Question 35844: 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, 8), (1, 8), (2, 28), (3, 1017)}. In the displayed order of S, the nonnegative probability weights are [1/(26), 20/(26), 5/(26), 0/(26)], summing exactly to one. Put Z=X+(6). 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, 8), (1, 8), (2, 28), (3, 1017)}. In the displayed order of S, the nonnegative probability weights are [1/(26), 20/(26), 5/(26), 0/(26)], summing exactly to one. Put Z=X+(6). 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, 20, 5, 0)

Variance ORIGINAL nonnegative RAW weights sum = 26

source roles = [[0, 1, 2, 3], powerset(S), [8, 8, 28, 1017], [1, 20, 5, 0], 26, null, 6]

derived = [Yes]

Variance translated ORIGINAL value i=0 := sum(8, 6)

Variance translated ORIGINAL value i=0 = 14

Variance weight times transformed value i=0 := mul(1, 14)

Variance weight times transformed value i=0 = 14

source roles = [0, 0, 8, 1, null, 6]

derived = [14, 1/26]

Variance translated ORIGINAL value i=1 := sum(8, 6)

Variance translated ORIGINAL value i=1 = 14

Variance weight times transformed value i=1 := mul(20, 14)

Variance weight times transformed value i=1 = 280

source roles = [1, 1, 8, 20, null, 6]

derived = [14, 10/13]

Variance translated ORIGINAL value i=2 := sum(28, 6)

Variance translated ORIGINAL value i=2 = 34

Variance weight times transformed value i=2 := mul(5, 34)

Variance weight times transformed value i=2 = 170

source roles = [2, 2, 28, 5, null, 6]

derived = [34, 5/26]

Variance translated ORIGINAL value i=3 := sum(1017, 6)

Variance translated ORIGINAL value i=3 = 1023

Variance weight times transformed value i=3 := mul(0, 1023)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1017, 0, null, 6]

derived = [1023, 0]

Variance mean numerator sum wY := sum(14, 280, 170, 0)

Variance mean numerator sum wY = 464

Variance exact expectation M over q := div(464, 26)

Variance exact expectation M over q = 232/13

Variance i=0 Y times q := mul(14, 26)

Variance i=0 Y times q = 364

Variance i=0 centered integer numerator := sub(364, 464)

Variance i=0 centered integer numerator = -100

Variance i=0 centered numerator squared := mul(-100, -100)

Variance i=0 centered numerator squared = 10000

Variance i=0 weight times squared numerator := mul(1, 10000)

Variance i=0 weight times squared numerator = 10000

Variance i=1 Y times q := mul(14, 26)

Variance i=1 Y times q = 364

Variance i=1 centered integer numerator := sub(364, 464)

Variance i=1 centered integer numerator = -100

Variance i=1 centered numerator squared := mul(-100, -100)

Variance i=1 centered numerator squared = 10000

Variance i=1 weight times squared numerator := mul(20, 10000)

Variance i=1 weight times squared numerator = 200000

Variance i=2 Y times q := mul(34, 26)

Variance i=2 Y times q = 884

Variance i=2 centered integer numerator := sub(884, 464)

Variance i=2 centered integer numerator = 420

Variance i=2 centered numerator squared := mul(420, 420)

Variance i=2 centered numerator squared = 176400

Variance i=2 weight times squared numerator := mul(5, 176400)

Variance i=2 weight times squared numerator = 882000

Variance i=3 Y times q := mul(1023, 26)

Variance i=3 Y times q = 26598

Variance i=3 centered integer numerator := sub(26598, 464)

Variance i=3 centered integer numerator = 26134

Variance i=3 centered numerator squared := mul(26134, 26134)

Variance i=3 centered numerator squared = 682985956

Variance i=3 weight times squared numerator := mul(0, 682985956)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(10000, 200000, 882000, 0)

Variance sum ALL weighted squared numerators = 1092000

Variance denominator q squared := mul(26, 26)

Variance denominator q squared = 676

Variance denominator q cubed binary second factor := mul(676, 26)

Variance denominator q cubed binary second factor = 17576

Variance probability population centered sum over q cubed := div(1092000, 17576)

Variance probability population centered sum over q cubed = 10500/169

source roles = [[14, 14, 34, 1023], [1, 20, 5, 0], 26, 232/13, 1092000, 17576]

derived = [10500/169, [14, 34], No]

source roles = [shift_variance, null, 6]

derived = [10500/169]

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

10500/169

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.