第 19800 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-second-moment-identity；資料={"laws":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],"statistic":[0,1,26,27],"parameter":0}。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-second-moment-identity；資料={"laws":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],"statistic":[0,1,26,27],"parameter":0}。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-second-moment-identity;original snapshot and mathematics retained

explicit ordered source output fields=[parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal]

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 second moment outcome0 T squared := mul(0, 0)

variance second moment outcome0 T squared = 0

variance second moment outcome0 weighted T squared := mul(1/34, 0)

variance second moment outcome0 weighted T squared = 0

variance second moment outcome1 T squared := mul(1, 1)

variance second moment outcome1 T squared = 1

variance second moment outcome1 weighted T squared := mul(1/17, 1)

variance second moment outcome1 weighted T squared = 1/17

variance second moment outcome2 T squared := mul(26, 26)

variance second moment outcome2 T squared = 676

variance second moment outcome2 weighted T squared := mul(3/34, 676)

variance second moment outcome2 weighted T squared = 1014/17

variance second moment outcome3 T squared := mul(27, 27)

variance second moment outcome3 T squared = 729

variance second moment outcome3 weighted T squared := mul(14/17, 729)

variance second moment outcome3 weighted T squared = 10206/17

variance full SAME-law second moment := sum(0, 1/17, 1014/17, 10206/17)

variance full SAME-law second moment = 11221/17

variance square of SAME-law mean := mul(418/17, 418/17)

variance square of SAME-law mean = 174724/289

variance second moment minus mean squared := sub(11221/17, 174724/289)

variance second moment minus mean squared = 16033/289

FIXED original law at SLOT=0;mass sum1;E[T]=418/17;E[T^2]=11221/17;mean squared=174724/289

expand sum mass*(T-mean)^2=E[T^2]-2*mean*E[T]+mean^2*1=16033/289=centered variance=16033/289

逐步反向核對

• S036 · 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。
• S035 · premise_and_result_audit — 以精確有理數重算 variance second moment minus mean squared，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 variance square of SAME-law mean，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 variance full SAME-law second moment，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 variance second moment outcome3 weighted T squared，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 variance second moment outcome3 T squared，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 variance second moment outcome2 weighted T squared，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 variance second moment outcome2 T squared，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 variance second moment outcome1 weighted T squared，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 variance second moment outcome1 T squared，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 variance second moment outcome0 weighted T squared，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 variance second moment outcome0 T squared，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, 11221/17, 418/17, 16033/289, 16033/289, 是)

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。