Question 35888: Zero Variance on Positive-Probability Support

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, 2), (1, 2), (2, 16), (3, 1011)}. In the displayed order of S, the nonnegative probability weights are [1/(18), 14/(18), 3/(18), 0/(18)], summing exactly to one. Is Var(X)=0 under this probability law? Constancy is required only at positive-weight outcomes.

• 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, 2), (1, 2), (2, 16), (3, 1011)}. In the displayed order of S, the nonnegative probability weights are [1/(18), 14/(18), 3/(18), 0/(18)], summing exactly to one. Is Var(X)=0 under this probability law? Constancy is required only at positive-weight outcomes.

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-zero-support, [F, S, denominator, operation, values, weights]]

derived = [variance, zero_variance, null]

Variance ORIGINAL nonnegative RAW weights sum := sum(1, 14, 3, 0)

Variance ORIGINAL nonnegative RAW weights sum = 18

source roles = [[0, 1, 2, 3], powerset(S), [2, 2, 16, 1011], [1, 14, 3, 0], 18, null, null]

derived = [Yes]

Variance weight times transformed value i=0 := mul(1, 2)

Variance weight times transformed value i=0 = 2

source roles = [0, 0, 2, 1, null, null]

derived = [2, 1/18]

Variance weight times transformed value i=1 := mul(14, 2)

Variance weight times transformed value i=1 = 28

source roles = [1, 1, 2, 14, null, null]

derived = [2, 7/9]

Variance weight times transformed value i=2 := mul(3, 16)

Variance weight times transformed value i=2 = 48

source roles = [2, 2, 16, 3, null, null]

derived = [16, 1/6]

Variance weight times transformed value i=3 := mul(0, 1011)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1011, 0, null, null]

derived = [1011, 0]

Variance mean numerator sum wY := sum(2, 28, 48, 0)

Variance mean numerator sum wY = 78

Variance exact expectation M over q := div(78, 18)

Variance exact expectation M over q = 13/3

Variance i=0 Y times q := mul(2, 18)

Variance i=0 Y times q = 36

Variance i=0 centered integer numerator := sub(36, 78)

Variance i=0 centered integer numerator = -42

Variance i=0 centered numerator squared := mul(-42, -42)

Variance i=0 centered numerator squared = 1764

Variance i=0 weight times squared numerator := mul(1, 1764)

Variance i=0 weight times squared numerator = 1764

Variance i=1 Y times q := mul(2, 18)

Variance i=1 Y times q = 36

Variance i=1 centered integer numerator := sub(36, 78)

Variance i=1 centered integer numerator = -42

Variance i=1 centered numerator squared := mul(-42, -42)

Variance i=1 centered numerator squared = 1764

Variance i=1 weight times squared numerator := mul(14, 1764)

Variance i=1 weight times squared numerator = 24696

Variance i=2 Y times q := mul(16, 18)

Variance i=2 Y times q = 288

Variance i=2 centered integer numerator := sub(288, 78)

Variance i=2 centered integer numerator = 210

Variance i=2 centered numerator squared := mul(210, 210)

Variance i=2 centered numerator squared = 44100

Variance i=2 weight times squared numerator := mul(3, 44100)

Variance i=2 weight times squared numerator = 132300

Variance i=3 Y times q := mul(1011, 18)

Variance i=3 Y times q = 18198

Variance i=3 centered integer numerator := sub(18198, 78)

Variance i=3 centered integer numerator = 18120

Variance i=3 centered numerator squared := mul(18120, 18120)

Variance i=3 centered numerator squared = 328334400

Variance i=3 weight times squared numerator := mul(0, 328334400)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(1764, 24696, 132300, 0)

Variance sum ALL weighted squared numerators = 158760

Variance denominator q squared := mul(18, 18)

Variance denominator q squared = 324

Variance denominator q cubed binary second factor := mul(324, 18)

Variance denominator q cubed binary second factor = 5832

Variance probability population centered sum over q cubed := div(158760, 5832)

Variance probability population centered sum over q cubed = 245/9

source roles = [[2, 2, 16, 1011], [1, 14, 3, 0], 18, 13/3, 158760, 5832]

derived = [245/9, [2, 16], No]

source roles = [zero_variance, null, null]

derived = [245/9]

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

No

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.