第 19757 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":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],"statistic":[0,1,8,9]}。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":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],"statistic":[0,1,8,9]}。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(1/16, 1/8, 3/16, 5/8)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/16, 1/8, 3/16, 5/8];ONE fixed statistic=[0, 1, 8, 9]

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

variance parameter1 complete law mass := sum(1/10, 3/20, 1/5, 11/20)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[1/10, 3/20, 1/5, 11/20];ONE fixed statistic=[0, 1, 8, 9]

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

variance parameter0 outcome0 weighted T := mul(1/16, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(1/8, 1)

variance parameter0 outcome1 weighted T = 1/8

variance parameter0 outcome2 weighted T := mul(3/16, 8)

variance parameter0 outcome2 weighted T = 3/2

variance parameter0 outcome3 weighted T := mul(5/8, 9)

variance parameter0 outcome3 weighted T = 45/8

variance parameter0 SAME-law mean := sum(0, 1/8, 3/2, 45/8)

variance parameter0 SAME-law mean = 29/4

variance parameter0 outcome0 T minus SAME mean := sub(0, 29/4)

variance parameter0 outcome0 T minus SAME mean = -29/4

variance parameter0 outcome0 centered square := mul(-29/4, -29/4)

variance parameter0 outcome0 centered square = 841/16

variance parameter0 outcome0 weighted centered square := mul(1/16, 841/16)

variance parameter0 outcome0 weighted centered square = 841/256

variance parameter0 outcome1 T minus SAME mean := sub(1, 29/4)

variance parameter0 outcome1 T minus SAME mean = -25/4

variance parameter0 outcome1 centered square := mul(-25/4, -25/4)

variance parameter0 outcome1 centered square = 625/16

variance parameter0 outcome1 weighted centered square := mul(1/8, 625/16)

variance parameter0 outcome1 weighted centered square = 625/128

variance parameter0 outcome2 T minus SAME mean := sub(8, 29/4)

variance parameter0 outcome2 T minus SAME mean = 3/4

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

variance parameter0 outcome2 centered square = 9/16

variance parameter0 outcome2 weighted centered square := mul(3/16, 9/16)

variance parameter0 outcome2 weighted centered square = 27/256

variance parameter0 outcome3 T minus SAME mean := sub(9, 29/4)

variance parameter0 outcome3 T minus SAME mean = 7/4

variance parameter0 outcome3 centered square := mul(7/4, 7/4)

variance parameter0 outcome3 centered square = 49/16

variance parameter0 outcome3 weighted centered square := mul(5/8, 49/16)

variance parameter0 outcome3 weighted centered square = 245/128

variance parameter0 full centered variance := sum(841/256, 625/128, 27/256, 245/128)

variance parameter0 full centered variance = 163/16

FIXED parameter SLOT=0;original same law=[1/16, 1/8, 3/16, 5/8];same statistic=[0, 1, 8, 9];its own mean=29/4;ALL centered deviations=[-29/4, -25/4, 3/4, 7/4]

ALL nonnegative weighted centered squares=[841/256, 625/128, 27/256, 245/128];variance=163/16;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(1/10, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/20, 1)

variance parameter1 outcome1 weighted T = 3/20

variance parameter1 outcome2 weighted T := mul(1/5, 8)

variance parameter1 outcome2 weighted T = 8/5

variance parameter1 outcome3 weighted T := mul(11/20, 9)

variance parameter1 outcome3 weighted T = 99/20

variance parameter1 SAME-law mean := sum(0, 3/20, 8/5, 99/20)

variance parameter1 SAME-law mean = 67/10

variance parameter1 outcome0 T minus SAME mean := sub(0, 67/10)

variance parameter1 outcome0 T minus SAME mean = -67/10

variance parameter1 outcome0 centered square := mul(-67/10, -67/10)

variance parameter1 outcome0 centered square = 4489/100

variance parameter1 outcome0 weighted centered square := mul(1/10, 4489/100)

variance parameter1 outcome0 weighted centered square = 4489/1000

variance parameter1 outcome1 T minus SAME mean := sub(1, 67/10)

variance parameter1 outcome1 T minus SAME mean = -57/10

variance parameter1 outcome1 centered square := mul(-57/10, -57/10)

variance parameter1 outcome1 centered square = 3249/100

variance parameter1 outcome1 weighted centered square := mul(3/20, 3249/100)

variance parameter1 outcome1 weighted centered square = 9747/2000

variance parameter1 outcome2 T minus SAME mean := sub(8, 67/10)

variance parameter1 outcome2 T minus SAME mean = 13/10

variance parameter1 outcome2 centered square := mul(13/10, 13/10)

variance parameter1 outcome2 centered square = 169/100

variance parameter1 outcome2 weighted centered square := mul(1/5, 169/100)

variance parameter1 outcome2 weighted centered square = 169/500

variance parameter1 outcome3 T minus SAME mean := sub(9, 67/10)

variance parameter1 outcome3 T minus SAME mean = 23/10

variance parameter1 outcome3 centered square := mul(23/10, 23/10)

variance parameter1 outcome3 centered square = 529/100

variance parameter1 outcome3 weighted centered square := mul(11/20, 529/100)

variance parameter1 outcome3 weighted centered square = 5819/2000

variance parameter1 full centered variance := sum(4489/1000, 9747/2000, 169/500, 5819/2000)

variance parameter1 full centered variance = 1261/100

FIXED parameter SLOT=1;original same law=[1/10, 3/20, 1/5, 11/20];same statistic=[0, 1, 8, 9];its own mean=67/10;ALL centered deviations=[-67/10, -57/10, 13/10, 23/10]

ALL nonnegative weighted centered squares=[4489/1000, 9747/2000, 169/500, 5819/2000];variance=1261/100;variance0 iff EVERY positive-mass outcome has T=THIS mean

ALL original parameter variance rows=[[0, 163/16], [1, 1261/100]];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, 163/16], [1, 1261/100]], 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。