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Question 31628: Expectation of an Affine Transformation

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0062-31628 · q-20261004-advance-20261005-r06-expectation-0028

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, -4), (3, 1001)}. In the displayed order of S, the nonnegative probability weights are [1/(10), 4/(10), 5/(10), 0/(10)], summing exactly to one. Let Z=(3)X+(-3). Find E[Z] exactly as a reduced fraction with positive 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, -4), (3, 1001)}. In the displayed order of S, the nonnegative probability weights are [1/(10), 4/(10), 5/(10), 0/(10)], summing exactly to one. Let Z=(3)X+(-3). Find E[Z] exactly as a reduced fraction with positive 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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-expectation-affine, no-domain-field, affine_expectation, no-form-field]

After

derived = [[reduced_fraction_RETAIN1_STRING]]

Conditions: none

Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction.

Step identifiers and dependencies

Rule: R62_EXACT_source_keys_domain_operation_form_AND_output_types · Uses: G001

S002 · calculation

Before

Expectation ORIGINAL weight total through slot0 := sum(0, 1)

After

Expectation ORIGINAL weight total through slot0 = 1

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Expectation ORIGINAL weight total through slot1 := sum(1, 4)

After

Expectation ORIGINAL weight total through slot1 = 5

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Expectation ORIGINAL weight total through slot2 := sum(5, 5)

After

Expectation ORIGINAL weight total through slot2 = 10

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Expectation ORIGINAL weight total through slot3 := sum(10, 0)

After

Expectation ORIGINAL weight total through slot3 = 10

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [[0, 1, 2, 3], powerset(S), [1, 4, 5, 0], 10, [-8, -8, -4, 1001], not-supplied]

After

derived = [[0, 1, 2, 3], 10, Yes]

Conditions: none

S is nonempty,distinct and≤100;weights,X/Y align to ORIGINAL S slots.F is exactly powerset(S),weights nonnegative,denominator positive,and total EXACTLYden.EVEN at a=0 or null weights validate ALL raw fields;never silently uniformize.

Step identifiers and dependencies

Rule: Expectation_FULL_model_RAW_slots_SAME_alignment_and_normalized_law · Uses: G001, S005

S007 · calculation

Before

Expectation affine ORIGINAL slot0 a times X := mul(3, -8)

After

Expectation affine ORIGINAL slot0 a times X = -24

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Expectation affine ORIGINAL slot0 plus b := sum(-24, -3)

After

Expectation affine ORIGINAL slot0 plus b = -27

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

source roles = [0, 0, affine_expectation, -8, not-supplied, 3, -3, not-supplied]

After

derived = [-27]

Conditions: none

Transform at EACH ORIGINAL outcome BEFORE using its aligned weight.Never sort S,X,Y,weights independently.Null-weight values remain original data;do not delete them to repair inputs or infer a range.

Step identifiers and dependencies

Rule: Expectation_EACH_original_outcome_transformation_BEFORE_weight · Uses: G001, S008

S010 · calculation

Before

Expectation ORIGINAL slot0 weight times transformed value := mul(1, -27)

After

Expectation ORIGINAL slot0 weight times transformed value = -27

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Expectation ORIGINAL weighted numerator through slot0 := sum(0, -27)

After

Expectation ORIGINAL weighted numerator through slot0 = -27

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Expectation affine ORIGINAL slot1 a times X := mul(3, -8)

After

Expectation affine ORIGINAL slot1 a times X = -24

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Expectation affine ORIGINAL slot1 plus b := sum(-24, -3)

After

Expectation affine ORIGINAL slot1 plus b = -27

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

source roles = [1, 1, affine_expectation, -8, not-supplied, 3, -3, not-supplied]

After

derived = [-27]

Conditions: none

Transform at EACH ORIGINAL outcome BEFORE using its aligned weight.Never sort S,X,Y,weights independently.Null-weight values remain original data;do not delete them to repair inputs or infer a range.

Step identifiers and dependencies

Rule: Expectation_EACH_original_outcome_transformation_BEFORE_weight · Uses: G001, S013

S015 · calculation

Before

Expectation ORIGINAL slot1 weight times transformed value := mul(4, -27)

After

Expectation ORIGINAL slot1 weight times transformed value = -108

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Expectation ORIGINAL weighted numerator through slot1 := sum(-27, -108)

After

Expectation ORIGINAL weighted numerator through slot1 = -135

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

Expectation affine ORIGINAL slot2 a times X := mul(3, -4)

After

Expectation affine ORIGINAL slot2 a times X = -12

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

Expectation affine ORIGINAL slot2 plus b := sum(-12, -3)

After

Expectation affine ORIGINAL slot2 plus b = -15

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

source roles = [2, 2, affine_expectation, -4, not-supplied, 3, -3, not-supplied]

After

derived = [-15]

Conditions: none

Transform at EACH ORIGINAL outcome BEFORE using its aligned weight.Never sort S,X,Y,weights independently.Null-weight values remain original data;do not delete them to repair inputs or infer a range.

Step identifiers and dependencies

Rule: Expectation_EACH_original_outcome_transformation_BEFORE_weight · Uses: G001, S018

S020 · calculation

Before

Expectation ORIGINAL slot2 weight times transformed value := mul(5, -15)

After

Expectation ORIGINAL slot2 weight times transformed value = -75

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

Expectation ORIGINAL weighted numerator through slot2 := sum(-135, -75)

After

Expectation ORIGINAL weighted numerator through slot2 = -210

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

Expectation affine ORIGINAL slot3 a times X := mul(3, 1001)

After

Expectation affine ORIGINAL slot3 a times X = 3003

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

Expectation affine ORIGINAL slot3 plus b := sum(3003, -3)

After

Expectation affine ORIGINAL slot3 plus b = 3000

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

source roles = [3, 3, affine_expectation, 1001, not-supplied, 3, -3, not-supplied]

After

derived = [3000]

Conditions: none

Transform at EACH ORIGINAL outcome BEFORE using its aligned weight.Never sort S,X,Y,weights independently.Null-weight values remain original data;do not delete them to repair inputs or infer a range.

Step identifiers and dependencies

Rule: Expectation_EACH_original_outcome_transformation_BEFORE_weight · Uses: G001, S023

S025 · calculation

Before

Expectation ORIGINAL slot3 weight times transformed value := mul(0, 3000)

After

Expectation ORIGINAL slot3 weight times transformed value = 0

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

Expectation ORIGINAL weighted numerator through slot3 := sum(-210, 0)

After

Expectation ORIGINAL weighted numerator through slot3 = -210

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

Expectation ORIGINAL weighted numerator over denominator := div(-210, 10)

After

Expectation ORIGINAL weighted numerator over denominator = -21

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

source roles = [[0, 1, 2, 3], [1, 4, 5, 0], 10, affine_expectation, [-27, -27, -15, 3000]]

After

derived = [[-27, -108, -75, 0], -210, -21/1]

Conditions: none

In the QUALIFIED SAME probability law,E[Z]=Σw_iZ(s_i)/q.Affine gives aE[X]+b and a same-model sum gives E[X]+E[Y]WITHOUT independence;this does not imply a product-expectation formula.A constant has mean c because total mass is1.Negative/unattained means are legal;retain/1.

Step identifiers and dependencies

Rule: Expectation_LINEARITY_qualified_SAME_law_NOT_independence_or_sample_mean · Uses: G001, S027

Reverse checks

  • S028 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S027 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator through slot3 with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute Expectation ORIGINAL slot3 weight times transformed value with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S023 · premise_and_result_audit — Recompute Expectation affine ORIGINAL slot3 plus b with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute Expectation affine ORIGINAL slot3 a times X with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator through slot2 with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute Expectation ORIGINAL slot2 weight times transformed value with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S018 · premise_and_result_audit — Recompute Expectation affine ORIGINAL slot2 plus b with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute Expectation affine ORIGINAL slot2 a times X with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator through slot1 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Expectation ORIGINAL slot1 weight times transformed value with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S013 · premise_and_result_audit — Recompute Expectation affine ORIGINAL slot1 plus b with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Expectation affine ORIGINAL slot1 a times X with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator through slot0 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Expectation ORIGINAL slot0 weight times transformed value with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S008 · premise_and_result_audit — Recompute Expectation affine ORIGINAL slot0 plus b with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Expectation affine ORIGINAL slot0 a times X with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S005 · premise_and_result_audit — Recompute Expectation ORIGINAL weight total through slot3 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Expectation ORIGINAL weight total through slot2 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Expectation ORIGINAL weight total through slot1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Expectation ORIGINAL weight total through slot0 with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.

Result

-21/1

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 0dd6340b9cacd8725ed8310ed43d7e30c08d5fe2aed52f8869b2df14779f9abf
  • candidate record SHA-256: ecd292820c96c0c7e4964ddc4c7ae096756866c43321a0b80ecbb7b5a8b92854
  • semantic review receipt SHA-256: 963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd
  • question-first witness SHA-256: d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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