Question 35845: 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, -25), (1, -25), (2, -4), (3, 1018)}. In the displayed order of S, the nonnegative probability weights are [1/(28), 21/(28), 6/(28), 0/(28)], summing exactly to one. Put Z=X+(7). 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, -25), (1, -25), (2, -4), (3, 1018)}. In the displayed order of S, the nonnegative probability weights are [1/(28), 21/(28), 6/(28), 0/(28)], summing exactly to one. Put Z=X+(7). 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, 21, 6, 0)

Variance ORIGINAL nonnegative RAW weights sum = 28

source roles = [[0, 1, 2, 3], powerset(S), [-25, -25, -4, 1018], [1, 21, 6, 0], 28, null, 7]

derived = [Yes]

Variance translated ORIGINAL value i=0 := sum(-25, 7)

Variance translated ORIGINAL value i=0 = -18

Variance weight times transformed value i=0 := mul(1, -18)

Variance weight times transformed value i=0 = -18

source roles = [0, 0, -25, 1, null, 7]

derived = [-18, 1/28]

Variance translated ORIGINAL value i=1 := sum(-25, 7)

Variance translated ORIGINAL value i=1 = -18

Variance weight times transformed value i=1 := mul(21, -18)

Variance weight times transformed value i=1 = -378

source roles = [1, 1, -25, 21, null, 7]

derived = [-18, 3/4]

Variance translated ORIGINAL value i=2 := sum(-4, 7)

Variance translated ORIGINAL value i=2 = 3

Variance weight times transformed value i=2 := mul(6, 3)

Variance weight times transformed value i=2 = 18

source roles = [2, 2, -4, 6, null, 7]

derived = [3, 3/14]

Variance translated ORIGINAL value i=3 := sum(1018, 7)

Variance translated ORIGINAL value i=3 = 1025

Variance weight times transformed value i=3 := mul(0, 1025)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1018, 0, null, 7]

derived = [1025, 0]

Variance mean numerator sum wY := sum(-18, -378, 18, 0)

Variance mean numerator sum wY = -378

Variance exact expectation M over q := div(-378, 28)

Variance exact expectation M over q = -27/2

Variance i=0 Y times q := mul(-18, 28)

Variance i=0 Y times q = -504

Variance i=0 centered integer numerator := sub(-504, -378)

Variance i=0 centered integer numerator = -126

Variance i=0 centered numerator squared := mul(-126, -126)

Variance i=0 centered numerator squared = 15876

Variance i=0 weight times squared numerator := mul(1, 15876)

Variance i=0 weight times squared numerator = 15876

Variance i=1 Y times q := mul(-18, 28)

Variance i=1 Y times q = -504

Variance i=1 centered integer numerator := sub(-504, -378)

Variance i=1 centered integer numerator = -126

Variance i=1 centered numerator squared := mul(-126, -126)

Variance i=1 centered numerator squared = 15876

Variance i=1 weight times squared numerator := mul(21, 15876)

Variance i=1 weight times squared numerator = 333396

Variance i=2 Y times q := mul(3, 28)

Variance i=2 Y times q = 84

Variance i=2 centered integer numerator := sub(84, -378)

Variance i=2 centered integer numerator = 462

Variance i=2 centered numerator squared := mul(462, 462)

Variance i=2 centered numerator squared = 213444

Variance i=2 weight times squared numerator := mul(6, 213444)

Variance i=2 weight times squared numerator = 1280664

Variance i=3 Y times q := mul(1025, 28)

Variance i=3 Y times q = 28700

Variance i=3 centered integer numerator := sub(28700, -378)

Variance i=3 centered integer numerator = 29078

Variance i=3 centered numerator squared := mul(29078, 29078)

Variance i=3 centered numerator squared = 845530084

Variance i=3 weight times squared numerator := mul(0, 845530084)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(15876, 333396, 1280664, 0)

Variance sum ALL weighted squared numerators = 1629936

Variance denominator q squared := mul(28, 28)

Variance denominator q squared = 784

Variance denominator q cubed binary second factor := mul(784, 28)

Variance denominator q cubed binary second factor = 21952

Variance probability population centered sum over q cubed := div(1629936, 21952)

Variance probability population centered sum over q cubed = 297/4

source roles = [[-18, -18, 3, 1025], [1, 21, 6, 0], 28, -27/2, 1629936, 21952]

derived = [297/4, [-18, 3], No]

source roles = [shift_variance, null, 7]

derived = [297/4]

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

297/4

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.