# Question 19126: An Observed Estimate Is Not the Whole Sampling Law [Return to catalogue page 96](https://ocme.evemisslab.com/en/questions/page/96/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261105-r06-sampling-distribution-foundations/#q-20261014-advance-20261105-r06-sampling-distribution-0051) 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. R0037-19126 · q-20261014-advance-20261105-r06-sampling-distribution-0051 ## Original question and givens Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/8, 2/8, 3/8, 2/8], [2/12, 3/12, 4/12, 3/12]], and one total parameter-independent statistic T=[0, 0, 1, 2] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0; supplied outcome index=0. Give (outcome, estimate, law, boundary). - G001 — Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/8, 2/8, 3/8, 2/8], [2/12, 3/12, 4/12, 3/12]], and one total parameter-independent statistic T=[0, 0, 1, 2] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0; supplied outcome index=0. Give (outcome, estimate, law, boundary). ## 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-sampling-distribution](https://ocme.evemisslab.com/en/mko/mko-sampling-distribution/) ## Step-by-step derivation S001 · impliesBefore source operation=sampling-observed-estimate-boundary;original snapshot and mathematics retained After explicit ordered source output fields=[outcome, estimate, law, boundary] Conditions: none Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field names are not mathematical evidence,and No retains the operation's specific claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore sampling parameter0 full law mass := sum(1/8, 1/4, 3/8, 1/4) After sampling parameter0 full law mass = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore sampling:original parameter SLOT=0;same outcome carrier Fin(4);ALL aligned law masses=[1/8, 1/4, 3/8, 1/4] After full sum=1;all masses nonnegative=Yes;qualified probability law;zero-mass outcomes retained Conditions: none Each complete parameter law must separately be nonnegative and total1.Keep zero-mass outcomes in the original table;do not normalize across parameter rows to invent a prior or posterior. Step identifiers and dependenciesRule: finite_statistical_full_law_not_partial_or_prior · Uses: G001, S002 S004 · calculationBefore sampling parameter1 full law mass := sum(1/6, 1/4, 1/3, 1/4) After sampling parameter1 full law mass = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore sampling:original parameter SLOT=1;same outcome carrier Fin(4);ALL aligned law masses=[1/6, 1/4, 1/3, 1/4] After full sum=1;all masses nonnegative=Yes;qualified probability law;zero-mass outcomes retained Conditions: none Each complete parameter law must separately be nonnegative and total1.Keep zero-mass outcomes in the original table;do not normalize across parameter rows to invent a prior or posterior. Step identifiers and dependenciesRule: finite_statistical_full_law_not_partial_or_prior · Uses: G001, S004 S006 · impliesBefore sampling:parameter SLOT carrier Fin(2);common outcome carrier Fin(4);ONE total original statistic array=[0, 0, 1, 2] After T takes only the observation index,not the unknown parameter;SAME function at all parameter slots;its sampling law may still vary with the slot Conditions: none Functional parameter-independence is different from a parameter-free distribution of T(X).Use the same original statistic value at every outcome rather than changing T between laws. Step identifiers and dependenciesRule: finite_statistic_definition_not_distribution_free · Uses: G001, S005 S007 · calculationBefore sampling parameter0 statistic value0 full fibre mass := sum(1/8, 1/4) After sampling parameter0 statistic value0 full fibre mass = 3/8 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore FIXED original parameter SLOT=0;statistic value=0;FULL original fibre outcome indices=[0, 1];aligned law masses=[1/8, 1/4] After pushforward mass at value0=3/8;retain this row even if mass0 Conditions: none Add masses from every outcome with the same statistic value,not one representative.Retain every value present in the original statistic even when its fibre has mass0 under this parameter. Step identifiers and dependenciesRule: sampling_complete_fibre_not_one_representative_or_null_deletion · Uses: G001, S007 S009 · calculationBefore sampling parameter0 statistic value1 full fibre mass := sum(3/8) After sampling parameter0 statistic value1 full fibre mass = 3/8 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore FIXED original parameter SLOT=0;statistic value=1;FULL original fibre outcome indices=[2];aligned law masses=[3/8] After pushforward mass at value1=3/8;retain this row even if mass0 Conditions: none Add masses from every outcome with the same statistic value,not one representative.Retain every value present in the original statistic even when its fibre has mass0 under this parameter. Step identifiers and dependenciesRule: sampling_complete_fibre_not_one_representative_or_null_deletion · Uses: G001, S009 S011 · calculationBefore sampling parameter0 statistic value2 full fibre mass := sum(1/4) After sampling parameter0 statistic value2 full fibre mass = 1/4 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore FIXED original parameter SLOT=0;statistic value=2;FULL original fibre outcome indices=[3];aligned law masses=[1/4] After pushforward mass at value2=1/4;retain this row even if mass0 Conditions: none Add masses from every outcome with the same statistic value,not one representative.Retain every value present in the original statistic even when its fibre has mass0 under this parameter. Step identifiers and dependenciesRule: sampling_complete_fibre_not_one_representative_or_null_deletion · Uses: G001, S011 S013 · calculationBefore sampling parameter0 complete pushforward mass := sum(3/8, 3/8, 1/4) After sampling parameter0 complete pushforward mass = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore fixed parameter SLOT=0;ALL increasingly ordered statistic values=[0, 1, 2];disjoint complete fibres=[[0, 1], [2], [3]] After pushforward law=[[0, 3/8], [1, 3/8], [2, 1/4]];total mass=1;ANY set of statistic values has preimage equal to the union of its fibres Conditions: none The disjoint fibres cover the entire common sample carrier,so the preimage probability of any set of statistic values is the sum of its rows.This gives the complete discrete sampling law,not observed frequencies or a distribution over parameters. Step identifiers and dependenciesRule: sampling_pushforward_full_partition_probability_law · Uses: G001, S013 S015 · impliesBefore supplied original outcome index=0;observed T(o)=0;SEPARATE full law at parameter SLOT=0 After estimate VALUE=0;sampling LAW=[[0, 3/8], [1, 3/8], [2, 1/4]];observed_estimate_is_not_sampling_distribution Conditions: none Read one number at the original outcome index.Do not replace the estimator function,the full sampling law or the unknown parameter with that number;coinciding numeric labels do not merge their roles. Step identifiers and dependenciesRule: sampling_observed_number_not_function_or_full_law · Uses: G001, S014 ## Reverse checks - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Recompute sampling parameter0 complete pushforward mass with exact rational arithmetic, not rounded decimals. - 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 sampling parameter0 statistic value2 full fibre mass with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute sampling parameter0 statistic value1 full fibre mass with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute sampling parameter0 statistic value0 full fibre mass 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 sampling parameter1 full 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 sampling parameter0 full 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, 0, [[0, 3/8], [1, 3/8], [2, 1/4]], observed_estimate_is_not_sampling_distribution) 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: baca53481c4c88d4f46e8c44e2cb294f13bd0527c31d2078d20b996494e2d738 - candidate record SHA-256: 27eb5b4d39907bd35fa64ba9e4188b387be0f02df2776606a78a100d8add0a52 - semantic review receipt SHA-256: 065507b602ceee60cf47a700d387a51940e0c6724c35a3336a7a7948a199ed80 - question-first witness SHA-256: 7203f33b030792559673b0fee520a8c0e70903c4dd970b35293aa40852f333d6 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261105-r06-sampling-distribution-0051.json)