# Question 31659: Expectation of a Constant in a Normalized Model [Return to catalogue page 159](https://ocme.evemisslab.com/en/questions/page/159/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261004-advance-20261005-r06-expectation-foundations/#q-20261004-advance-20261005-r06-expectation-0084) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0062-31659 · q-20261004-advance-20261005-r06-expectation-0084 ## Original question and givens The nonempty finite sample space is S={0, 1}, with F=𝒫(S). In the displayed order of S, the nonnegative probability weights are [1/(4), 3/(4)], summing exactly to one. For X(s)=-4 at every outcome, compute E[X] as a reduced fraction with positive denominator. - G001 — The nonempty finite sample space is S={0, 1}, with F=𝒫(S). In the displayed order of S, the nonnegative probability weights are [1/(4), 3/(4)], summing exactly to one. For X(s)=-4 at every outcome, compute E[X] 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. - [mko-expectation](https://ocme.evemisslab.com/en/mko/mko-expectation/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-expectation-constant, no-domain-field, constant_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 dependenciesRule: R62_EXACT_source_keys_domain_operation_form_AND_output_types · Uses: G001 S002 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Expectation ORIGINAL weight total through slot1 := sum(1, 3) After Expectation ORIGINAL weight total through slot1 = 4 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [[0, 1], powerset(S), [1, 3], 4, not-supplied, not-supplied] After derived = [[0, 1], 4, 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 dependenciesRule: Expectation_FULL_model_RAW_slots_SAME_alignment_and_normalized_law · Uses: G001, S003 S005 · impliesBefore source roles = [0, 0, constant_expectation, not-supplied, not-supplied, not-supplied, not-supplied, -4] After derived = [-4] 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 dependenciesRule: Expectation_EACH_original_outcome_transformation_BEFORE_weight · Uses: G001, S004 S006 · calculationBefore Expectation ORIGINAL slot0 weight times transformed value := mul(1, -4) After Expectation ORIGINAL slot0 weight times transformed value = -4 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Expectation ORIGINAL weighted numerator through slot0 := sum(0, -4) After Expectation ORIGINAL weighted numerator through slot0 = -4 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore source roles = [1, 1, constant_expectation, not-supplied, not-supplied, not-supplied, not-supplied, -4] After derived = [-4] 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 dependenciesRule: Expectation_EACH_original_outcome_transformation_BEFORE_weight · Uses: G001, S007 S009 · calculationBefore Expectation ORIGINAL slot1 weight times transformed value := mul(3, -4) After Expectation ORIGINAL slot1 weight times transformed value = -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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Expectation ORIGINAL weighted numerator through slot1 := sum(-4, -12) After Expectation ORIGINAL weighted numerator through slot1 = -16 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Expectation ORIGINAL weighted numerator over denominator := div(-16, 4) After Expectation ORIGINAL weighted numerator over denominator = -4 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore source roles = [[0, 1], [1, 3], 4, constant_expectation, [-4, -4]] After derived = [[-4, -12], -16, -4/1] 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 dependenciesRule: Expectation_LINEARITY_qualified_SAME_law_NOT_independence_or_sample_mean · Uses: G001, S011 ## Reverse checks - S012 · 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. - S011 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator over denominator with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator through slot1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Expectation ORIGINAL slot1 weight times transformed value with exact rational arithmetic, not rounded decimals. - S008 · 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. - S007 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator through slot0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Expectation ORIGINAL slot0 weight times transformed value with exact rational arithmetic, not rounded decimals. - S005 · 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. - S004 · 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. - 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 -4/1 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: 3b72dc9606ad024187c90bc45983ecaa454f5d17b223ed8575404c21b86c0992 - candidate record SHA-256: 81a0227a5cf5b304cab0af9227946951ad974733b66e603161f45e616400a2b9 - semantic review receipt SHA-256: 963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd - question-first witness SHA-256: d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261004-advance-20261005-r06-expectation-0084.json)