第 19767 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":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],"statistic":[0,1,18,19]}。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":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],"statistic":[0,1,18,19]}。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(1/26, 1/13, 3/26, 10/13)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/26, 1/13, 3/26, 10/13];ONE fixed statistic=[0, 1, 18, 19]

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

variance parameter1 complete law mass := sum(1/15, 1/10, 2/15, 7/10)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[1/15, 1/10, 2/15, 7/10];ONE fixed statistic=[0, 1, 18, 19]

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

variance parameter0 outcome0 weighted T := mul(1/26, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(1/13, 1)

variance parameter0 outcome1 weighted T = 1/13

variance parameter0 outcome2 weighted T := mul(3/26, 18)

variance parameter0 outcome2 weighted T = 27/13

variance parameter0 outcome3 weighted T := mul(10/13, 19)

variance parameter0 outcome3 weighted T = 190/13

variance parameter0 SAME-law mean := sum(0, 1/13, 27/13, 190/13)

variance parameter0 SAME-law mean = 218/13

variance parameter0 outcome0 T minus SAME mean := sub(0, 218/13)

variance parameter0 outcome0 T minus SAME mean = -218/13

variance parameter0 outcome0 centered square := mul(-218/13, -218/13)

variance parameter0 outcome0 centered square = 47524/169

variance parameter0 outcome0 weighted centered square := mul(1/26, 47524/169)

variance parameter0 outcome0 weighted centered square = 23762/2197

variance parameter0 outcome1 T minus SAME mean := sub(1, 218/13)

variance parameter0 outcome1 T minus SAME mean = -205/13

variance parameter0 outcome1 centered square := mul(-205/13, -205/13)

variance parameter0 outcome1 centered square = 42025/169

variance parameter0 outcome1 weighted centered square := mul(1/13, 42025/169)

variance parameter0 outcome1 weighted centered square = 42025/2197

variance parameter0 outcome2 T minus SAME mean := sub(18, 218/13)

variance parameter0 outcome2 T minus SAME mean = 16/13

variance parameter0 outcome2 centered square := mul(16/13, 16/13)

variance parameter0 outcome2 centered square = 256/169

variance parameter0 outcome2 weighted centered square := mul(3/26, 256/169)

variance parameter0 outcome2 weighted centered square = 384/2197

variance parameter0 outcome3 T minus SAME mean := sub(19, 218/13)

variance parameter0 outcome3 T minus SAME mean = 29/13

variance parameter0 outcome3 centered square := mul(29/13, 29/13)

variance parameter0 outcome3 centered square = 841/169

variance parameter0 outcome3 weighted centered square := mul(10/13, 841/169)

variance parameter0 outcome3 weighted centered square = 8410/2197

variance parameter0 full centered variance := sum(23762/2197, 42025/2197, 384/2197, 8410/2197)

variance parameter0 full centered variance = 5737/169

FIXED parameter SLOT=0;original same law=[1/26, 1/13, 3/26, 10/13];same statistic=[0, 1, 18, 19];its own mean=218/13;ALL centered deviations=[-218/13, -205/13, 16/13, 29/13]

ALL nonnegative weighted centered squares=[23762/2197, 42025/2197, 384/2197, 8410/2197];variance=5737/169;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(1/15, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(1/10, 1)

variance parameter1 outcome1 weighted T = 1/10

variance parameter1 outcome2 weighted T := mul(2/15, 18)

variance parameter1 outcome2 weighted T = 12/5

variance parameter1 outcome3 weighted T := mul(7/10, 19)

variance parameter1 outcome3 weighted T = 133/10

variance parameter1 SAME-law mean := sum(0, 1/10, 12/5, 133/10)

variance parameter1 SAME-law mean = 79/5

variance parameter1 outcome0 T minus SAME mean := sub(0, 79/5)

variance parameter1 outcome0 T minus SAME mean = -79/5

variance parameter1 outcome0 centered square := mul(-79/5, -79/5)

variance parameter1 outcome0 centered square = 6241/25

variance parameter1 outcome0 weighted centered square := mul(1/15, 6241/25)

variance parameter1 outcome0 weighted centered square = 6241/375

variance parameter1 outcome1 T minus SAME mean := sub(1, 79/5)

variance parameter1 outcome1 T minus SAME mean = -74/5

variance parameter1 outcome1 centered square := mul(-74/5, -74/5)

variance parameter1 outcome1 centered square = 5476/25

variance parameter1 outcome1 weighted centered square := mul(1/10, 5476/25)

variance parameter1 outcome1 weighted centered square = 2738/125

variance parameter1 outcome2 T minus SAME mean := sub(18, 79/5)

variance parameter1 outcome2 T minus SAME mean = 11/5

variance parameter1 outcome2 centered square := mul(11/5, 11/5)

variance parameter1 outcome2 centered square = 121/25

variance parameter1 outcome2 weighted centered square := mul(2/15, 121/25)

variance parameter1 outcome2 weighted centered square = 242/375

variance parameter1 outcome3 T minus SAME mean := sub(19, 79/5)

variance parameter1 outcome3 T minus SAME mean = 16/5

variance parameter1 outcome3 centered square := mul(16/5, 16/5)

variance parameter1 outcome3 centered square = 256/25

variance parameter1 outcome3 weighted centered square := mul(7/10, 256/25)

variance parameter1 outcome3 weighted centered square = 896/125

variance parameter1 full centered variance := sum(6241/375, 2738/125, 242/375, 896/125)

variance parameter1 full centered variance = 1159/25

FIXED parameter SLOT=1;original same law=[1/15, 1/10, 2/15, 7/10];same statistic=[0, 1, 18, 19];its own mean=79/5;ALL centered deviations=[-79/5, -74/5, 11/5, 16/5]

ALL nonnegative weighted centered squares=[6241/375, 2738/125, 242/375, 896/125];variance=1159/25;variance0 iff EVERY positive-mass outcome has T=THIS mean

ALL original parameter variance rows=[[0, 5737/169], [1, 1159/25]];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, 5737/169], [1, 1159/25]], 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。