第 19756 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":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],"statistic":[0,1,7,8]}。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":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],"statistic":[0,1,7,8]}。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(1/15, 2/15, 1/5, 3/5)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/15, 2/15, 1/5, 3/5];ONE fixed statistic=[0, 1, 7, 8]

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

variance parameter1 complete law mass := sum(2/19, 3/19, 4/19, 10/19)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[2/19, 3/19, 4/19, 10/19];ONE fixed statistic=[0, 1, 7, 8]

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

variance parameter0 outcome0 weighted T := mul(1/15, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(2/15, 1)

variance parameter0 outcome1 weighted T = 2/15

variance parameter0 outcome2 weighted T := mul(1/5, 7)

variance parameter0 outcome2 weighted T = 7/5

variance parameter0 outcome3 weighted T := mul(3/5, 8)

variance parameter0 outcome3 weighted T = 24/5

variance parameter0 SAME-law mean := sum(0, 2/15, 7/5, 24/5)

variance parameter0 SAME-law mean = 19/3

variance parameter0 outcome0 T minus SAME mean := sub(0, 19/3)

variance parameter0 outcome0 T minus SAME mean = -19/3

variance parameter0 outcome0 centered square := mul(-19/3, -19/3)

variance parameter0 outcome0 centered square = 361/9

variance parameter0 outcome0 weighted centered square := mul(1/15, 361/9)

variance parameter0 outcome0 weighted centered square = 361/135

variance parameter0 outcome1 T minus SAME mean := sub(1, 19/3)

variance parameter0 outcome1 T minus SAME mean = -16/3

variance parameter0 outcome1 centered square := mul(-16/3, -16/3)

variance parameter0 outcome1 centered square = 256/9

variance parameter0 outcome1 weighted centered square := mul(2/15, 256/9)

variance parameter0 outcome1 weighted centered square = 512/135

variance parameter0 outcome2 T minus SAME mean := sub(7, 19/3)

variance parameter0 outcome2 T minus SAME mean = 2/3

variance parameter0 outcome2 centered square := mul(2/3, 2/3)

variance parameter0 outcome2 centered square = 4/9

variance parameter0 outcome2 weighted centered square := mul(1/5, 4/9)

variance parameter0 outcome2 weighted centered square = 4/45

variance parameter0 outcome3 T minus SAME mean := sub(8, 19/3)

variance parameter0 outcome3 T minus SAME mean = 5/3

variance parameter0 outcome3 centered square := mul(5/3, 5/3)

variance parameter0 outcome3 centered square = 25/9

variance parameter0 outcome3 weighted centered square := mul(3/5, 25/9)

variance parameter0 outcome3 weighted centered square = 5/3

variance parameter0 full centered variance := sum(361/135, 512/135, 4/45, 5/3)

variance parameter0 full centered variance = 74/9

FIXED parameter SLOT=0;original same law=[1/15, 2/15, 1/5, 3/5];same statistic=[0, 1, 7, 8];its own mean=19/3;ALL centered deviations=[-19/3, -16/3, 2/3, 5/3]

ALL nonnegative weighted centered squares=[361/135, 512/135, 4/45, 5/3];variance=74/9;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(2/19, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/19, 1)

variance parameter1 outcome1 weighted T = 3/19

variance parameter1 outcome2 weighted T := mul(4/19, 7)

variance parameter1 outcome2 weighted T = 28/19

variance parameter1 outcome3 weighted T := mul(10/19, 8)

variance parameter1 outcome3 weighted T = 80/19

variance parameter1 SAME-law mean := sum(0, 3/19, 28/19, 80/19)

variance parameter1 SAME-law mean = 111/19

variance parameter1 outcome0 T minus SAME mean := sub(0, 111/19)

variance parameter1 outcome0 T minus SAME mean = -111/19

variance parameter1 outcome0 centered square := mul(-111/19, -111/19)

variance parameter1 outcome0 centered square = 12321/361

variance parameter1 outcome0 weighted centered square := mul(2/19, 12321/361)

variance parameter1 outcome0 weighted centered square = 24642/6859

variance parameter1 outcome1 T minus SAME mean := sub(1, 111/19)

variance parameter1 outcome1 T minus SAME mean = -92/19

variance parameter1 outcome1 centered square := mul(-92/19, -92/19)

variance parameter1 outcome1 centered square = 8464/361

variance parameter1 outcome1 weighted centered square := mul(3/19, 8464/361)

variance parameter1 outcome1 weighted centered square = 25392/6859

variance parameter1 outcome2 T minus SAME mean := sub(7, 111/19)

variance parameter1 outcome2 T minus SAME mean = 22/19

variance parameter1 outcome2 centered square := mul(22/19, 22/19)

variance parameter1 outcome2 centered square = 484/361

variance parameter1 outcome2 weighted centered square := mul(4/19, 484/361)

variance parameter1 outcome2 weighted centered square = 1936/6859

variance parameter1 outcome3 T minus SAME mean := sub(8, 111/19)

variance parameter1 outcome3 T minus SAME mean = 41/19

variance parameter1 outcome3 centered square := mul(41/19, 41/19)

variance parameter1 outcome3 centered square = 1681/361

variance parameter1 outcome3 weighted centered square := mul(10/19, 1681/361)

variance parameter1 outcome3 weighted centered square = 16810/6859

variance parameter1 full centered variance := sum(24642/6859, 25392/6859, 1936/6859, 16810/6859)

variance parameter1 full centered variance = 3620/361

FIXED parameter SLOT=1;original same law=[2/19, 3/19, 4/19, 10/19];same statistic=[0, 1, 7, 8];its own mean=111/19;ALL centered deviations=[-111/19, -92/19, 22/19, 41/19]

ALL nonnegative weighted centered squares=[24642/6859, 25392/6859, 1936/6859, 16810/6859];variance=3620/361;variance0 iff EVERY positive-mass outcome has T=THIS mean

ALL original parameter variance rows=[[0, 74/9], [1, 3620/361]];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, 74/9], [1, 3620/361]], 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。