Question 35858: 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, -13), (1, -13), (2, -4), (3, 1006)}. In the displayed order of S, the nonnegative probability weights are [1/(16), 9/(16), 6/(16), 0/(16)], 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, -13), (1, -13), (2, -4), (3, 1006)}. In the displayed order of S, the nonnegative probability weights are [1/(16), 9/(16), 6/(16), 0/(16)], 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, 9, 6, 0)

Variance ORIGINAL nonnegative RAW weights sum = 16

source roles = [[0, 1, 2, 3], powerset(S), [-13, -13, -4, 1006], [1, 9, 6, 0], 16, null, null]

derived = [Yes]

Variance weight times transformed value i=0 := mul(1, -13)

Variance weight times transformed value i=0 = -13

source roles = [0, 0, -13, 1, null, null]

derived = [-13, 1/16]

Variance weight times transformed value i=1 := mul(9, -13)

Variance weight times transformed value i=1 = -117

source roles = [1, 1, -13, 9, null, null]

derived = [-13, 9/16]

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, 3/8]

Variance weight times transformed value i=3 := mul(0, 1006)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1006, 0, null, null]

derived = [1006, 0]

Variance mean numerator sum wY := sum(-13, -117, -24, 0)

Variance mean numerator sum wY = -154

Variance exact expectation M over q := div(-154, 16)

Variance exact expectation M over q = -77/8

Variance i=0 Y times q := mul(-13, 16)

Variance i=0 Y times q = -208

Variance i=0 centered integer numerator := sub(-208, -154)

Variance i=0 centered integer numerator = -54

Variance i=0 centered numerator squared := mul(-54, -54)

Variance i=0 centered numerator squared = 2916

Variance i=0 weight times squared numerator := mul(1, 2916)

Variance i=0 weight times squared numerator = 2916

Variance i=1 Y times q := mul(-13, 16)

Variance i=1 Y times q = -208

Variance i=1 centered integer numerator := sub(-208, -154)

Variance i=1 centered integer numerator = -54

Variance i=1 centered numerator squared := mul(-54, -54)

Variance i=1 centered numerator squared = 2916

Variance i=1 weight times squared numerator := mul(9, 2916)

Variance i=1 weight times squared numerator = 26244

Variance i=2 Y times q := mul(-4, 16)

Variance i=2 Y times q = -64

Variance i=2 centered integer numerator := sub(-64, -154)

Variance i=2 centered integer numerator = 90

Variance i=2 centered numerator squared := mul(90, 90)

Variance i=2 centered numerator squared = 8100

Variance i=2 weight times squared numerator := mul(6, 8100)

Variance i=2 weight times squared numerator = 48600

Variance i=3 Y times q := mul(1006, 16)

Variance i=3 Y times q = 16096

Variance i=3 centered integer numerator := sub(16096, -154)

Variance i=3 centered integer numerator = 16250

Variance i=3 centered numerator squared := mul(16250, 16250)

Variance i=3 centered numerator squared = 264062500

Variance i=3 weight times squared numerator := mul(0, 264062500)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(2916, 26244, 48600, 0)

Variance sum ALL weighted squared numerators = 77760

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(77760, 4096)

Variance probability population centered sum over q cubed = 1215/64

source roles = [[-13, -13, -4, 1006], [1, 9, 6, 0], 16, -77/8, 77760, 4096]

derived = [1215/64, [-13, -4], No]

source roles = [variance, null, null]

derived = [1215/64]

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

1215/64

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.