# Question 4154: Positive-Mass Value Recovered from a Mean [Return to catalogue page 21](https://ocme.evemisslab.com/en/questions/page/21/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261010-r05-expectation-foundations/#q-20261008-advance-20261010-r05-expectation-0079) 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. R0007-04154 · q-20261008-advance-20261010-r05-expectation-0079 ## Original question and givens S={0,…,2}, F=P(S), masses=[4, 2, 1], D=7, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 5, -1] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-3/7. Find that slot’s integer value, written as p/1. - G001 — S={0,…,2}, F=P(S), masses=[4, 2, 1], D=7, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 5, -1] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-3/7. Find that slot’s integer value, written as p/1. ## 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 · calculationBefore total mass numerator := sum(4, 2, 1) After total mass numerator = 7 Conditions: none Nonnegative integer masses must sum to D for normalized weights. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · impliesBefore indexed masses=[4, 2, 1];D=7 After probabilities=[4/7, 2/7, 1/7] Conditions: none Outcome indices start at0 and align masses with X and Y values.Retain zero-mass outcomes;negative random-variable values are allowed. Step identifiers and dependenciesRule: indexed_normalized_expectation_model · Uses: G001, S001 S003 · calculationBefore known numerator1 := mul(2, 5) After known numerator1 = 10 Conditions: none Include mass times value for known slots only;leave the unknown slot as a variable. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore known numerator2 := mul(1, -1) After known numerator2 = -1 Conditions: none Include mass times value for known slots only;leave the unknown slot as a variable. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore known weighted numerator := sum(10, -1) After known weighted numerator = 9 Conditions: none Add all known mass-value terms. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore target expectation numerator := mul(-3/7, 7) After target expectation numerator = -3 Conditions: none Multiply the target mean by D to obtain the target weighted numerator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore unknown weighted contribution := sub(-3, 9) After unknown weighted contribution = -12 Conditions: none Subtract known terms to leave mass_k times the unknown value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore recovered unknown value := div(-12, 4) After recovered unknown value = -3 Conditions: none Positive unknown-slot mass makes the value uniquely recoverable;this exercise slice requires an integer result. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore recovered weighted numerator := mul(4, -3) After recovered weighted numerator = -12 Conditions: none Multiply the recovered value by its original slot mass to audit alignment and sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore complete reconstructed numerator := sum(9, -12) After complete reconstructed numerator = -3 Conditions: none Restore the missing term to obtain the complete numerator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore reconstructed expectation := div(-3, 7) After reconstructed expectation = -3/7 Conditions: none Divide by D to substitute back;the result must equal the stated mean. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore k=0;mass_k=4>0;target=-3/7 After unique recovered value=-3;substitution mean=-3/7 Conditions: none With mass0 the mean would not constrain this slot,so this inversion would fail.Null marks the unique unknown and is not numeric0. Step identifiers and dependenciesRule: positive_mass_missing_value_uniqueness · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute reconstructed expectation with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute complete reconstructed numerator with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute recovered weighted numerator with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute recovered unknown value with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute unknown weighted contribution with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute target expectation numerator with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute known weighted numerator with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute known numerator2 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute known numerator1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Recompute total mass numerator with exact rational arithmetic, not rounded decimals. ## Result -3/1 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: 900767b2262b770878361dd80e2b7611274a7079d7b872b174d802bdc187e195 - candidate record SHA-256: 9eeb7210cfbe6daface78f34d1d10d3d6ed5b1c4151e7e61a69586277c5013a5 - semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a - question-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261010-r05-expectation-0079.json)