第 19309 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-at-parameter；資料={"theta_values":[[1,2],[9,11]],"laws":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],"statistic":[0,1,10,11],"parameter":0}。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-at-parameter；資料={"theta_values":[[1,2],[9,11]],"laws":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],"statistic":[0,1,10,11],"parameter":0}。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-at-parameter;original snapshot and mathematics retained

explicit ordered source output fields=[parameter_index, theta_value, expectation, bias, certificate]

bias parameter0 full law mass := sum(1/18, 1/9, 1/6, 2/3)

bias parameter0 full law mass = 1

bias:original parameter SLOT=0;same outcome carrier Fin(4);ALL aligned law masses=[1/18, 1/9, 1/6, 2/3]

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

bias parameter1 full law mass := sum(1/11, 3/22, 2/11, 13/22)

bias parameter1 full law mass = 1

bias:original parameter SLOT=1;same outcome carrier Fin(4);ALL aligned law masses=[1/11, 3/22, 2/11, 13/22]

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, 10, 11]

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, 9/11]

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(1/18, 0)

bias parameter0 original outcome0 weighted statistic = 0

bias parameter0 original outcome1 weighted statistic := mul(1/9, 1)

bias parameter0 original outcome1 weighted statistic = 1/9

bias parameter0 original outcome2 weighted statistic := mul(1/6, 10)

bias parameter0 original outcome2 weighted statistic = 5/3

bias parameter0 original outcome3 weighted statistic := mul(2/3, 11)

bias parameter0 original outcome3 weighted statistic = 22/3

bias parameter0 same-law expectation := sum(0, 1/9, 5/3, 22/3)

bias parameter0 same-law expectation = 82/9

bias parameter0 expectation minus actual target := sub(82/9, 1/2)

bias parameter0 expectation minus actual target = 155/18

fixed original parameter SLOT=0;actual aligned theta VALUE=1/2;SAME complete law=[1/18, 1/9, 1/6, 2/3];same statistic=[0, 1, 10, 11];weighted terms=[0, 1/9, 5/3, 22/3]

E_at_slot[T]=82/9;Bias=E_at_slot[T]-ACTUAL theta value=155/18

逐步反向核對

• 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, 1/2, 82/9, 155/18, expectation_from_same_theta_sampling_distribution)

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。