第 19771 題：估計量變異數

原題與已知條件

共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={"laws":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],"statistic":[0,1,22,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={"laws":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],"statistic":[0,1,22,23]}。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-variance

逐步推導

source operation=estvar-property-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary]

variance parameter0 complete law mass := sum(1/30, 1/15, 1/10, 4/5)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/30, 1/15, 1/10, 4/5];ONE fixed statistic=[0, 1, 22, 23]

complete same-law mass=1;all masses nonnegative;positive-mass support=[0, 1, 2, 3];null outcomes RETAINED

variance parameter1 complete law mass := sum(1/17, 3/34, 2/17, 25/34)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[1/17, 3/34, 2/17, 25/34];ONE fixed statistic=[0, 1, 22, 23]

complete same-law mass=1;all masses nonnegative;positive-mass support=[0, 1, 2, 3];null outcomes RETAINED

variance parameter0 outcome0 weighted T := mul(1/30, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(1/15, 1)

variance parameter0 outcome1 weighted T = 1/15

variance parameter0 outcome2 weighted T := mul(1/10, 22)

variance parameter0 outcome2 weighted T = 11/5

variance parameter0 outcome3 weighted T := mul(4/5, 23)

variance parameter0 outcome3 weighted T = 92/5

variance parameter0 SAME-law mean := sum(0, 1/15, 11/5, 92/5)

variance parameter0 SAME-law mean = 62/3

variance parameter0 outcome0 T minus SAME mean := sub(0, 62/3)

variance parameter0 outcome0 T minus SAME mean = -62/3

variance parameter0 outcome0 centered square := mul(-62/3, -62/3)

variance parameter0 outcome0 centered square = 3844/9

variance parameter0 outcome0 weighted centered square := mul(1/30, 3844/9)

variance parameter0 outcome0 weighted centered square = 1922/135

variance parameter0 outcome1 T minus SAME mean := sub(1, 62/3)

variance parameter0 outcome1 T minus SAME mean = -59/3

variance parameter0 outcome1 centered square := mul(-59/3, -59/3)

variance parameter0 outcome1 centered square = 3481/9

variance parameter0 outcome1 weighted centered square := mul(1/15, 3481/9)

variance parameter0 outcome1 weighted centered square = 3481/135

variance parameter0 outcome2 T minus SAME mean := sub(22, 62/3)

variance parameter0 outcome2 T minus SAME mean = 4/3

variance parameter0 outcome2 centered square := mul(4/3, 4/3)

variance parameter0 outcome2 centered square = 16/9

variance parameter0 outcome2 weighted centered square := mul(1/10, 16/9)

variance parameter0 outcome2 weighted centered square = 8/45

variance parameter0 outcome3 T minus SAME mean := sub(23, 62/3)

variance parameter0 outcome3 T minus SAME mean = 7/3

variance parameter0 outcome3 centered square := mul(7/3, 7/3)

variance parameter0 outcome3 centered square = 49/9

variance parameter0 outcome3 weighted centered square := mul(4/5, 49/9)

variance parameter0 outcome3 weighted centered square = 196/45

variance parameter0 full centered variance := sum(1922/135, 3481/135, 8/45, 196/45)

variance parameter0 full centered variance = 401/9

FIXED parameter SLOT=0;original same law=[1/30, 1/15, 1/10, 4/5];same statistic=[0, 1, 22, 23];its own mean=62/3;ALL centered deviations=[-62/3, -59/3, 4/3, 7/3]

ALL nonnegative weighted centered squares=[1922/135, 3481/135, 8/45, 196/45];variance=401/9;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(1/17, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/34, 1)

variance parameter1 outcome1 weighted T = 3/34

variance parameter1 outcome2 weighted T := mul(2/17, 22)

variance parameter1 outcome2 weighted T = 44/17

variance parameter1 outcome3 weighted T := mul(25/34, 23)

variance parameter1 outcome3 weighted T = 575/34

variance parameter1 SAME-law mean := sum(0, 3/34, 44/17, 575/34)

variance parameter1 SAME-law mean = 333/17

variance parameter1 outcome0 T minus SAME mean := sub(0, 333/17)

variance parameter1 outcome0 T minus SAME mean = -333/17

variance parameter1 outcome0 centered square := mul(-333/17, -333/17)

variance parameter1 outcome0 centered square = 110889/289

variance parameter1 outcome0 weighted centered square := mul(1/17, 110889/289)

variance parameter1 outcome0 weighted centered square = 110889/4913

variance parameter1 outcome1 T minus SAME mean := sub(1, 333/17)

variance parameter1 outcome1 T minus SAME mean = -316/17

variance parameter1 outcome1 centered square := mul(-316/17, -316/17)

variance parameter1 outcome1 centered square = 99856/289

variance parameter1 outcome1 weighted centered square := mul(3/34, 99856/289)

variance parameter1 outcome1 weighted centered square = 149784/4913

variance parameter1 outcome2 T minus SAME mean := sub(22, 333/17)

variance parameter1 outcome2 T minus SAME mean = 41/17

variance parameter1 outcome2 centered square := mul(41/17, 41/17)

variance parameter1 outcome2 centered square = 1681/289

variance parameter1 outcome2 weighted centered square := mul(2/17, 1681/289)

variance parameter1 outcome2 weighted centered square = 3362/4913

variance parameter1 outcome3 T minus SAME mean := sub(23, 333/17)

variance parameter1 outcome3 T minus SAME mean = 58/17

variance parameter1 outcome3 centered square := mul(58/17, 58/17)

variance parameter1 outcome3 centered square = 3364/289

variance parameter1 outcome3 weighted centered square := mul(25/34, 3364/289)

variance parameter1 outcome3 weighted centered square = 42050/4913

variance parameter1 full centered variance := sum(110889/4913, 149784/4913, 3362/4913, 42050/4913)

variance parameter1 full centered variance = 18005/289

FIXED parameter SLOT=1;original same law=[1/17, 3/34, 2/17, 25/34];same statistic=[0, 1, 22, 23];its own mean=333/17;ALL centered deviations=[-333/17, -316/17, 41/17, 58/17]

ALL nonnegative weighted centered squares=[110889/4913, 149784/4913, 3362/4913, 42050/4913];variance=18005/289;variance0 iff EVERY positive-mass outcome has T=THIS mean

ALL original parameter variance rows=[[0, 401/9], [1, 18005/289]];no target,bias or sequence of statistics supplied

three literal No markers WITHHOLD unbiasedness,consistency or MSE claims;not proved negations and NOT an assumption all listed variances0

逐步反向核對

• S044 · 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。
• S043 · 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。
• S042 · premise_and_result_audit — 以精確有理數重算 variance parameter1 full centered variance，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome3 weighted centered square，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome3 centered square，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome3 T minus SAME mean，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome2 weighted centered square，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome2 centered square，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome2 T minus SAME mean，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome1 weighted centered square，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome1 centered square，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome1 T minus SAME mean，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome0 weighted centered square，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome0 centered square，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome0 T minus SAME mean，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 variance parameter1 SAME-law mean，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome3 weighted T，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome2 weighted T，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome1 weighted T，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 variance parameter1 outcome0 weighted T，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 variance parameter0 full centered variance，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome3 weighted centered square，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome3 centered square，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome3 T minus SAME mean，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome2 weighted centered square，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome2 centered square，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome2 T minus SAME mean，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome1 weighted centered square，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome1 centered square，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome1 T minus SAME mean，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome0 weighted centered square，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome0 centered square，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome0 T minus SAME mean，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 variance parameter0 SAME-law mean，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome3 weighted T，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome2 weighted T，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome1 weighted T，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 variance parameter0 outcome0 weighted T，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 variance parameter1 complete 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 variance parameter0 complete 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, 401/9], [1, 18005/289]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)

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。