第 19762 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":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],"statistic":[0,1,13,14]}。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":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],"statistic":[0,1,13,14]}。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(1/21, 2/21, 1/7, 5/7)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/21, 2/21, 1/7, 5/7];ONE fixed statistic=[0, 1, 13, 14]

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

variance parameter1 complete law mass := sum(2/25, 3/25, 4/25, 16/25)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[2/25, 3/25, 4/25, 16/25];ONE fixed statistic=[0, 1, 13, 14]

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

variance parameter0 outcome0 weighted T := mul(1/21, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(2/21, 1)

variance parameter0 outcome1 weighted T = 2/21

variance parameter0 outcome2 weighted T := mul(1/7, 13)

variance parameter0 outcome2 weighted T = 13/7

variance parameter0 outcome3 weighted T := mul(5/7, 14)

variance parameter0 outcome3 weighted T = 10

variance parameter0 SAME-law mean := sum(0, 2/21, 13/7, 10)

variance parameter0 SAME-law mean = 251/21

variance parameter0 outcome0 T minus SAME mean := sub(0, 251/21)

variance parameter0 outcome0 T minus SAME mean = -251/21

variance parameter0 outcome0 centered square := mul(-251/21, -251/21)

variance parameter0 outcome0 centered square = 63001/441

variance parameter0 outcome0 weighted centered square := mul(1/21, 63001/441)

variance parameter0 outcome0 weighted centered square = 63001/9261

variance parameter0 outcome1 T minus SAME mean := sub(1, 251/21)

variance parameter0 outcome1 T minus SAME mean = -230/21

variance parameter0 outcome1 centered square := mul(-230/21, -230/21)

variance parameter0 outcome1 centered square = 52900/441

variance parameter0 outcome1 weighted centered square := mul(2/21, 52900/441)

variance parameter0 outcome1 weighted centered square = 105800/9261

variance parameter0 outcome2 T minus SAME mean := sub(13, 251/21)

variance parameter0 outcome2 T minus SAME mean = 22/21

variance parameter0 outcome2 centered square := mul(22/21, 22/21)

variance parameter0 outcome2 centered square = 484/441

variance parameter0 outcome2 weighted centered square := mul(1/7, 484/441)

variance parameter0 outcome2 weighted centered square = 484/3087

variance parameter0 outcome3 T minus SAME mean := sub(14, 251/21)

variance parameter0 outcome3 T minus SAME mean = 43/21

variance parameter0 outcome3 centered square := mul(43/21, 43/21)

variance parameter0 outcome3 centered square = 1849/441

variance parameter0 outcome3 weighted centered square := mul(5/7, 1849/441)

variance parameter0 outcome3 weighted centered square = 9245/3087

variance parameter0 full centered variance := sum(63001/9261, 105800/9261, 484/3087, 9245/3087)

variance parameter0 full centered variance = 9428/441

FIXED parameter SLOT=0;original same law=[1/21, 2/21, 1/7, 5/7];same statistic=[0, 1, 13, 14];its own mean=251/21;ALL centered deviations=[-251/21, -230/21, 22/21, 43/21]

ALL nonnegative weighted centered squares=[63001/9261, 105800/9261, 484/3087, 9245/3087];variance=9428/441;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(2/25, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/25, 1)

variance parameter1 outcome1 weighted T = 3/25

variance parameter1 outcome2 weighted T := mul(4/25, 13)

variance parameter1 outcome2 weighted T = 52/25

variance parameter1 outcome3 weighted T := mul(16/25, 14)

variance parameter1 outcome3 weighted T = 224/25

variance parameter1 SAME-law mean := sum(0, 3/25, 52/25, 224/25)

variance parameter1 SAME-law mean = 279/25

variance parameter1 outcome0 T minus SAME mean := sub(0, 279/25)

variance parameter1 outcome0 T minus SAME mean = -279/25

variance parameter1 outcome0 centered square := mul(-279/25, -279/25)

variance parameter1 outcome0 centered square = 77841/625

variance parameter1 outcome0 weighted centered square := mul(2/25, 77841/625)

variance parameter1 outcome0 weighted centered square = 155682/15625

variance parameter1 outcome1 T minus SAME mean := sub(1, 279/25)

variance parameter1 outcome1 T minus SAME mean = -254/25

variance parameter1 outcome1 centered square := mul(-254/25, -254/25)

variance parameter1 outcome1 centered square = 64516/625

variance parameter1 outcome1 weighted centered square := mul(3/25, 64516/625)

variance parameter1 outcome1 weighted centered square = 193548/15625

variance parameter1 outcome2 T minus SAME mean := sub(13, 279/25)

variance parameter1 outcome2 T minus SAME mean = 46/25

variance parameter1 outcome2 centered square := mul(46/25, 46/25)

variance parameter1 outcome2 centered square = 2116/625

variance parameter1 outcome2 weighted centered square := mul(4/25, 2116/625)

variance parameter1 outcome2 weighted centered square = 8464/15625

variance parameter1 outcome3 T minus SAME mean := sub(14, 279/25)

variance parameter1 outcome3 T minus SAME mean = 71/25

variance parameter1 outcome3 centered square := mul(71/25, 71/25)

variance parameter1 outcome3 centered square = 5041/625

variance parameter1 outcome3 weighted centered square := mul(16/25, 5041/625)

variance parameter1 outcome3 weighted centered square = 80656/15625

variance parameter1 full centered variance := sum(155682/15625, 193548/15625, 8464/15625, 80656/15625)

variance parameter1 full centered variance = 17534/625

FIXED parameter SLOT=1;original same law=[2/25, 3/25, 4/25, 16/25];same statistic=[0, 1, 13, 14];its own mean=279/25;ALL centered deviations=[-279/25, -254/25, 46/25, 71/25]

ALL nonnegative weighted centered squares=[155682/15625, 193548/15625, 8464/15625, 80656/15625];variance=17534/625;variance0 iff EVERY positive-mass outcome has T=THIS mean

ALL original parameter variance rows=[[0, 9428/441], [1, 17534/625]];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, 9428/441], [1, 17534/625]], 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。