第 19799 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":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],"statistic":[0,1,25,26],"parameter":1}。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":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],"statistic":[0,1,25,26],"parameter":1}。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(1/33, 2/33, 1/11, 9/11)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/33, 2/33, 1/11, 9/11];ONE fixed statistic=[0, 1, 25, 26]

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

variance parameter1 complete law mass := sum(2/37, 3/37, 4/37, 28/37)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[2/37, 3/37, 4/37, 28/37];ONE fixed statistic=[0, 1, 25, 26]

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

variance parameter1 outcome0 weighted T := mul(2/37, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/37, 1)

variance parameter1 outcome1 weighted T = 3/37

variance parameter1 outcome2 weighted T := mul(4/37, 25)

variance parameter1 outcome2 weighted T = 100/37

variance parameter1 outcome3 weighted T := mul(28/37, 26)

variance parameter1 outcome3 weighted T = 728/37

variance parameter1 SAME-law mean := sum(0, 3/37, 100/37, 728/37)

variance parameter1 SAME-law mean = 831/37

variance parameter1 outcome0 T minus SAME mean := sub(0, 831/37)

variance parameter1 outcome0 T minus SAME mean = -831/37

variance parameter1 outcome0 centered square := mul(-831/37, -831/37)

variance parameter1 outcome0 centered square = 690561/1369

variance parameter1 outcome0 weighted centered square := mul(2/37, 690561/1369)

variance parameter1 outcome0 weighted centered square = 1381122/50653

variance parameter1 outcome1 T minus SAME mean := sub(1, 831/37)

variance parameter1 outcome1 T minus SAME mean = -794/37

variance parameter1 outcome1 centered square := mul(-794/37, -794/37)

variance parameter1 outcome1 centered square = 630436/1369

variance parameter1 outcome1 weighted centered square := mul(3/37, 630436/1369)

variance parameter1 outcome1 weighted centered square = 1891308/50653

variance parameter1 outcome2 T minus SAME mean := sub(25, 831/37)

variance parameter1 outcome2 T minus SAME mean = 94/37

variance parameter1 outcome2 centered square := mul(94/37, 94/37)

variance parameter1 outcome2 centered square = 8836/1369

variance parameter1 outcome2 weighted centered square := mul(4/37, 8836/1369)

variance parameter1 outcome2 weighted centered square = 35344/50653

variance parameter1 outcome3 T minus SAME mean := sub(26, 831/37)

variance parameter1 outcome3 T minus SAME mean = 131/37

variance parameter1 outcome3 centered square := mul(131/37, 131/37)

variance parameter1 outcome3 centered square = 17161/1369

variance parameter1 outcome3 weighted centered square := mul(28/37, 17161/1369)

variance parameter1 outcome3 weighted centered square = 480508/50653

variance parameter1 full centered variance := sum(1381122/50653, 1891308/50653, 35344/50653, 480508/50653)

variance parameter1 full centered variance = 102386/1369

FIXED parameter SLOT=1;original same law=[2/37, 3/37, 4/37, 28/37];same statistic=[0, 1, 25, 26];its own mean=831/37;ALL centered deviations=[-831/37, -794/37, 94/37, 131/37]

ALL nonnegative weighted centered squares=[1381122/50653, 1891308/50653, 35344/50653, 480508/50653];variance=102386/1369;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(2/37, 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(3/37, 1)

variance second moment outcome1 weighted T squared = 3/37

variance second moment outcome2 T squared := mul(25, 25)

variance second moment outcome2 T squared = 625

variance second moment outcome2 weighted T squared := mul(4/37, 625)

variance second moment outcome2 weighted T squared = 2500/37

variance second moment outcome3 T squared := mul(26, 26)

variance second moment outcome3 T squared = 676

variance second moment outcome3 weighted T squared := mul(28/37, 676)

variance second moment outcome3 weighted T squared = 18928/37

variance full SAME-law second moment := sum(0, 3/37, 2500/37, 18928/37)

variance full SAME-law second moment = 21431/37

variance square of SAME-law mean := mul(831/37, 831/37)

variance square of SAME-law mean = 690561/1369

variance second moment minus mean squared := sub(21431/37, 690561/1369)

variance second moment minus mean squared = 102386/1369

FIXED original law at SLOT=1;mass sum1;E[T]=831/37;E[T^2]=21431/37;mean squared=690561/1369

expand sum mass*(T-mean)^2=E[T^2]-2*mean*E[T]+mean^2*1=102386/1369=centered variance=102386/1369

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 parameter1 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 parameter1 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 parameter1 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 parameter1 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 parameter1 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 parameter1 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 parameter1 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 parameter1 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 parameter1 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 parameter1 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 parameter1 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 parameter1 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 parameter1 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 parameter1 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 parameter1 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 parameter1 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 parameter1 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 parameter1 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

(1, 21431/37, 831/37, 102386/1369, 102386/1369, 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。