第 19770 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":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],"statistic":[0,1,21,22]}。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":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],"statistic":[0,1,21,22]}。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(1/29, 2/29, 3/29, 23/29)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/29, 2/29, 3/29, 23/29];ONE fixed statistic=[0, 1, 21, 22]

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

variance parameter1 complete law mass := sum(2/33, 1/11, 4/33, 8/11)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[2/33, 1/11, 4/33, 8/11];ONE fixed statistic=[0, 1, 21, 22]

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

variance parameter0 outcome0 weighted T := mul(1/29, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(2/29, 1)

variance parameter0 outcome1 weighted T = 2/29

variance parameter0 outcome2 weighted T := mul(3/29, 21)

variance parameter0 outcome2 weighted T = 63/29

variance parameter0 outcome3 weighted T := mul(23/29, 22)

variance parameter0 outcome3 weighted T = 506/29

variance parameter0 SAME-law mean := sum(0, 2/29, 63/29, 506/29)

variance parameter0 SAME-law mean = 571/29

variance parameter0 outcome0 T minus SAME mean := sub(0, 571/29)

variance parameter0 outcome0 T minus SAME mean = -571/29

variance parameter0 outcome0 centered square := mul(-571/29, -571/29)

variance parameter0 outcome0 centered square = 326041/841

variance parameter0 outcome0 weighted centered square := mul(1/29, 326041/841)

variance parameter0 outcome0 weighted centered square = 326041/24389

variance parameter0 outcome1 T minus SAME mean := sub(1, 571/29)

variance parameter0 outcome1 T minus SAME mean = -542/29

variance parameter0 outcome1 centered square := mul(-542/29, -542/29)

variance parameter0 outcome1 centered square = 293764/841

variance parameter0 outcome1 weighted centered square := mul(2/29, 293764/841)

variance parameter0 outcome1 weighted centered square = 587528/24389

variance parameter0 outcome2 T minus SAME mean := sub(21, 571/29)

variance parameter0 outcome2 T minus SAME mean = 38/29

variance parameter0 outcome2 centered square := mul(38/29, 38/29)

variance parameter0 outcome2 centered square = 1444/841

variance parameter0 outcome2 weighted centered square := mul(3/29, 1444/841)

variance parameter0 outcome2 weighted centered square = 4332/24389

variance parameter0 outcome3 T minus SAME mean := sub(22, 571/29)

variance parameter0 outcome3 T minus SAME mean = 67/29

variance parameter0 outcome3 centered square := mul(67/29, 67/29)

variance parameter0 outcome3 centered square = 4489/841

variance parameter0 outcome3 weighted centered square := mul(23/29, 4489/841)

variance parameter0 outcome3 weighted centered square = 103247/24389

variance parameter0 full centered variance := sum(326041/24389, 587528/24389, 4332/24389, 103247/24389)

variance parameter0 full centered variance = 35212/841

FIXED parameter SLOT=0;original same law=[1/29, 2/29, 3/29, 23/29];same statistic=[0, 1, 21, 22];its own mean=571/29;ALL centered deviations=[-571/29, -542/29, 38/29, 67/29]

ALL nonnegative weighted centered squares=[326041/24389, 587528/24389, 4332/24389, 103247/24389];variance=35212/841;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(2/33, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(1/11, 1)

variance parameter1 outcome1 weighted T = 1/11

variance parameter1 outcome2 weighted T := mul(4/33, 21)

variance parameter1 outcome2 weighted T = 28/11

variance parameter1 outcome3 weighted T := mul(8/11, 22)

variance parameter1 outcome3 weighted T = 16

variance parameter1 SAME-law mean := sum(0, 1/11, 28/11, 16)

variance parameter1 SAME-law mean = 205/11

variance parameter1 outcome0 T minus SAME mean := sub(0, 205/11)

variance parameter1 outcome0 T minus SAME mean = -205/11

variance parameter1 outcome0 centered square := mul(-205/11, -205/11)

variance parameter1 outcome0 centered square = 42025/121

variance parameter1 outcome0 weighted centered square := mul(2/33, 42025/121)

variance parameter1 outcome0 weighted centered square = 84050/3993

variance parameter1 outcome1 T minus SAME mean := sub(1, 205/11)

variance parameter1 outcome1 T minus SAME mean = -194/11

variance parameter1 outcome1 centered square := mul(-194/11, -194/11)

variance parameter1 outcome1 centered square = 37636/121

variance parameter1 outcome1 weighted centered square := mul(1/11, 37636/121)

variance parameter1 outcome1 weighted centered square = 37636/1331

variance parameter1 outcome2 T minus SAME mean := sub(21, 205/11)

variance parameter1 outcome2 T minus SAME mean = 26/11

variance parameter1 outcome2 centered square := mul(26/11, 26/11)

variance parameter1 outcome2 centered square = 676/121

variance parameter1 outcome2 weighted centered square := mul(4/33, 676/121)

variance parameter1 outcome2 weighted centered square = 2704/3993

variance parameter1 outcome3 T minus SAME mean := sub(22, 205/11)

variance parameter1 outcome3 T minus SAME mean = 37/11

variance parameter1 outcome3 centered square := mul(37/11, 37/11)

variance parameter1 outcome3 centered square = 1369/121

variance parameter1 outcome3 weighted centered square := mul(8/11, 1369/121)

variance parameter1 outcome3 weighted centered square = 10952/1331

variance parameter1 full centered variance := sum(84050/3993, 37636/1331, 2704/3993, 10952/1331)

variance parameter1 full centered variance = 7046/121

FIXED parameter SLOT=1;original same law=[2/33, 1/11, 4/33, 8/11];same statistic=[0, 1, 21, 22];its own mean=205/11;ALL centered deviations=[-205/11, -194/11, 26/11, 37/11]

ALL nonnegative weighted centered squares=[84050/3993, 37636/1331, 2704/3993, 10952/1331];variance=7046/121;variance0 iff EVERY positive-mass outcome has T=THIS mean

ALL original parameter variance rows=[[0, 35212/841], [1, 7046/121]];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, 35212/841], [1, 7046/121]], 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。