Question 35832: 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, -4), (1, -4), (2, 4), (3, 1005)}. In the displayed order of S, the nonnegative probability weights are [1/(14), 8/(14), 5/(14), 0/(14)], 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, -4), (1, -4), (2, 4), (3, 1005)}. In the displayed order of S, the nonnegative probability weights are [1/(14), 8/(14), 5/(14), 0/(14)], 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, 8, 5, 0)

Variance ORIGINAL nonnegative RAW weights sum = 14

source roles = [[0, 1, 2, 3], powerset(S), [-4, -4, 4, 1005], [1, 8, 5, 0], 14, null, -6]

derived = [Yes]

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

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, -4, 1, null, -6]

derived = [-10, 1/14]

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

Variance translated ORIGINAL value i=1 = -10

Variance weight times transformed value i=1 := mul(8, -10)

Variance weight times transformed value i=1 = -80

source roles = [1, 1, -4, 8, null, -6]

derived = [-10, 4/7]

Variance translated ORIGINAL value i=2 := sum(4, -6)

Variance translated ORIGINAL value i=2 = -2

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

Variance weight times transformed value i=2 = -10

source roles = [2, 2, 4, 5, null, -6]

derived = [-2, 5/14]

Variance translated ORIGINAL value i=3 := sum(1005, -6)

Variance translated ORIGINAL value i=3 = 999

Variance weight times transformed value i=3 := mul(0, 999)

Variance weight times transformed value i=3 = 0

source roles = [3, 3, 1005, 0, null, -6]

derived = [999, 0]

Variance mean numerator sum wY := sum(-10, -80, -10, 0)

Variance mean numerator sum wY = -100

Variance exact expectation M over q := div(-100, 14)

Variance exact expectation M over q = -50/7

Variance i=0 Y times q := mul(-10, 14)

Variance i=0 Y times q = -140

Variance i=0 centered integer numerator := sub(-140, -100)

Variance i=0 centered integer numerator = -40

Variance i=0 centered numerator squared := mul(-40, -40)

Variance i=0 centered numerator squared = 1600

Variance i=0 weight times squared numerator := mul(1, 1600)

Variance i=0 weight times squared numerator = 1600

Variance i=1 Y times q := mul(-10, 14)

Variance i=1 Y times q = -140

Variance i=1 centered integer numerator := sub(-140, -100)

Variance i=1 centered integer numerator = -40

Variance i=1 centered numerator squared := mul(-40, -40)

Variance i=1 centered numerator squared = 1600

Variance i=1 weight times squared numerator := mul(8, 1600)

Variance i=1 weight times squared numerator = 12800

Variance i=2 Y times q := mul(-2, 14)

Variance i=2 Y times q = -28

Variance i=2 centered integer numerator := sub(-28, -100)

Variance i=2 centered integer numerator = 72

Variance i=2 centered numerator squared := mul(72, 72)

Variance i=2 centered numerator squared = 5184

Variance i=2 weight times squared numerator := mul(5, 5184)

Variance i=2 weight times squared numerator = 25920

Variance i=3 Y times q := mul(999, 14)

Variance i=3 Y times q = 13986

Variance i=3 centered integer numerator := sub(13986, -100)

Variance i=3 centered integer numerator = 14086

Variance i=3 centered numerator squared := mul(14086, 14086)

Variance i=3 centered numerator squared = 198415396

Variance i=3 weight times squared numerator := mul(0, 198415396)

Variance i=3 weight times squared numerator = 0

Variance sum ALL weighted squared numerators := sum(1600, 12800, 25920, 0)

Variance sum ALL weighted squared numerators = 40320

Variance denominator q squared := mul(14, 14)

Variance denominator q squared = 196

Variance denominator q cubed binary second factor := mul(196, 14)

Variance denominator q cubed binary second factor = 2744

Variance probability population centered sum over q cubed := div(40320, 2744)

Variance probability population centered sum over q cubed = 720/49

source roles = [[-10, -10, -2, 999], [1, 8, 5, 0], 14, -50/7, 40320, 2744]

derived = [720/49, [-10, -2], No]

source roles = [shift_variance, null, -6]

derived = [720/49]

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

720/49

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.