第 19760 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":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],"statistic":[0,1,11,12]}。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":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],"statistic":[0,1,11,12]}。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(1/19, 2/19, 3/19, 13/19)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/19, 2/19, 3/19, 13/19];ONE fixed statistic=[0, 1, 11, 12]

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

variance parameter1 complete law mass := sum(2/23, 3/23, 4/23, 14/23)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[2/23, 3/23, 4/23, 14/23];ONE fixed statistic=[0, 1, 11, 12]

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

variance parameter0 outcome0 weighted T := mul(1/19, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(2/19, 1)

variance parameter0 outcome1 weighted T = 2/19

variance parameter0 outcome2 weighted T := mul(3/19, 11)

variance parameter0 outcome2 weighted T = 33/19

variance parameter0 outcome3 weighted T := mul(13/19, 12)

variance parameter0 outcome3 weighted T = 156/19

variance parameter0 SAME-law mean := sum(0, 2/19, 33/19, 156/19)

variance parameter0 SAME-law mean = 191/19

variance parameter0 outcome0 T minus SAME mean := sub(0, 191/19)

variance parameter0 outcome0 T minus SAME mean = -191/19

variance parameter0 outcome0 centered square := mul(-191/19, -191/19)

variance parameter0 outcome0 centered square = 36481/361

variance parameter0 outcome0 weighted centered square := mul(1/19, 36481/361)

variance parameter0 outcome0 weighted centered square = 36481/6859

variance parameter0 outcome1 T minus SAME mean := sub(1, 191/19)

variance parameter0 outcome1 T minus SAME mean = -172/19

variance parameter0 outcome1 centered square := mul(-172/19, -172/19)

variance parameter0 outcome1 centered square = 29584/361

variance parameter0 outcome1 weighted centered square := mul(2/19, 29584/361)

variance parameter0 outcome1 weighted centered square = 59168/6859

variance parameter0 outcome2 T minus SAME mean := sub(11, 191/19)

variance parameter0 outcome2 T minus SAME mean = 18/19

variance parameter0 outcome2 centered square := mul(18/19, 18/19)

variance parameter0 outcome2 centered square = 324/361

variance parameter0 outcome2 weighted centered square := mul(3/19, 324/361)

variance parameter0 outcome2 weighted centered square = 972/6859

variance parameter0 outcome3 T minus SAME mean := sub(12, 191/19)

variance parameter0 outcome3 T minus SAME mean = 37/19

variance parameter0 outcome3 centered square := mul(37/19, 37/19)

variance parameter0 outcome3 centered square = 1369/361

variance parameter0 outcome3 weighted centered square := mul(13/19, 1369/361)

variance parameter0 outcome3 weighted centered square = 17797/6859

variance parameter0 full centered variance := sum(36481/6859, 59168/6859, 972/6859, 17797/6859)

variance parameter0 full centered variance = 6022/361

FIXED parameter SLOT=0;original same law=[1/19, 2/19, 3/19, 13/19];same statistic=[0, 1, 11, 12];its own mean=191/19;ALL centered deviations=[-191/19, -172/19, 18/19, 37/19]

ALL nonnegative weighted centered squares=[36481/6859, 59168/6859, 972/6859, 17797/6859];variance=6022/361;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(2/23, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/23, 1)

variance parameter1 outcome1 weighted T = 3/23

variance parameter1 outcome2 weighted T := mul(4/23, 11)

variance parameter1 outcome2 weighted T = 44/23

variance parameter1 outcome3 weighted T := mul(14/23, 12)

variance parameter1 outcome3 weighted T = 168/23

variance parameter1 SAME-law mean := sum(0, 3/23, 44/23, 168/23)

variance parameter1 SAME-law mean = 215/23

variance parameter1 outcome0 T minus SAME mean := sub(0, 215/23)

variance parameter1 outcome0 T minus SAME mean = -215/23

variance parameter1 outcome0 centered square := mul(-215/23, -215/23)

variance parameter1 outcome0 centered square = 46225/529

variance parameter1 outcome0 weighted centered square := mul(2/23, 46225/529)

variance parameter1 outcome0 weighted centered square = 92450/12167

variance parameter1 outcome1 T minus SAME mean := sub(1, 215/23)

variance parameter1 outcome1 T minus SAME mean = -192/23

variance parameter1 outcome1 centered square := mul(-192/23, -192/23)

variance parameter1 outcome1 centered square = 36864/529

variance parameter1 outcome1 weighted centered square := mul(3/23, 36864/529)

variance parameter1 outcome1 weighted centered square = 110592/12167

variance parameter1 outcome2 T minus SAME mean := sub(11, 215/23)

variance parameter1 outcome2 T minus SAME mean = 38/23

variance parameter1 outcome2 centered square := mul(38/23, 38/23)

variance parameter1 outcome2 centered square = 1444/529

variance parameter1 outcome2 weighted centered square := mul(4/23, 1444/529)

variance parameter1 outcome2 weighted centered square = 5776/12167

variance parameter1 outcome3 T minus SAME mean := sub(12, 215/23)

variance parameter1 outcome3 T minus SAME mean = 61/23

variance parameter1 outcome3 centered square := mul(61/23, 61/23)

variance parameter1 outcome3 centered square = 3721/529

variance parameter1 outcome3 weighted centered square := mul(14/23, 3721/529)

variance parameter1 outcome3 weighted centered square = 52094/12167

variance parameter1 full centered variance := sum(92450/12167, 110592/12167, 5776/12167, 52094/12167)

variance parameter1 full centered variance = 11344/529

FIXED parameter SLOT=1;original same law=[2/23, 3/23, 4/23, 14/23];same statistic=[0, 1, 11, 12];its own mean=215/23;ALL centered deviations=[-215/23, -192/23, 38/23, 61/23]

ALL nonnegative weighted centered squares=[92450/12167, 110592/12167, 5776/12167, 52094/12167];variance=11344/529;variance0 iff EVERY positive-mass outcome has T=THIS mean

ALL original parameter variance rows=[[0, 6022/361], [1, 11344/529]];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, 6022/361], [1, 11344/529]], 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。