第 25141 題：估計量均方誤差

原題與已知條件

scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={"theta":3,"support":[2,3,4],"probabilities":[[1,4],[1,2],[1,4]],"observed":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={"theta":3,"support":[2,3,4],"probabilities":[[1,4],[1,2],[1,4]],"observed":2}。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-mean-squared-error

逐步推導

source roles = [mse-observed-unbiased-boundary]

derived = [[theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary]]

MSE ORIGINAL single theta ORIGINAL probability cumulative0 := sum(0, 1/4)

MSE ORIGINAL single theta ORIGINAL probability cumulative0 = 1/4

MSE ORIGINAL single theta ORIGINAL probability cumulative1 := sum(1/4, 1/2)

MSE ORIGINAL single theta ORIGINAL probability cumulative1 = 3/4

MSE ORIGINAL single theta ORIGINAL probability cumulative2 := sum(3/4, 1/4)

MSE ORIGINAL single theta ORIGINAL probability cumulative2 = 1

source roles = [MSE ORIGINAL single theta, 3, [2, 3, 4], [[1, 4], [1, 2], [1, 4]]]

derived = [[[2, 1/4], [3, 1/2], [4, 1/4]], 1]

MSE ORIGINAL single theta exact mean product0 := mul(2, 1/4)

MSE ORIGINAL single theta exact mean product0 = 1/2

MSE ORIGINAL single theta exact mean cumulative0 := sum(0, 1/2)

MSE ORIGINAL single theta exact mean cumulative0 = 1/2

MSE ORIGINAL single theta exact mean product1 := mul(3, 1/2)

MSE ORIGINAL single theta exact mean product1 = 3/2

MSE ORIGINAL single theta exact mean cumulative1 := sum(1/2, 3/2)

MSE ORIGINAL single theta exact mean cumulative1 = 2

MSE ORIGINAL single theta exact mean product2 := mul(4, 1/4)

MSE ORIGINAL single theta exact mean product2 = 1

MSE ORIGINAL single theta exact mean cumulative2 := sum(2, 1)

MSE ORIGINAL single theta exact mean cumulative2 = 3

MSE ORIGINAL single theta error row0 := sub(2, 3)

MSE ORIGINAL single theta error row0 = -1

MSE ORIGINAL single theta squared error row0 := mul(-1, -1)

MSE ORIGINAL single theta squared error row0 = 1

MSE ORIGINAL single theta weighted error row0 := mul(1/4, 1)

MSE ORIGINAL single theta weighted error row0 = 1/4

MSE ORIGINAL single theta MSE cumulative0 := sum(0, 1/4)

MSE ORIGINAL single theta MSE cumulative0 = 1/4

MSE ORIGINAL single theta error row1 := sub(3, 3)

MSE ORIGINAL single theta error row1 = 0

MSE ORIGINAL single theta squared error row1 := mul(0, 0)

MSE ORIGINAL single theta squared error row1 = 0

MSE ORIGINAL single theta weighted error row1 := mul(1/2, 0)

MSE ORIGINAL single theta weighted error row1 = 0

MSE ORIGINAL single theta MSE cumulative1 := sum(1/4, 0)

MSE ORIGINAL single theta MSE cumulative1 = 1/4

MSE ORIGINAL single theta error row2 := sub(4, 3)

MSE ORIGINAL single theta error row2 = 1

MSE ORIGINAL single theta squared error row2 := mul(1, 1)

MSE ORIGINAL single theta squared error row2 = 1

MSE ORIGINAL single theta weighted error row2 := mul(1/4, 1)

MSE ORIGINAL single theta weighted error row2 = 1/4

MSE ORIGINAL single theta MSE cumulative2 := sum(1/4, 1/4)

MSE ORIGINAL single theta MSE cumulative2 = 1/2

source roles = [MSE ORIGINAL single theta, 3, [[2, 1/4, 1, 1/4], [3, 1/2, 0, 0], [4, 1/4, 1, 1/4]]]

derived = [3, 1/2, 否, 否, 是]

MSE ORIGINAL observed minus theta := sub(2, 3)

MSE ORIGINAL observed minus theta = -1

MSE ORIGINAL one observed squared error := mul(-1, -1)

MSE ORIGINAL one observed squared error = 1

source roles = [3, 2, 1, 3, 1/2]

derived = [是, 否, 是, 否]

逐步反向核對

• S027 · 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。
• S026 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL one observed squared error，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL observed minus theta，不以小數近似替代。
• S024 · 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。
• S023 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta MSE cumulative2，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta weighted error row2，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta squared error row2，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta error row2，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta MSE cumulative1，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta weighted error row1，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta squared error row1，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta error row1，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta MSE cumulative0，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta weighted error row0，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta squared error row0，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta error row0，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta exact mean cumulative2，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta exact mean product2，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta exact mean cumulative1，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta exact mean product1，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta exact mean cumulative0，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta exact mean product0，不以小數近似替代。
• 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 MSE ORIGINAL single theta ORIGINAL probability cumulative2，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta ORIGINAL probability cumulative1，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 MSE ORIGINAL single theta ORIGINAL probability cumulative0，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

(3, 2, 1, 3, 是, 1/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)

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。