第 19750 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":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],"statistic":[0,1,26,27]}。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":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],"statistic":[0,1,26,27]}。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(1/34, 1/17, 3/34, 14/17)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/34, 1/17, 3/34, 14/17];ONE fixed statistic=[0, 1, 26, 27]

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

variance parameter1 complete law mass := sum(1/19, 3/38, 2/19, 29/38)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[1/19, 3/38, 2/19, 29/38];ONE fixed statistic=[0, 1, 26, 27]

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

variance parameter0 outcome0 weighted T := mul(1/34, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(1/17, 1)

variance parameter0 outcome1 weighted T = 1/17

variance parameter0 outcome2 weighted T := mul(3/34, 26)

variance parameter0 outcome2 weighted T = 39/17

variance parameter0 outcome3 weighted T := mul(14/17, 27)

variance parameter0 outcome3 weighted T = 378/17

variance parameter0 SAME-law mean := sum(0, 1/17, 39/17, 378/17)

variance parameter0 SAME-law mean = 418/17

variance parameter0 outcome0 T minus SAME mean := sub(0, 418/17)

variance parameter0 outcome0 T minus SAME mean = -418/17

variance parameter0 outcome0 centered square := mul(-418/17, -418/17)

variance parameter0 outcome0 centered square = 174724/289

variance parameter0 outcome0 weighted centered square := mul(1/34, 174724/289)

variance parameter0 outcome0 weighted centered square = 87362/4913

variance parameter0 outcome1 T minus SAME mean := sub(1, 418/17)

variance parameter0 outcome1 T minus SAME mean = -401/17

variance parameter0 outcome1 centered square := mul(-401/17, -401/17)

variance parameter0 outcome1 centered square = 160801/289

variance parameter0 outcome1 weighted centered square := mul(1/17, 160801/289)

variance parameter0 outcome1 weighted centered square = 160801/4913

variance parameter0 outcome2 T minus SAME mean := sub(26, 418/17)

variance parameter0 outcome2 T minus SAME mean = 24/17

variance parameter0 outcome2 centered square := mul(24/17, 24/17)

variance parameter0 outcome2 centered square = 576/289

variance parameter0 outcome2 weighted centered square := mul(3/34, 576/289)

variance parameter0 outcome2 weighted centered square = 864/4913

variance parameter0 outcome3 T minus SAME mean := sub(27, 418/17)

variance parameter0 outcome3 T minus SAME mean = 41/17

variance parameter0 outcome3 centered square := mul(41/17, 41/17)

variance parameter0 outcome3 centered square = 1681/289

variance parameter0 outcome3 weighted centered square := mul(14/17, 1681/289)

variance parameter0 outcome3 weighted centered square = 23534/4913

variance parameter0 full centered variance := sum(87362/4913, 160801/4913, 864/4913, 23534/4913)

variance parameter0 full centered variance = 16033/289

FIXED parameter SLOT=0;original same law=[1/34, 1/17, 3/34, 14/17];same statistic=[0, 1, 26, 27];its own mean=418/17;ALL centered deviations=[-418/17, -401/17, 24/17, 41/17]

ALL nonnegative weighted centered squares=[87362/4913, 160801/4913, 864/4913, 23534/4913];variance=16033/289;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(1/19, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/38, 1)

variance parameter1 outcome1 weighted T = 3/38

variance parameter1 outcome2 weighted T := mul(2/19, 26)

variance parameter1 outcome2 weighted T = 52/19

variance parameter1 outcome3 weighted T := mul(29/38, 27)

variance parameter1 outcome3 weighted T = 783/38

variance parameter1 SAME-law mean := sum(0, 3/38, 52/19, 783/38)

variance parameter1 SAME-law mean = 445/19

variance parameter1 outcome0 T minus SAME mean := sub(0, 445/19)

variance parameter1 outcome0 T minus SAME mean = -445/19

variance parameter1 outcome0 centered square := mul(-445/19, -445/19)

variance parameter1 outcome0 centered square = 198025/361

variance parameter1 outcome0 weighted centered square := mul(1/19, 198025/361)

variance parameter1 outcome0 weighted centered square = 198025/6859

variance parameter1 outcome1 T minus SAME mean := sub(1, 445/19)

variance parameter1 outcome1 T minus SAME mean = -426/19

variance parameter1 outcome1 centered square := mul(-426/19, -426/19)

variance parameter1 outcome1 centered square = 181476/361

variance parameter1 outcome1 weighted centered square := mul(3/38, 181476/361)

variance parameter1 outcome1 weighted centered square = 272214/6859

variance parameter1 outcome2 T minus SAME mean := sub(26, 445/19)

variance parameter1 outcome2 T minus SAME mean = 49/19

variance parameter1 outcome2 centered square := mul(49/19, 49/19)

variance parameter1 outcome2 centered square = 2401/361

variance parameter1 outcome2 weighted centered square := mul(2/19, 2401/361)

variance parameter1 outcome2 weighted centered square = 4802/6859

variance parameter1 outcome3 T minus SAME mean := sub(27, 445/19)

variance parameter1 outcome3 T minus SAME mean = 68/19

variance parameter1 outcome3 centered square := mul(68/19, 68/19)

variance parameter1 outcome3 centered square = 4624/361

variance parameter1 outcome3 weighted centered square := mul(29/38, 4624/361)

variance parameter1 outcome3 weighted centered square = 67048/6859

variance parameter1 full centered variance := sum(198025/6859, 272214/6859, 4802/6859, 67048/6859)

variance parameter1 full centered variance = 28531/361

FIXED parameter SLOT=1;original same law=[1/19, 3/38, 2/19, 29/38];same statistic=[0, 1, 26, 27];its own mean=445/19;ALL centered deviations=[-445/19, -426/19, 49/19, 68/19]

ALL nonnegative weighted centered squares=[198025/6859, 272214/6859, 4802/6859, 67048/6859];variance=28531/361;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, 418/17, 16033/289], [1, 445/19, 28531/361]], 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。