第 19759 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":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],"statistic":[0,1,10,11]}。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":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],"statistic":[0,1,10,11]}。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(1/18, 1/9, 1/6, 2/3)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/18, 1/9, 1/6, 2/3];ONE fixed statistic=[0, 1, 10, 11]

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

variance parameter1 complete law mass := sum(1/11, 3/22, 2/11, 13/22)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[1/11, 3/22, 2/11, 13/22];ONE fixed statistic=[0, 1, 10, 11]

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

variance parameter0 outcome0 weighted T := mul(1/18, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(1/9, 1)

variance parameter0 outcome1 weighted T = 1/9

variance parameter0 outcome2 weighted T := mul(1/6, 10)

variance parameter0 outcome2 weighted T = 5/3

variance parameter0 outcome3 weighted T := mul(2/3, 11)

variance parameter0 outcome3 weighted T = 22/3

variance parameter0 SAME-law mean := sum(0, 1/9, 5/3, 22/3)

variance parameter0 SAME-law mean = 82/9

variance parameter0 outcome0 T minus SAME mean := sub(0, 82/9)

variance parameter0 outcome0 T minus SAME mean = -82/9

variance parameter0 outcome0 centered square := mul(-82/9, -82/9)

variance parameter0 outcome0 centered square = 6724/81

variance parameter0 outcome0 weighted centered square := mul(1/18, 6724/81)

variance parameter0 outcome0 weighted centered square = 3362/729

variance parameter0 outcome1 T minus SAME mean := sub(1, 82/9)

variance parameter0 outcome1 T minus SAME mean = -73/9

variance parameter0 outcome1 centered square := mul(-73/9, -73/9)

variance parameter0 outcome1 centered square = 5329/81

variance parameter0 outcome1 weighted centered square := mul(1/9, 5329/81)

variance parameter0 outcome1 weighted centered square = 5329/729

variance parameter0 outcome2 T minus SAME mean := sub(10, 82/9)

variance parameter0 outcome2 T minus SAME mean = 8/9

variance parameter0 outcome2 centered square := mul(8/9, 8/9)

variance parameter0 outcome2 centered square = 64/81

variance parameter0 outcome2 weighted centered square := mul(1/6, 64/81)

variance parameter0 outcome2 weighted centered square = 32/243

variance parameter0 outcome3 T minus SAME mean := sub(11, 82/9)

variance parameter0 outcome3 T minus SAME mean = 17/9

variance parameter0 outcome3 centered square := mul(17/9, 17/9)

variance parameter0 outcome3 centered square = 289/81

variance parameter0 outcome3 weighted centered square := mul(2/3, 289/81)

variance parameter0 outcome3 weighted centered square = 578/243

variance parameter0 full centered variance := sum(3362/729, 5329/729, 32/243, 578/243)

variance parameter0 full centered variance = 1169/81

FIXED parameter SLOT=0;original same law=[1/18, 1/9, 1/6, 2/3];same statistic=[0, 1, 10, 11];its own mean=82/9;ALL centered deviations=[-82/9, -73/9, 8/9, 17/9]

ALL nonnegative weighted centered squares=[3362/729, 5329/729, 32/243, 578/243];variance=1169/81;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(1/11, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/22, 1)

variance parameter1 outcome1 weighted T = 3/22

variance parameter1 outcome2 weighted T := mul(2/11, 10)

variance parameter1 outcome2 weighted T = 20/11

variance parameter1 outcome3 weighted T := mul(13/22, 11)

variance parameter1 outcome3 weighted T = 13/2

variance parameter1 SAME-law mean := sum(0, 3/22, 20/11, 13/2)

variance parameter1 SAME-law mean = 93/11

variance parameter1 outcome0 T minus SAME mean := sub(0, 93/11)

variance parameter1 outcome0 T minus SAME mean = -93/11

variance parameter1 outcome0 centered square := mul(-93/11, -93/11)

variance parameter1 outcome0 centered square = 8649/121

variance parameter1 outcome0 weighted centered square := mul(1/11, 8649/121)

variance parameter1 outcome0 weighted centered square = 8649/1331

variance parameter1 outcome1 T minus SAME mean := sub(1, 93/11)

variance parameter1 outcome1 T minus SAME mean = -82/11

variance parameter1 outcome1 centered square := mul(-82/11, -82/11)

variance parameter1 outcome1 centered square = 6724/121

variance parameter1 outcome1 weighted centered square := mul(3/22, 6724/121)

variance parameter1 outcome1 weighted centered square = 10086/1331

variance parameter1 outcome2 T minus SAME mean := sub(10, 93/11)

variance parameter1 outcome2 T minus SAME mean = 17/11

variance parameter1 outcome2 centered square := mul(17/11, 17/11)

variance parameter1 outcome2 centered square = 289/121

variance parameter1 outcome2 weighted centered square := mul(2/11, 289/121)

variance parameter1 outcome2 weighted centered square = 578/1331

variance parameter1 outcome3 T minus SAME mean := sub(11, 93/11)

variance parameter1 outcome3 T minus SAME mean = 28/11

variance parameter1 outcome3 centered square := mul(28/11, 28/11)

variance parameter1 outcome3 centered square = 784/121

variance parameter1 outcome3 weighted centered square := mul(13/22, 784/121)

variance parameter1 outcome3 weighted centered square = 5096/1331

variance parameter1 full centered variance := sum(8649/1331, 10086/1331, 578/1331, 5096/1331)

variance parameter1 full centered variance = 2219/121

FIXED parameter SLOT=1;original same law=[1/11, 3/22, 2/11, 13/22];same statistic=[0, 1, 10, 11];its own mean=93/11;ALL centered deviations=[-93/11, -82/11, 17/11, 28/11]

ALL nonnegative weighted centered squares=[8649/1331, 10086/1331, 578/1331, 5096/1331];variance=2219/121;variance0 iff EVERY positive-mass outcome has T=THIS mean

ALL original parameter variance rows=[[0, 1169/81], [1, 2219/121]];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, 1169/81], [1, 2219/121]], 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。