_audit — Recompute Variance weight times transformed 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

867/16

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.