# Question 18116: Compute an Estimator Expectation under One Fixed Parameter Law [Return to catalogue page 91](https://ocme.evemisslab.com/en/questions/page/91/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261104-r04-statistical-estimator-foundations/#q-20261014-advance-20261104-r04-statistical-estimator-0066) 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. R0035-18116 · q-20261014-advance-20261104-r04-statistical-estimator-0066 ## Original question and givens Use one common finite model with parameter indices=Fin(2)={0,…,1}, outcomes=Fin(3)={0,…,2}, complete candidate laws=[[1/19, 1/19, 17/19], [1/2, 1/3, 1/6]], and the single total statistic T=[0, 16, 32] in outcome order. Each law must be nonnegative and sum to 1 before model qualification. T depends on the observation, not the unknown parameter. An observed value is an estimate, while T is the estimator function; no prior or posterior is supplied. Qualified-model parameter index=1. Give (parameter, expectation, statistic). - G001 — Use one common finite model with parameter indices=Fin(2)={0,…,1}, outcomes=Fin(3)={0,…,2}, complete candidate laws=[[1/19, 1/19, 17/19], [1/2, 1/3, 1/6]], and the single total statistic T=[0, 16, 32] in outcome order. Each law must be nonnegative and sum to 1 before model qualification. T depends on the observation, not the unknown parameter. An observed value is an estimate, while T is the estimator function; no prior or posterior is supplied. Qualified-model parameter index=1. Give (parameter, expectation, statistic). ## 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-statistical-estimator](https://ocme.evemisslab.com/en/mko/mko-statistical-estimator/) ## Step-by-step derivation S001 · impliesBefore source operation=estimator-expectation-at-parameter;original snapshot and mathematics retained After explicit ordered source output fields=[parameter, expectation, statistic] Conditions: none Keep the historical interface marker in the original snapshot while giving ordered fields here,without JavaScript or an external table.Labels are not mathematical evidence,and each No marker retains the operation's claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore estimator parameter0 complete law sum := sum(1/19, 1/19, 17/19) After estimator parameter0 complete law sum = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore original parameter=0;SAME outcome carrier Fin(3);aligned reduced candidate law=[1/19, 1/19, 17/19] After sum=1;nonnegative=Yes;law qualified=Yes Conditions: none Every model row must be nonnegative and total1.Invalid candidate rows retain false flags;a total statistic does not repair a probability model. Step identifiers and dependenciesRule: estimator_complete_laws_separate_from_total_statistic · Uses: G001, S002 S004 · calculationBefore estimator parameter1 complete law sum := sum(1/2, 1/3, 1/6) After estimator parameter1 complete law sum = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore original parameter=1;SAME outcome carrier Fin(3);aligned reduced candidate law=[1/2, 1/3, 1/6] After sum=1;nonnegative=Yes;law qualified=Yes Conditions: none Every model row must be nonnegative and total1.Invalid candidate rows retain false flags;a total statistic does not repair a probability model. Step identifiers and dependenciesRule: estimator_complete_laws_separate_from_total_statistic · Uses: G001, S004 S006 · impliesBefore original parameter carrier Fin(2);common outcome carrier Fin(3);ONE original statistic array=[0, 16, 32] After all model laws qualified=Yes;one value per outcome=Yes;SAME statistic map T(o) at EVERY parameter Conditions: none The statistic definition takes an observation index,not the unknown parameter.This is functional parameter-independence,not a claim that the law or expectation of T(X) is parameter-free.No estimand,prior or posterior has been supplied. Step identifiers and dependenciesRule: estimator_definition_parameter_independence_not_distribution_free · Uses: G001, S005 S007 · calculationBefore estimator selected law weighted outcome0 := mul(1/2, 0) After estimator selected law weighted outcome0 = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore estimator selected law weighted outcome1 := mul(1/3, 16) After estimator selected law weighted outcome1 = 16/3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore estimator selected law weighted outcome2 := mul(1/6, 32) After estimator selected law weighted outcome2 = 16/3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore estimator selected parameter expectation := sum(0, 16/3, 16/3) After estimator selected parameter expectation = 32/3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore FIXED original parameter=1;complete law=[1/2, 1/3, 1/6];SAME statistic=[0, 16, 32];weighted terms=[0, 16/3, 16/3] After E_parameter[T]=32/3;no unbiasedness or posterior conclusion Conditions: none Weight the same statistic by the complete law for one fixed parameter.Without a declared estimand,comparing this expectation with the parameter index does not establish bias or unbiasedness;no prior supplies a posterior. Step identifiers and dependenciesRule: estimator_expectation_fixed_law_no_undeclared_estimand · Uses: G001, S010 ## Reverse checks - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute estimator selected parameter expectation with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute estimator selected law weighted outcome2 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute estimator selected law weighted outcome1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute estimator selected law weighted outcome0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute estimator parameter1 complete law sum with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Recompute estimator parameter0 complete law sum with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (1, 32/3, [0, 16, 32]) 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: 794cf4bb15159c6574d3b66e428bd09ee1a0458f96184fe59578d825958ff8b0 - candidate record SHA-256: 76ae78d4c6fec97732e96cc2247e81eeceee1fc8d231576c61c684f6faa52e15 - semantic review receipt SHA-256: 713db6309b22bf8b31f0253d4aaa7aca9011ab908587ae0e9d4a365edce1f9bc - question-first witness SHA-256: 0338a3db344ab237113fc63be29153c3684da96e388109e9dcd913a0a693aed2 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261104-r04-statistical-estimator-0066.json)