# Question 19707: Use One Parameter Law and Its Own Mean for Variance [Return to catalogue page 99](https://ocme.evemisslab.com/en/questions/page/99/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261106-r04-estimator-variance-foundations/#q-20261014-advance-20261106-r04-estimator-variance-0007) 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. R0038-19707 · q-20261014-advance-20261106-r04-estimator-variance-0007 ## Original question and givens Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate). - G001 — Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate). ## 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-estimator-variance](https://ocme.evemisslab.com/en/mko/mko-estimator-variance/) ## Step-by-step derivation S001 · impliesBefore source operation=estvar-at-parameter;original snapshot and mathematics retained After explicit ordered source output fields=[parameter_index, expectation, variance, variance_certificate, certificate] Conditions: none Keep the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not mathematical evidence,and No retains the operation's claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore variance parameter0 complete law mass := sum(1/16, 1/8, 3/16, 5/8) After variance parameter0 complete law mass = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore original parameter SLOT=0;ALL original outcome masses=[1/16, 1/8, 3/16, 5/8];ONE fixed statistic=[0, 1, 8, 9] After complete same-law mass=1;all masses nonnegative;positive-mass support=[0, 1, 2, 3];null outcomes RETAINED Conditions: none Qualify each complete parameter law separately with total1.The same statistic definition is parameter-independent,but laws,means and variances remain row-specific.Keep null outcomes in the original table even though they are outside that law's positive-mass support. Step identifiers and dependenciesRule: estvar_full_law_statistic_null_support_roles · Uses: G001, S002 S004 · calculationBefore variance parameter1 complete law mass := sum(1/10, 3/20, 1/5, 11/20) After variance parameter1 complete law mass = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore original parameter SLOT=1;ALL original outcome masses=[1/10, 3/20, 1/5, 11/20];ONE fixed statistic=[0, 1, 8, 9] After complete same-law mass=1;all masses nonnegative;positive-mass support=[0, 1, 2, 3];null outcomes RETAINED Conditions: none Qualify each complete parameter law separately with total1.The same statistic definition is parameter-independent,but laws,means and variances remain row-specific.Keep null outcomes in the original table even though they are outside that law's positive-mass support. Step identifiers and dependenciesRule: estvar_full_law_statistic_null_support_roles · Uses: G001, S004 S006 · calculationBefore variance parameter0 outcome0 weighted T := mul(1/16, 0) After variance parameter0 outcome0 weighted T = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore variance parameter0 outcome1 weighted T := mul(1/8, 1) After variance parameter0 outcome1 weighted T = 1/8 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore variance parameter0 outcome2 weighted T := mul(3/16, 8) After variance parameter0 outcome2 weighted T = 3/2 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore variance parameter0 outcome3 weighted T := mul(5/8, 9) After variance parameter0 outcome3 weighted T = 45/8 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore variance parameter0 SAME-law mean := sum(0, 1/8, 3/2, 45/8) After variance parameter0 SAME-law mean = 29/4 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore variance parameter0 outcome0 T minus SAME mean := sub(0, 29/4) After variance parameter0 outcome0 T minus SAME mean = -29/4 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore variance parameter0 outcome0 centered square := mul(-29/4, -29/4) After variance parameter0 outcome0 centered square = 841/16 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore variance parameter0 outcome0 weighted centered square := mul(1/16, 841/16) After variance parameter0 outcome0 weighted centered square = 841/256 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore variance parameter0 outcome1 T minus SAME mean := sub(1, 29/4) After variance parameter0 outcome1 T minus SAME mean = -25/4 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore variance parameter0 outcome1 centered square := mul(-25/4, -25/4) After variance parameter0 outcome1 centered square = 625/16 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore variance parameter0 outcome1 weighted centered square := mul(1/8, 625/16) After variance parameter0 outcome1 weighted centered square = 625/128 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore variance parameter0 outcome2 T minus SAME mean := sub(8, 29/4) After variance parameter0 outcome2 T minus SAME mean = 3/4 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore variance parameter0 outcome2 centered square := mul(3/4, 3/4) After variance parameter0 outcome2 centered square = 9/16 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore variance parameter0 outcome2 weighted centered square := mul(3/16, 9/16) After variance parameter0 outcome2 weighted centered square = 27/256 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore variance parameter0 outcome3 T minus SAME mean := sub(9, 29/4) After variance parameter0 outcome3 T minus SAME mean = 7/4 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore variance parameter0 outcome3 centered square := mul(7/4, 7/4) After variance parameter0 outcome3 centered square = 49/16 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore variance parameter0 outcome3 weighted centered square := mul(5/8, 49/16) After variance parameter0 outcome3 weighted centered square = 245/128 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore variance parameter0 full centered variance := sum(841/256, 625/128, 27/256, 245/128) After variance parameter0 full centered variance = 163/16 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore FIXED parameter SLOT=0;original same law=[1/16, 1/8, 3/16, 5/8];same statistic=[0, 1, 8, 9];its own mean=29/4;ALL centered deviations=[-29/4, -25/4, 3/4, 7/4] After ALL nonnegative weighted centered squares=[841/256, 625/128, 27/256, 245/128];variance=163/16;variance0 iff EVERY positive-mass outcome has T=THIS mean Conditions: none A finite sum of nonnegative terms is0 iff each term is0.At positive masses this forces T to equal this mean;null outcomes impose no restriction.This is almost-sure constancy under the same law,not literal table constancy,unbiasedness or constancy under another parameter. Step identifiers and dependenciesRule: estvar_zero_iff_same_law_almost_sure_not_null_table_constancy · Uses: G001, S023 ## Reverse checks - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Recompute variance parameter0 full centered variance with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute variance parameter0 outcome3 weighted centered square with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute variance parameter0 outcome3 centered square with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute variance parameter0 outcome3 T minus SAME mean with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute variance parameter0 outcome2 weighted centered square with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute variance parameter0 outcome2 centered square with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute variance parameter0 outcome2 T minus SAME mean with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute variance parameter0 outcome1 weighted centered square with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute variance parameter0 outcome1 centered square with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute variance parameter0 outcome1 T minus SAME mean with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute variance parameter0 outcome0 weighted centered square with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute variance parameter0 outcome0 centered square with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute variance parameter0 outcome0 T minus SAME mean with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute variance parameter0 SAME-law mean with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute variance parameter0 outcome3 weighted T with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute variance parameter0 outcome2 weighted T with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute variance parameter0 outcome1 weighted T with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute variance parameter0 outcome0 weighted T with exact rational arithmetic, not rounded decimals. - 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 variance parameter1 complete law mass 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 variance parameter0 complete law mass 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 (0, 29/4, 163/16, Yes, same_theta_sampling_law_support_and_mean) 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: b701da4474407943f6f70abfd45e46598e8c65cca3f9a474ba2e22a8f26a86a0 - candidate record SHA-256: df5b5c650c99d1dac9898e392ae78a62d8dda1430252a86680e43086b54865ad - semantic review receipt SHA-256: d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612 - question-first witness SHA-256: c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261106-r04-estimator-variance-0007.json)