第 19400 題：估計量偏誤

原題與已知條件

共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={"theta_values":[[1,2],[25,27]],"laws":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],"statistic":[0,1,26,27]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={"theta_values":[[1,2],[25,27]],"laws":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],"statistic":[0,1,26,27]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-estimator-bias

逐步推導

source operation=bias-unbiased-audit;original snapshot and mathematics retained

explicit ordered source output fields=[parameter_biases, unbiased_all_parameters, boundary]

bias parameter0 full law mass := sum(1/34, 1/17, 3/34, 14/17)

bias parameter0 full law mass = 1

bias:original parameter SLOT=0;same outcome carrier Fin(4);ALL aligned law masses=[1/34, 1/17, 3/34, 14/17]

full sum=1;all masses nonnegative=是;qualified probability law;zero-mass outcomes retained

bias parameter1 full law mass := sum(1/19, 3/38, 2/19, 29/38)

bias parameter1 full law mass = 1

bias:original parameter SLOT=1;same outcome carrier Fin(4);ALL aligned law masses=[1/19, 3/38, 2/19, 29/38]

full sum=1;all masses nonnegative=是;qualified probability law;zero-mass outcomes retained

bias:parameter SLOT carrier Fin(2);common outcome carrier Fin(4);ONE total original statistic array=[0, 1, 26, 27]

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

original parameter SLOT order0..1;aligned exact theta_values=[1/2, 25/27]

declared target g(theta)=theta;indices are not targets;claims range over THIS supplied finite parameter model only

bias parameter0 original outcome0 weighted statistic := mul(1/34, 0)

bias parameter0 original outcome0 weighted statistic = 0

bias parameter0 original outcome1 weighted statistic := mul(1/17, 1)

bias parameter0 original outcome1 weighted statistic = 1/17

bias parameter0 original outcome2 weighted statistic := mul(3/34, 26)

bias parameter0 original outcome2 weighted statistic = 39/17

bias parameter0 original outcome3 weighted statistic := mul(14/17, 27)

bias parameter0 original outcome3 weighted statistic = 378/17

bias parameter0 same-law expectation := sum(0, 1/17, 39/17, 378/17)

bias parameter0 same-law expectation = 418/17

bias parameter0 expectation minus actual target := sub(418/17, 1/2)

bias parameter0 expectation minus actual target = 819/34

fixed original parameter SLOT=0;actual aligned theta VALUE=1/2;SAME complete law=[1/34, 1/17, 3/34, 14/17];same statistic=[0, 1, 26, 27];weighted terms=[0, 1/17, 39/17, 378/17]

E_at_slot[T]=418/17;Bias=E_at_slot[T]-ACTUAL theta value=819/34

bias parameter1 original outcome0 weighted statistic := mul(1/19, 0)

bias parameter1 original outcome0 weighted statistic = 0

bias parameter1 original outcome1 weighted statistic := mul(3/38, 1)

bias parameter1 original outcome1 weighted statistic = 3/38

bias parameter1 original outcome2 weighted statistic := mul(2/19, 26)

bias parameter1 original outcome2 weighted statistic = 52/19

bias parameter1 original outcome3 weighted statistic := mul(29/38, 27)

bias parameter1 original outcome3 weighted statistic = 783/38

bias parameter1 same-law expectation := sum(0, 3/38, 52/19, 783/38)

bias parameter1 same-law expectation = 445/19

bias parameter1 expectation minus actual target := sub(445/19, 25/27)

bias parameter1 expectation minus actual target = 11540/513

fixed original parameter SLOT=1;actual aligned theta VALUE=25/27;SAME complete law=[1/19, 3/38, 2/19, 29/38];same statistic=[0, 1, 26, 27];weighted terms=[0, 3/38, 52/19, 783/38]

E_at_slot[T]=445/19;Bias=E_at_slot[T]-ACTUAL theta value=11540/513

ALL original parameter bias rows=[[0, 819/34], [1, 11540/513]]

every declared parameter bias zero=否;unbiased_requires_every_theta

逐步反向核對

• S022 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S021 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S020 · premise_and_result_audit — 以精確有理數重算 bias parameter1 expectation minus actual target，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 bias parameter1 same-law expectation，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 bias parameter1 original outcome3 weighted statistic，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 bias parameter1 original outcome2 weighted statistic，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 bias parameter1 original outcome1 weighted statistic，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 bias parameter1 original outcome0 weighted statistic，不以小數近似替代。
• S014 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S013 · premise_and_result_audit — 以精確有理數重算 bias parameter0 expectation minus actual target，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 bias parameter0 same-law expectation，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 bias parameter0 original outcome3 weighted statistic，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 bias parameter0 original outcome2 weighted statistic，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 bias parameter0 original outcome1 weighted statistic，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 bias parameter0 original outcome0 weighted statistic，不以小數近似替代。
• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 以精確有理數重算 bias parameter1 full law mass，不以小數近似替代。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 以精確有理數重算 bias parameter0 full law mass，不以小數近似替代。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

([[0, 819/34], [1, 11540/513]], 否, unbiased_requires_every_theta)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。