第 25147 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":9,"support":[7,9,11],"probabilities":[[1,4],[1,2],[1,4]],"observed":7}。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":9,"support":[7,9,11],"probabilities":[[1,4],[1,2],[1,4]],"observed":7}。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, 9, [7, 9, 11], [[1, 4], [1, 2], [1, 4]]]

derived = [[[7, 1/4], [9, 1/2], [11, 1/4]], 1]

MSE ORIGINAL single theta exact mean product0 := mul(7, 1/4)

MSE ORIGINAL single theta exact mean product0 = 7/4

MSE ORIGINAL single theta exact mean cumulative0 := sum(0, 7/4)

MSE ORIGINAL single theta exact mean cumulative0 = 7/4

MSE ORIGINAL single theta exact mean product1 := mul(9, 1/2)

MSE ORIGINAL single theta exact mean product1 = 9/2

MSE ORIGINAL single theta exact mean cumulative1 := sum(7/4, 9/2)

MSE ORIGINAL single theta exact mean cumulative1 = 25/4

MSE ORIGINAL single theta exact mean product2 := mul(11, 1/4)

MSE ORIGINAL single theta exact mean product2 = 11/4

MSE ORIGINAL single theta exact mean cumulative2 := sum(25/4, 11/4)

MSE ORIGINAL single theta exact mean cumulative2 = 9

MSE ORIGINAL single theta error row0 := sub(7, 9)

MSE ORIGINAL single theta error row0 = -2

MSE ORIGINAL single theta squared error row0 := mul(-2, -2)

MSE ORIGINAL single theta squared error row0 = 4

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

MSE ORIGINAL single theta weighted error row0 = 1

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

MSE ORIGINAL single theta MSE cumulative0 = 1

MSE ORIGINAL single theta error row1 := sub(9, 9)

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, 0)

MSE ORIGINAL single theta MSE cumulative1 = 1

MSE ORIGINAL single theta error row2 := sub(11, 9)

MSE ORIGINAL single theta error row2 = 2

MSE ORIGINAL single theta squared error row2 := mul(2, 2)

MSE ORIGINAL single theta squared error row2 = 4

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

MSE ORIGINAL single theta weighted error row2 = 1

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

MSE ORIGINAL single theta MSE cumulative2 = 2

source roles = [MSE ORIGINAL single theta, 9, [[7, 1/4, 4, 1], [9, 1/2, 0, 0], [11, 1/4, 4, 1]]]

derived = [9, 2, 否, 否, 是]

MSE ORIGINAL observed minus theta := sub(7, 9)

MSE ORIGINAL observed minus theta = -2

MSE ORIGINAL one observed squared error := mul(-2, -2)

MSE ORIGINAL one observed squared error = 4

source roles = [9, 7, 4, 9, 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

(9, 7, 4, 9, 是, 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。