# Question 31721: Expectation in a Nonuniform Finite 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-0021) 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. R0062-31721 · q-20261004-advance-20261005-r06-expectation-0021 ## 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, 10), (1, 10), (2, 32), (3, 1019)}. In the displayed order of S, the nonnegative probability weights are [1/(26), 22/(26), 3/(26), 0/(26)], summing exactly to one. Compute E[X] exactly as a reduced positive-denominator fraction; do not replace it with the ordinary average of distinct values. - 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, 10), (1, 10), (2, 32), (3, 1019)}. In the displayed order of S, the nonnegative probability weights are [1/(26), 22/(26), 3/(26), 0/(26)], summing exactly to one. Compute E[X] exactly as a reduced positive-denominator fraction; do not replace it with the ordinary average of distinct values. ## 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-weighted, no-domain-field, 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, 22) After Expectation ORIGINAL weight total through slot1 = 23 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 · calculationBefore Expectation ORIGINAL weight total through slot2 := sum(23, 3) After Expectation ORIGINAL weight total through slot2 = 26 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, S003 S005 · calculationBefore Expectation ORIGINAL weight total through slot3 := sum(26, 0) After Expectation ORIGINAL weight total through slot3 = 26 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, S004 S006 · impliesBefore source roles = [[0, 1, 2, 3], powerset(S), [1, 22, 3, 0], 26, [10, 10, 32, 1019], not-supplied] After derived = [[0, 1, 2, 3], 26, 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, S005 S007 · impliesBefore source roles = [0, 0, expectation, 10, not-supplied, not-supplied, not-supplied, not-supplied] After derived = [10] 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, S006 S008 · calculationBefore Expectation ORIGINAL slot0 weight times transformed value := mul(1, 10) After Expectation ORIGINAL slot0 weight times transformed value = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Expectation ORIGINAL weighted numerator through slot0 := sum(0, 10) After Expectation ORIGINAL weighted numerator through slot0 = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore source roles = [1, 1, expectation, 10, not-supplied, not-supplied, not-supplied, not-supplied] After derived = [10] 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, S009 S011 · calculationBefore Expectation ORIGINAL slot1 weight times transformed value := mul(22, 10) After Expectation ORIGINAL slot1 weight times transformed value = 220 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 · calculationBefore Expectation ORIGINAL weighted numerator through slot1 := sum(10, 220) After Expectation ORIGINAL weighted numerator through slot1 = 230 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, S011 S013 · impliesBefore source roles = [2, 2, expectation, 32, not-supplied, not-supplied, not-supplied, not-supplied] After derived = [32] 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, S012 S014 · calculationBefore Expectation ORIGINAL slot2 weight times transformed value := mul(3, 32) After Expectation ORIGINAL slot2 weight times transformed value = 96 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, S013 S015 · calculationBefore Expectation ORIGINAL weighted numerator through slot2 := sum(230, 96) After Expectation ORIGINAL weighted numerator through slot2 = 326 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, S014 S016 · impliesBefore source roles = [3, 3, expectation, 1019, not-supplied, not-supplied, not-supplied, not-supplied] After derived = [1019] 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, S015 S017 · calculationBefore Expectation ORIGINAL slot3 weight times transformed value := mul(0, 1019) 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Expectation ORIGINAL weighted numerator through slot3 := sum(326, 0) After Expectation ORIGINAL weighted numerator through slot3 = 326 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, S017 S019 · calculationBefore Expectation ORIGINAL weighted numerator over denominator := div(326, 26) After Expectation ORIGINAL weighted numerator over denominator = 163/13 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, S018 S020 · impliesBefore source roles = [[0, 1, 2, 3], [1, 22, 3, 0], 26, expectation, [10, 10, 32, 1019]] After derived = [[10, 220, 96, 0], 326, 163/13] 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, S019 ## Reverse checks - S020 · 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. - S019 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator over denominator with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator through slot3 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Expectation ORIGINAL slot3 weight times transformed value with exact rational arithmetic, not rounded decimals. - S016 · 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. - S015 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator through slot2 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Expectation ORIGINAL slot2 weight times transformed value with exact rational arithmetic, not rounded decimals. - S013 · 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. - S012 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator through slot1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Expectation ORIGINAL slot1 weight times transformed value with exact rational arithmetic, not rounded decimals. - S010 · 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. - S009 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator through slot0 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Expectation ORIGINAL slot0 weight times transformed value with exact rational arithmetic, not rounded decimals. - S007 · 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. - 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 163/13 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: 7b8777cee94fa4b89094e4012bb0d753ca76d2eddda2c05692e03fb1cedab6e5 - candidate record SHA-256: ce67521626c32adfa42142a86a599b0e9ec70b659341e4f7b88bf173a5b8b853 - 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-0021.json)