第 19727 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-family-readback；資料={"laws":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],"statistic":[0,1,3,4]}。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-family-readback；資料={"laws":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],"statistic":[0,1,3,4]}。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-family-readback;original snapshot and mathematics retained

explicit ordered source output fields=[parameter_variance_rows, certificate]

variance parameter0 complete law mass := sum(1/11, 2/11, 3/11, 5/11)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/11, 2/11, 3/11, 5/11];ONE fixed statistic=[0, 1, 3, 4]

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

variance parameter1 complete law mass := sum(2/15, 1/5, 4/15, 2/5)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[2/15, 1/5, 4/15, 2/5];ONE fixed statistic=[0, 1, 3, 4]

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

variance parameter0 outcome0 weighted T := mul(1/11, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(2/11, 1)

variance parameter0 outcome1 weighted T = 2/11

variance parameter0 outcome2 weighted T := mul(3/11, 3)

variance parameter0 outcome2 weighted T = 9/11

variance parameter0 outcome3 weighted T := mul(5/11, 4)

variance parameter0 outcome3 weighted T = 20/11

variance parameter0 SAME-law mean := sum(0, 2/11, 9/11, 20/11)

variance parameter0 SAME-law mean = 31/11

variance parameter0 outcome0 T minus SAME mean := sub(0, 31/11)

variance parameter0 outcome0 T minus SAME mean = -31/11

variance parameter0 outcome0 centered square := mul(-31/11, -31/11)

variance parameter0 outcome0 centered square = 961/121

variance parameter0 outcome0 weighted centered square := mul(1/11, 961/121)

variance parameter0 outcome0 weighted centered square = 961/1331

variance parameter0 outcome1 T minus SAME mean := sub(1, 31/11)

variance parameter0 outcome1 T minus SAME mean = -20/11

variance parameter0 outcome1 centered square := mul(-20/11, -20/11)

variance parameter0 outcome1 centered square = 400/121

variance parameter0 outcome1 weighted centered square := mul(2/11, 400/121)

variance parameter0 outcome1 weighted centered square = 800/1331

variance parameter0 outcome2 T minus SAME mean := sub(3, 31/11)

variance parameter0 outcome2 T minus SAME mean = 2/11

variance parameter0 outcome2 centered square := mul(2/11, 2/11)

variance parameter0 outcome2 centered square = 4/121

variance parameter0 outcome2 weighted centered square := mul(3/11, 4/121)

variance parameter0 outcome2 weighted centered square = 12/1331

variance parameter0 outcome3 T minus SAME mean := sub(4, 31/11)

variance parameter0 outcome3 T minus SAME mean = 13/11

variance parameter0 outcome3 centered square := mul(13/11, 13/11)

variance parameter0 outcome3 centered square = 169/121

variance parameter0 outcome3 weighted centered square := mul(5/11, 169/121)

variance parameter0 outcome3 weighted centered square = 845/1331

variance parameter0 full centered variance := sum(961/1331, 800/1331, 12/1331, 845/1331)

variance parameter0 full centered variance = 238/121

FIXED parameter SLOT=0;original same law=[1/11, 2/11, 3/11, 5/11];same statistic=[0, 1, 3, 4];its own mean=31/11;ALL centered deviations=[-31/11, -20/11, 2/11, 13/11]

ALL nonnegative weighted centered squares=[961/1331, 800/1331, 12/1331, 845/1331];variance=238/121;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(2/15, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(1/5, 1)

variance parameter1 outcome1 weighted T = 1/5

variance parameter1 outcome2 weighted T := mul(4/15, 3)

variance parameter1 outcome2 weighted T = 4/5

variance parameter1 outcome3 weighted T := mul(2/5, 4)

variance parameter1 outcome3 weighted T = 8/5

variance parameter1 SAME-law mean := sum(0, 1/5, 4/5, 8/5)

variance parameter1 SAME-law mean = 13/5

variance parameter1 outcome0 T minus SAME mean := sub(0, 13/5)

variance parameter1 outcome0 T minus SAME mean = -13/5

variance parameter1 outcome0 centered square := mul(-13/5, -13/5)

variance parameter1 outcome0 centered square = 169/25

variance parameter1 outcome0 weighted centered square := mul(2/15, 169/25)

variance parameter1 outcome0 weighted centered square = 338/375

variance parameter1 outcome1 T minus SAME mean := sub(1, 13/5)

variance parameter1 outcome1 T minus SAME mean = -8/5

variance parameter1 outcome1 centered square := mul(-8/5, -8/5)

variance parameter1 outcome1 centered square = 64/25

variance parameter1 outcome1 weighted centered square := mul(1/5, 64/25)

variance parameter1 outcome1 weighted centered square = 64/125

variance parameter1 outcome2 T minus SAME mean := sub(3, 13/5)

variance parameter1 outcome2 T minus SAME mean = 2/5

variance parameter1 outcome2 centered square := mul(2/5, 2/5)

variance parameter1 outcome2 centered square = 4/25

variance parameter1 outcome2 weighted centered square := mul(4/15, 4/25)

variance parameter1 outcome2 weighted centered square = 16/375

variance parameter1 outcome3 T minus SAME mean := sub(4, 13/5)

variance parameter1 outcome3 T minus SAME mean = 7/5

variance parameter1 outcome3 centered square := mul(7/5, 7/5)

variance parameter1 outcome3 centered square = 49/25

variance parameter1 outcome3 weighted centered square := mul(2/5, 49/25)

variance parameter1 outcome3 weighted centered square = 98/125

variance parameter1 full centered variance := sum(338/375, 64/125, 16/375, 98/125)

variance parameter1 full centered variance = 56/25

FIXED parameter SLOT=1;original same law=[2/15, 1/5, 4/15, 2/5];same statistic=[0, 1, 3, 4];its own mean=13/5;ALL centered deviations=[-13/5, -8/5, 2/5, 7/5]

ALL nonnegative weighted centered squares=[338/375, 64/125, 16/375, 98/125];variance=56/25;variance0 iff EVERY positive-mass outcome has T=THIS mean

逐步反向核對

• 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, 31/11, 238/121], [1, 13/5, 56/25]], variance_is_parameter_indexed_family)

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。