第 19743 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":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],"statistic":[0,1,19,20]}。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":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],"statistic":[0,1,19,20]}。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(1/27, 2/27, 1/9, 7/9)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/27, 2/27, 1/9, 7/9];ONE fixed statistic=[0, 1, 19, 20]

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

variance parameter1 complete law mass := sum(2/31, 3/31, 4/31, 22/31)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[2/31, 3/31, 4/31, 22/31];ONE fixed statistic=[0, 1, 19, 20]

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

variance parameter0 outcome0 weighted T := mul(1/27, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(2/27, 1)

variance parameter0 outcome1 weighted T = 2/27

variance parameter0 outcome2 weighted T := mul(1/9, 19)

variance parameter0 outcome2 weighted T = 19/9

variance parameter0 outcome3 weighted T := mul(7/9, 20)

variance parameter0 outcome3 weighted T = 140/9

variance parameter0 SAME-law mean := sum(0, 2/27, 19/9, 140/9)

variance parameter0 SAME-law mean = 479/27

variance parameter0 outcome0 T minus SAME mean := sub(0, 479/27)

variance parameter0 outcome0 T minus SAME mean = -479/27

variance parameter0 outcome0 centered square := mul(-479/27, -479/27)

variance parameter0 outcome0 centered square = 229441/729

variance parameter0 outcome0 weighted centered square := mul(1/27, 229441/729)

variance parameter0 outcome0 weighted centered square = 229441/19683

variance parameter0 outcome1 T minus SAME mean := sub(1, 479/27)

variance parameter0 outcome1 T minus SAME mean = -452/27

variance parameter0 outcome1 centered square := mul(-452/27, -452/27)

variance parameter0 outcome1 centered square = 204304/729

variance parameter0 outcome1 weighted centered square := mul(2/27, 204304/729)

variance parameter0 outcome1 weighted centered square = 408608/19683

variance parameter0 outcome2 T minus SAME mean := sub(19, 479/27)

variance parameter0 outcome2 T minus SAME mean = 34/27

variance parameter0 outcome2 centered square := mul(34/27, 34/27)

variance parameter0 outcome2 centered square = 1156/729

variance parameter0 outcome2 weighted centered square := mul(1/9, 1156/729)

variance parameter0 outcome2 weighted centered square = 1156/6561

variance parameter0 outcome3 T minus SAME mean := sub(20, 479/27)

variance parameter0 outcome3 T minus SAME mean = 61/27

variance parameter0 outcome3 centered square := mul(61/27, 61/27)

variance parameter0 outcome3 centered square = 3721/729

variance parameter0 outcome3 weighted centered square := mul(7/9, 3721/729)

variance parameter0 outcome3 weighted centered square = 26047/6561

variance parameter0 full centered variance := sum(229441/19683, 408608/19683, 1156/6561, 26047/6561)

variance parameter0 full centered variance = 26654/729

FIXED parameter SLOT=0;original same law=[1/27, 2/27, 1/9, 7/9];same statistic=[0, 1, 19, 20];its own mean=479/27;ALL centered deviations=[-479/27, -452/27, 34/27, 61/27]

ALL nonnegative weighted centered squares=[229441/19683, 408608/19683, 1156/6561, 26047/6561];variance=26654/729;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(2/31, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/31, 1)

variance parameter1 outcome1 weighted T = 3/31

variance parameter1 outcome2 weighted T := mul(4/31, 19)

variance parameter1 outcome2 weighted T = 76/31

variance parameter1 outcome3 weighted T := mul(22/31, 20)

variance parameter1 outcome3 weighted T = 440/31

variance parameter1 SAME-law mean := sum(0, 3/31, 76/31, 440/31)

variance parameter1 SAME-law mean = 519/31

variance parameter1 outcome0 T minus SAME mean := sub(0, 519/31)

variance parameter1 outcome0 T minus SAME mean = -519/31

variance parameter1 outcome0 centered square := mul(-519/31, -519/31)

variance parameter1 outcome0 centered square = 269361/961

variance parameter1 outcome0 weighted centered square := mul(2/31, 269361/961)

variance parameter1 outcome0 weighted centered square = 538722/29791

variance parameter1 outcome1 T minus SAME mean := sub(1, 519/31)

variance parameter1 outcome1 T minus SAME mean = -488/31

variance parameter1 outcome1 centered square := mul(-488/31, -488/31)

variance parameter1 outcome1 centered square = 238144/961

variance parameter1 outcome1 weighted centered square := mul(3/31, 238144/961)

variance parameter1 outcome1 weighted centered square = 714432/29791

variance parameter1 outcome2 T minus SAME mean := sub(19, 519/31)

variance parameter1 outcome2 T minus SAME mean = 70/31

variance parameter1 outcome2 centered square := mul(70/31, 70/31)

variance parameter1 outcome2 centered square = 4900/961

variance parameter1 outcome2 weighted centered square := mul(4/31, 4900/961)

variance parameter1 outcome2 weighted centered square = 19600/29791

variance parameter1 outcome3 T minus SAME mean := sub(20, 519/31)

variance parameter1 outcome3 T minus SAME mean = 101/31

variance parameter1 outcome3 centered square := mul(101/31, 101/31)

variance parameter1 outcome3 centered square = 10201/961

variance parameter1 outcome3 weighted centered square := mul(22/31, 10201/961)

variance parameter1 outcome3 weighted centered square = 224422/29791

variance parameter1 full centered variance := sum(538722/29791, 714432/29791, 19600/29791, 224422/29791)

variance parameter1 full centered variance = 48296/961

FIXED parameter SLOT=1;original same law=[2/31, 3/31, 4/31, 22/31];same statistic=[0, 1, 19, 20];its own mean=519/31;ALL centered deviations=[-519/31, -488/31, 70/31, 101/31]

ALL nonnegative weighted centered squares=[538722/29791, 714432/29791, 19600/29791, 224422/29791];variance=48296/961;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, 479/27, 26654/729], [1, 519/31, 48296/961]], 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。