Question 35850: 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, 14), (1, 14), (2, 40), (3, 1023)}. In the displayed order of S, the nonnegative probability weights are [1/(30), 26/(30), 3/(30), 0/(30)], summing exactly to one. Put Z=X+(12). 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, 14), (1, 14), (2, 40), (3, 1023)}. In the displayed order of S, the nonnegative probability weights are [1/(30), 26/(30), 3/(30), 0/(30)], summing exactly to one. Put Z=X+(12). 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, 26, 3, 0)

Variance ORIGINAL nonnegative RAW weights sum = 30

source roles = [[0, 1, 2, 3], powerset(S), [14, 14, 40, 1023], [1, 26, 3, 0], 30, null, 12]

derived = [Yes]

Variance translated ORIGINAL value i=0 := sum(14, 12)

Variance translated ORIGINAL value i=0 = 26

Variance weight times transformed value i=0 := mul(1, 26)

Variance weight times transformed value i=0 = 26

source roles = [0, 0, 14, 1, null, 12]

derived = [26, 1/30]

Variance translated ORIGINAL value i=1 := sum(14, 12)

Variance translated ORIGINAL value i=1 = 26

Variance weight times transformed value i=1 := mul(26, 26)

Variance weight times transformed value i=1 = 676

source roles = [1, 1, 14, 26, null, 12]

derived = [26, 13/15]

Variance translated ORIGINAL value i=2 := sum(40, 12)

Variance translated ORIGINAL value i=2 = 52

Variance weight times transformed value i=2 := mul(3, 52)

Variance weight times transformed value i=2 = 156

source roles = [2, 2, 40, 3, null, 12]

derived = [52, 1/10]

Variance translated ORIGINAL value i=3 := sum(1023, 12)

Variance translated ORIGINAL value i=3 = 1035

Variance weight times transformed value i=3 := mul(0, 1035)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1023, 0, null, 12]

derived = [1035, 0]

Variance mean numerator sum wY := sum(26, 676, 156, 0)

Variance mean numerator sum wY = 858

Variance exact expectation M over q := div(858, 30)

Variance exact expectation M over q = 143/5

Variance i=0 Y times q := mul(26, 30)

Variance i=0 Y times q = 780

Variance i=0 centered integer numerator := sub(780, 858)

Variance i=0 centered integer numerator = -78

Variance i=0 centered numerator squared := mul(-78, -78)

Variance i=0 centered numerator squared = 6084

Variance i=0 weight times squared numerator := mul(1, 6084)

Variance i=0 weight times squared numerator = 6084

Variance i=1 Y times q := mul(26, 30)

Variance i=1 Y times q = 780

Variance i=1 centered integer numerator := sub(780, 858)

Variance i=1 centered integer numerator = -78

Variance i=1 centered numerator squared := mul(-78, -78)

Variance i=1 centered numerator squared = 6084

Variance i=1 weight times squared numerator := mul(26, 6084)

Variance i=1 weight times squared numerator = 158184

Variance i=2 Y times q := mul(52, 30)

Variance i=2 Y times q = 1560

Variance i=2 centered integer numerator := sub(1560, 858)

Variance i=2 centered integer numerator = 702

Variance i=2 centered numerator squared := mul(702, 702)

Variance i=2 centered numerator squared = 492804

Variance i=2 weight times squared numerator := mul(3, 492804)

Variance i=2 weight times squared numerator = 1478412

Variance i=3 Y times q := mul(1035, 30)

Variance i=3 Y times q = 31050

Variance i=3 centered integer numerator := sub(31050, 858)

Variance i=3 centered integer numerator = 30192

Variance i=3 centered numerator squared := mul(30192, 30192)

Variance i=3 centered numerator squared = 911556864

Variance i=3 weight times squared numerator := mul(0, 911556864)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(6084, 158184, 1478412, 0)

Variance sum ALL weighted squared numerators = 1642680

Variance denominator q squared := mul(30, 30)

Variance denominator q squared = 900

Variance denominator q cubed binary second factor := mul(900, 30)

Variance denominator q cubed binary second factor = 27000

Variance probability population centered sum over q cubed := div(1642680, 27000)

Variance probability population centered sum over q cubed = 1521/25

source roles = [[26, 26, 52, 1035], [1, 26, 3, 0], 30, 143/5, 1642680, 27000]

derived = [1521/25, [26, 52], No]

source roles = [shift_variance, null, 12]

derived = [1521/25]

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

1521/25

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.