第 19769 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":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],"statistic":[0,1,20,21]}。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":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],"statistic":[0,1,20,21]}。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(1/28, 1/14, 3/28, 11/14)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/28, 1/14, 3/28, 11/14];ONE fixed statistic=[0, 1, 20, 21]

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

variance parameter1 complete law mass := sum(1/16, 3/32, 1/8, 23/32)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[1/16, 3/32, 1/8, 23/32];ONE fixed statistic=[0, 1, 20, 21]

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

variance parameter0 outcome0 weighted T := mul(1/28, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(1/14, 1)

variance parameter0 outcome1 weighted T = 1/14

variance parameter0 outcome2 weighted T := mul(3/28, 20)

variance parameter0 outcome2 weighted T = 15/7

variance parameter0 outcome3 weighted T := mul(11/14, 21)

variance parameter0 outcome3 weighted T = 33/2

variance parameter0 SAME-law mean := sum(0, 1/14, 15/7, 33/2)

variance parameter0 SAME-law mean = 131/7

variance parameter0 outcome0 T minus SAME mean := sub(0, 131/7)

variance parameter0 outcome0 T minus SAME mean = -131/7

variance parameter0 outcome0 centered square := mul(-131/7, -131/7)

variance parameter0 outcome0 centered square = 17161/49

variance parameter0 outcome0 weighted centered square := mul(1/28, 17161/49)

variance parameter0 outcome0 weighted centered square = 17161/1372

variance parameter0 outcome1 T minus SAME mean := sub(1, 131/7)

variance parameter0 outcome1 T minus SAME mean = -124/7

variance parameter0 outcome1 centered square := mul(-124/7, -124/7)

variance parameter0 outcome1 centered square = 15376/49

variance parameter0 outcome1 weighted centered square := mul(1/14, 15376/49)

variance parameter0 outcome1 weighted centered square = 7688/343

variance parameter0 outcome2 T minus SAME mean := sub(20, 131/7)

variance parameter0 outcome2 T minus SAME mean = 9/7

variance parameter0 outcome2 centered square := mul(9/7, 9/7)

variance parameter0 outcome2 centered square = 81/49

variance parameter0 outcome2 weighted centered square := mul(3/28, 81/49)

variance parameter0 outcome2 weighted centered square = 243/1372

variance parameter0 outcome3 T minus SAME mean := sub(21, 131/7)

variance parameter0 outcome3 T minus SAME mean = 16/7

variance parameter0 outcome3 centered square := mul(16/7, 16/7)

variance parameter0 outcome3 centered square = 256/49

variance parameter0 outcome3 weighted centered square := mul(11/14, 256/49)

variance parameter0 outcome3 weighted centered square = 1408/343

variance parameter0 full centered variance := sum(17161/1372, 7688/343, 243/1372, 1408/343)

variance parameter0 full centered variance = 1921/49

FIXED parameter SLOT=0;original same law=[1/28, 1/14, 3/28, 11/14];same statistic=[0, 1, 20, 21];its own mean=131/7;ALL centered deviations=[-131/7, -124/7, 9/7, 16/7]

ALL nonnegative weighted centered squares=[17161/1372, 7688/343, 243/1372, 1408/343];variance=1921/49;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(1/16, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/32, 1)

variance parameter1 outcome1 weighted T = 3/32

variance parameter1 outcome2 weighted T := mul(1/8, 20)

variance parameter1 outcome2 weighted T = 5/2

variance parameter1 outcome3 weighted T := mul(23/32, 21)

variance parameter1 outcome3 weighted T = 483/32

variance parameter1 SAME-law mean := sum(0, 3/32, 5/2, 483/32)

variance parameter1 SAME-law mean = 283/16

variance parameter1 outcome0 T minus SAME mean := sub(0, 283/16)

variance parameter1 outcome0 T minus SAME mean = -283/16

variance parameter1 outcome0 centered square := mul(-283/16, -283/16)

variance parameter1 outcome0 centered square = 80089/256

variance parameter1 outcome0 weighted centered square := mul(1/16, 80089/256)

variance parameter1 outcome0 weighted centered square = 80089/4096

variance parameter1 outcome1 T minus SAME mean := sub(1, 283/16)

variance parameter1 outcome1 T minus SAME mean = -267/16

variance parameter1 outcome1 centered square := mul(-267/16, -267/16)

variance parameter1 outcome1 centered square = 71289/256

variance parameter1 outcome1 weighted centered square := mul(3/32, 71289/256)

variance parameter1 outcome1 weighted centered square = 213867/8192

variance parameter1 outcome2 T minus SAME mean := sub(20, 283/16)

variance parameter1 outcome2 T minus SAME mean = 37/16

variance parameter1 outcome2 centered square := mul(37/16, 37/16)

variance parameter1 outcome2 centered square = 1369/256

variance parameter1 outcome2 weighted centered square := mul(1/8, 1369/256)

variance parameter1 outcome2 weighted centered square = 1369/2048

variance parameter1 outcome3 T minus SAME mean := sub(21, 283/16)

variance parameter1 outcome3 T minus SAME mean = 53/16

variance parameter1 outcome3 centered square := mul(53/16, 53/16)

variance parameter1 outcome3 centered square = 2809/256

variance parameter1 outcome3 weighted centered square := mul(23/32, 2809/256)

variance parameter1 outcome3 weighted centered square = 64607/8192

variance parameter1 full centered variance := sum(80089/4096, 213867/8192, 1369/2048, 64607/8192)

variance parameter1 full centered variance = 13879/256

FIXED parameter SLOT=1;original same law=[1/16, 3/32, 1/8, 23/32];same statistic=[0, 1, 20, 21];its own mean=283/16;ALL centered deviations=[-283/16, -267/16, 37/16, 53/16]

ALL nonnegative weighted centered squares=[80089/4096, 213867/8192, 1369/2048, 64607/8192];variance=13879/256;variance0 iff EVERY positive-mass outcome has T=THIS mean

ALL original parameter variance rows=[[0, 1921/49], [1, 13879/256]];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, 1921/49], [1, 13879/256]], 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。