第 25590 題：偏誤－變異數分解

原題與已知條件

逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={"theta":2,"support":[2,5,8],"probabilities":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={"theta":2,"support":[2,5,8],"probabilities":[[1,4],[1,2],[1,4]]}。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-bias-variance-decomposition

逐步推導

source roles = [biasvar-single-theta-identity]

derived = [[theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate]]

Biasvar ORIGINAL single theta ORIGINAL mass sum := sum(1/4, 1/2, 1/4)

Biasvar ORIGINAL single theta ORIGINAL mass sum = 1

source roles = [Biasvar ORIGINAL single theta, 2, [2, 5, 8], [[1, 4], [1, 2], [1, 4]]]

derived = [[(2, 1/4), (5, 1/2), (8, 1/4)], 1]

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

Biasvar ORIGINAL single theta mean product0 = 1/2

Biasvar ORIGINAL single theta mean product1 := mul(5, 1/2)

Biasvar ORIGINAL single theta mean product1 = 5/2

Biasvar ORIGINAL single theta mean product2 := mul(8, 1/4)

Biasvar ORIGINAL single theta mean product2 = 2

Biasvar ORIGINAL single theta mean sum := sum(1/2, 5/2, 2)

Biasvar ORIGINAL single theta mean sum = 5

Biasvar ORIGINAL single theta bias mean minus theta := sub(5, 2)

Biasvar ORIGINAL single theta bias mean minus theta = 3

Biasvar ORIGINAL single theta centered deviation0 := sub(2, 5)

Biasvar ORIGINAL single theta centered deviation0 = -3

Biasvar ORIGINAL single theta centered square0 := mul(-3, -3)

Biasvar ORIGINAL single theta centered square0 = 9

Biasvar ORIGINAL single theta variance weighted term0 := mul(1/4, 9)

Biasvar ORIGINAL single theta variance weighted term0 = 9/4

Biasvar ORIGINAL single theta centered deviation1 := sub(5, 5)

Biasvar ORIGINAL single theta centered deviation1 = 0

Biasvar ORIGINAL single theta centered square1 := mul(0, 0)

Biasvar ORIGINAL single theta centered square1 = 0

Biasvar ORIGINAL single theta variance weighted term1 := mul(1/2, 0)

Biasvar ORIGINAL single theta variance weighted term1 = 0

Biasvar ORIGINAL single theta centered deviation2 := sub(8, 5)

Biasvar ORIGINAL single theta centered deviation2 = 3

Biasvar ORIGINAL single theta centered square2 := mul(3, 3)

Biasvar ORIGINAL single theta centered square2 = 9

Biasvar ORIGINAL single theta variance weighted term2 := mul(1/4, 9)

Biasvar ORIGINAL single theta variance weighted term2 = 9/4

Biasvar ORIGINAL single theta variance sum := sum(9/4, 0, 9/4)

Biasvar ORIGINAL single theta variance sum = 9/2

Biasvar ORIGINAL single theta theta error0 := sub(2, 2)

Biasvar ORIGINAL single theta theta error0 = 0

Biasvar ORIGINAL single theta theta square0 := mul(0, 0)

Biasvar ORIGINAL single theta theta square0 = 0

Biasvar ORIGINAL single theta direct MSE term0 := mul(1/4, 0)

Biasvar ORIGINAL single theta direct MSE term0 = 0

Biasvar ORIGINAL single theta theta error1 := sub(5, 2)

Biasvar ORIGINAL single theta theta error1 = 3

Biasvar ORIGINAL single theta theta square1 := mul(3, 3)

Biasvar ORIGINAL single theta theta square1 = 9

Biasvar ORIGINAL single theta direct MSE term1 := mul(1/2, 9)

Biasvar ORIGINAL single theta direct MSE term1 = 9/2

Biasvar ORIGINAL single theta theta error2 := sub(8, 2)

Biasvar ORIGINAL single theta theta error2 = 6

Biasvar ORIGINAL single theta theta square2 := mul(6, 6)

Biasvar ORIGINAL single theta theta square2 = 36

Biasvar ORIGINAL single theta direct MSE term2 := mul(1/4, 36)

Biasvar ORIGINAL single theta direct MSE term2 = 9

Biasvar ORIGINAL single theta direct MSE sum := sum(0, 9/2, 9)

Biasvar ORIGINAL single theta direct MSE sum = 27/2

Biasvar ORIGINAL single theta centered mean product0 := mul(1/4, -3)

Biasvar ORIGINAL single theta centered mean product0 = -3/4

Biasvar ORIGINAL single theta centered mean product1 := mul(1/2, 0)

Biasvar ORIGINAL single theta centered mean product1 = 0

Biasvar ORIGINAL single theta centered mean product2 := mul(1/4, 3)

Biasvar ORIGINAL single theta centered mean product2 = 3/4

Biasvar ORIGINAL single theta centered mean sum := sum(-3/4, 0, 3/4)

Biasvar ORIGINAL single theta centered mean sum = 0

Biasvar ORIGINAL single theta bias times centered mean := mul(3, 0)

Biasvar ORIGINAL single theta bias times centered mean = 0

Biasvar ORIGINAL single theta cross twice inner := mul(2, 0)

Biasvar ORIGINAL single theta cross twice inner = 0

Biasvar ORIGINAL single theta squared bias := mul(3, 3)

Biasvar ORIGINAL single theta squared bias = 9

Biasvar ORIGINAL single theta variance plus squared bias := sum(9/2, 9)

Biasvar ORIGINAL single theta variance plus squared bias = 27/2

source roles = [Biasvar ORIGINAL single theta, 2, 5, 3, [(2, 1/4), (5, 1/2), (8, 1/4)]]

derived = [[[2, 1/4, -3, 3, 0, 9, -18, 9, 0], [5, 1/2, 0, 3, 3, 0, 0, 9, 9], [8, 1/4, 3, 3, 6, 9, 18, 9, 36]], 0, 0, 9/2, 27/2, 27/2, 是]

逐步反向核對

• S037 · 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。
• S036 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta variance plus squared bias，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta squared bias，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta cross twice inner，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta bias times centered mean，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta centered mean sum，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta centered mean product2，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta centered mean product1，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta centered mean product0，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta direct MSE sum，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta direct MSE term2，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta theta square2，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta theta error2，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta direct MSE term1，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta theta square1，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta theta error1，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta direct MSE term0，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta theta square0，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta theta error0，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta variance sum，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta variance weighted term2，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta centered square2，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta centered deviation2，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta variance weighted term1，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta centered square1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta centered deviation1，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta variance weighted term0，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta centered square0，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta centered deviation0，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta bias mean minus theta，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta mean sum，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta mean product2，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta mean product1，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL single theta mean product0，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 Biasvar ORIGINAL single theta ORIGINAL mass sum，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

(2, [[2, 1/4], [5, 1/2], [8, 1/4]], 5, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)

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。