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R0057-29139 · q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0064
Use the explicit finite discrete outcome list=[13, 14, 16] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
G001 — Use the explicit finite discrete outcome list=[13, 14, 16] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.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.
Before
source roles = [qf-str22-distribution-expectation, distribution-expectation]
After
derived = [[outcomes, masses, expectation]]
Conditions: none
Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning.
Rule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001
Before
source roles = [PMF raw mass slot0, [1, 4]]
After
derived = [1/4, 1, 4]
Conditions: none
A raw rational is a PAIR with each component within±1e6 and nonzero denominator.Normalize denominator sign and gcd.Do not accept a scalar or reduce before checking the RAW input bounds.
Rule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S001
Before
source roles = [PMF raw mass slot1, [1, 4]]
After
derived = [1/4, 1, 4]
Conditions: none
A raw rational is a PAIR with each component within±1e6 and nonzero denominator.Normalize denominator sign and gcd.Do not accept a scalar or reduce before checking the RAW input bounds.
Rule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S002
Before
source roles = [PMF raw mass slot2, [1, 2]]
After
derived = [1/2, 1, 2]
Conditions: none
A raw rational is a PAIR with each component within±1e6 and nonzero denominator.Normalize denominator sign and gcd.Do not accept a scalar or reduce before checking the RAW input bounds.
Rule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S003
Before
PMF ORIGINAL normalization running sum0 := sum(0, 1/4)
After
PMF ORIGINAL normalization running sum0 = 1/4
Conditions: none
Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
source roles = [PMF ORIGINAL normalization running sum0, sum, [0, 1/4]]
After
derived = [[1], [1, 4], 1/4]
Conditions: none
SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate.
Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S005
Before
PMF ORIGINAL normalization running sum1 := sum(1/4, 1/4)
After
PMF ORIGINAL normalization running sum1 = 1/2
Conditions: none
Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
source roles = [PMF ORIGINAL normalization running sum1, sum, [1/4, 1/4]]
After
derived = [[4], [2, 4], 1/2]
Conditions: none
SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate.
Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S007
Before
PMF ORIGINAL normalization running sum2 := sum(1/2, 1/2)
After
PMF ORIGINAL normalization running sum2 = 1
Conditions: none
Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
source roles = [PMF ORIGINAL normalization running sum2, sum, [1/2, 1/2]]
After
derived = [[2], [2, 2], 1]
Conditions: none
SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate.
Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S009
Before
source roles = [[13, 14, 16], [1/4, 1/4, 1/2]]
After
derived = [[[13, 1/4], [14, 1/4], [16, 1/2]], Yes, 1, Yes, Yes]
Conditions: none
Preserve ORIGINAL outcome-mass slot alignment and sum every mass in original order.Nonnegativity and total1 are separate obligations.Qualification may legally be No;do not renormalize or clip negative masses.
Rule: PMF_ALL_original_outcome_mass_alignment_NONNEG_AND_EXACTsum1 · Uses: G001, S010
Before
PMF ORIGINAL value times mass slot0 := mul(13, 1/4)
After
PMF ORIGINAL value times mass slot0 = 13/4
Conditions: none
Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
source roles = [PMF ORIGINAL value times mass slot0, mul, [13, 1/4]]
After
derived = [[1, 1], [13, 4], 13/4]
Conditions: none
SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate.
Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S012
Before
PMF ORIGINAL expectation running sum0 := sum(0, 13/4)
After
PMF ORIGINAL expectation running sum0 = 13/4
Conditions: none
Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
source roles = [PMF ORIGINAL expectation running sum0, sum, [0, 13/4]]
After
derived = [[1], [13, 4], 13/4]
Conditions: none
SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate.
Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S014
Before
PMF ORIGINAL value times mass slot1 := mul(14, 1/4)
After
PMF ORIGINAL value times mass slot1 = 7/2
Conditions: none
Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
source roles = [PMF ORIGINAL value times mass slot1, mul, [14, 1/4]]
After
derived = [[2, 1], [7, 2], 7/2]
Conditions: none
SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate.
Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S016
Before
PMF ORIGINAL expectation running sum1 := sum(13/4, 7/2)
After
PMF ORIGINAL expectation running sum1 = 27/4
Conditions: none
Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
source roles = [PMF ORIGINAL expectation running sum1, sum, [13/4, 7/2]]
After
derived = [[2], [27, 4], 27/4]
Conditions: none
SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate.
Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S018
Before
PMF ORIGINAL value times mass slot2 := mul(16, 1/2)
After
PMF ORIGINAL value times mass slot2 = 8
Conditions: none
Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
source roles = [PMF ORIGINAL value times mass slot2, mul, [16, 1/2]]
After
derived = [[2, 1], [8, 1], 8]
Conditions: none
SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate.
Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S020
Before
PMF ORIGINAL expectation running sum2 := sum(27/4, 8)
After
PMF ORIGINAL expectation running sum2 = 59/4
Conditions: none
Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
source roles = [PMF ORIGINAL expectation running sum2, sum, [27/4, 8]]
After
derived = [[1], [59, 4], 59/4]
Conditions: none
SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate.
Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S022
Before
source roles = [[13, 14, 16], [1/4, 1/4, 1/2]]
After
derived = [59/4]
Conditions: none
E[Y]=sum y_i*p_i uses EVERY original value-mass pair.Negative outcomes/expectation are legal.This is expectation under the GIVEN mathematical law,not empirical fit or population validation.
Rule: PMF_expected_value_COMPLETE_signed_weighted_sum · Uses: G001, S023
S024 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.S023 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.S022 · premise_and_result_audit — Recompute PMF ORIGINAL expectation running sum2 with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.S020 · premise_and_result_audit — Recompute PMF ORIGINAL value times mass slot2 with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.S018 · premise_and_result_audit — Recompute PMF ORIGINAL expectation running sum1 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.S016 · premise_and_result_audit — Recompute PMF ORIGINAL value times mass slot1 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.S014 · premise_and_result_audit — Recompute PMF ORIGINAL expectation running sum0 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.S012 · premise_and_result_audit — Recompute PMF ORIGINAL value times mass slot0 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.S010 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.S009 · premise_and_result_audit — Recompute PMF ORIGINAL normalization running sum2 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.S007 · premise_and_result_audit — Recompute PMF ORIGINAL normalization running sum1 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.S005 · premise_and_result_audit — Recompute PMF ORIGINAL normalization running sum0 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.S003 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.S002 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.S001 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.([13, 14, 16], [1/4, 1/4, 1/2], 59/4)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: e56e9c0e5d2d19a7f64ac710fbf733502771abbee24a6e5ecbbd305475f30e49candidate record SHA-256: d82d3bd74354c1ec08d2569f932e48690f8a89dc8e260ac752a0032d3e598350semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5Lean: NOT_RUN · human review: NOT_PERFORMED