Question 35842: Probability Variance after Translation

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, 6), (1, 6), (2, 24), (3, 1015)}. In the displayed order of S, the nonnegative probability weights are [1/(22), 18/(22), 3/(22), 0/(22)], summing exactly to one. Put Z=X+(4). 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, 6), (1, 6), (2, 24), (3, 1015)}. In the displayed order of S, the nonnegative probability weights are [1/(22), 18/(22), 3/(22), 0/(22)], summing exactly to one. Put Z=X+(4). 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-shift, [F, S, c, denominator, operation, values, weights]]

derived = [variance, shift_variance, null]

Variance ORIGINAL nonnegative RAW weights sum := sum(1, 18, 3, 0)

Variance ORIGINAL nonnegative RAW weights sum = 22

source roles = [[0, 1, 2, 3], powerset(S), [6, 6, 24, 1015], [1, 18, 3, 0], 22, null, 4]

derived = [Yes]

Variance translated ORIGINAL value i=0 := sum(6, 4)

Variance translated ORIGINAL value i=0 = 10

Variance weight times transformed value i=0 := mul(1, 10)

Variance weight times transformed value i=0 = 10

source roles = [0, 0, 6, 1, null, 4]

derived = [10, 1/22]

Variance translated ORIGINAL value i=1 := sum(6, 4)

Variance translated ORIGINAL value i=1 = 10

Variance weight times transformed value i=1 := mul(18, 10)

Variance weight times transformed value i=1 = 180

source roles = [1, 1, 6, 18, null, 4]

derived = [10, 9/11]

Variance translated ORIGINAL value i=2 := sum(24, 4)

Variance translated ORIGINAL value i=2 = 28

Variance weight times transformed value i=2 := mul(3, 28)

Variance weight times transformed value i=2 = 84

source roles = [2, 2, 24, 3, null, 4]

derived = [28, 3/22]

Variance translated ORIGINAL value i=3 := sum(1015, 4)

Variance translated ORIGINAL value i=3 = 1019

Variance weight times transformed value i=3 := mul(0, 1019)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1015, 0, null, 4]

derived = [1019, 0]

Variance mean numerator sum wY := sum(10, 180, 84, 0)

Variance mean numerator sum wY = 274

Variance exact expectation M over q := div(274, 22)

Variance exact expectation M over q = 137/11

Variance i=0 Y times q := mul(10, 22)

Variance i=0 Y times q = 220

Variance i=0 centered integer numerator := sub(220, 274)

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(10, 22)

Variance i=1 Y times q = 220

Variance i=1 centered integer numerator := sub(220, 274)

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(18, 2916)

Variance i=1 weight times squared numerator = 52488

Variance i=2 Y times q := mul(28, 22)

Variance i=2 Y times q = 616

Variance i=2 centered integer numerator := sub(616, 274)

Variance i=2 centered integer numerator = 342

Variance i=2 centered numerator squared := mul(342, 342)

Variance i=2 centered numerator squared = 116964

Variance i=2 weight times squared numerator := mul(3, 116964)

Variance i=2 weight times squared numerator = 350892

Variance i=3 Y times q := mul(1019, 22)

Variance i=3 Y times q = 22418

Variance i=3 centered integer numerator := sub(22418, 274)

Variance i=3 centered integer numerator = 22144

Variance i=3 centered numerator squared := mul(22144, 22144)

Variance i=3 centered numerator squared = 490356736

Variance i=3 weight times squared numerator := mul(0, 490356736)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(2916, 52488, 350892, 0)

Variance sum ALL weighted squared numerators = 406296

Variance denominator q squared := mul(22, 22)

Variance denominator q squared = 484

Variance denominator q cubed binary second factor := mul(484, 22)

Variance denominator q cubed binary second factor = 10648

Variance probability population centered sum over q cubed := div(406296, 10648)

Variance probability population centered sum over q cubed = 4617/121

source roles = [[10, 10, 28, 1019], [1, 18, 3, 0], 22, 137/11, 406296, 10648]

derived = [4617/121, [10, 28], No]

source roles = [shift_variance, null, 4]

derived = [4617/121]

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 translated 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 translated 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 translated 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 translated 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

4617/121

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.