第 19791 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":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],"statistic":[0,1,17,18],"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":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],"statistic":[0,1,17,18],"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(1/25, 2/25, 3/25, 19/25)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/25, 2/25, 3/25, 19/25];ONE fixed statistic=[0, 1, 17, 18]

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

variance parameter1 complete law mass := sum(2/29, 3/29, 4/29, 20/29)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[2/29, 3/29, 4/29, 20/29];ONE fixed statistic=[0, 1, 17, 18]

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

variance parameter1 outcome0 weighted T := mul(2/29, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/29, 1)

variance parameter1 outcome1 weighted T = 3/29

variance parameter1 outcome2 weighted T := mul(4/29, 17)

variance parameter1 outcome2 weighted T = 68/29

variance parameter1 outcome3 weighted T := mul(20/29, 18)

variance parameter1 outcome3 weighted T = 360/29

variance parameter1 SAME-law mean := sum(0, 3/29, 68/29, 360/29)

variance parameter1 SAME-law mean = 431/29

variance parameter1 outcome0 T minus SAME mean := sub(0, 431/29)

variance parameter1 outcome0 T minus SAME mean = -431/29

variance parameter1 outcome0 centered square := mul(-431/29, -431/29)

variance parameter1 outcome0 centered square = 185761/841

variance parameter1 outcome0 weighted centered square := mul(2/29, 185761/841)

variance parameter1 outcome0 weighted centered square = 371522/24389

variance parameter1 outcome1 T minus SAME mean := sub(1, 431/29)

variance parameter1 outcome1 T minus SAME mean = -402/29

variance parameter1 outcome1 centered square := mul(-402/29, -402/29)

variance parameter1 outcome1 centered square = 161604/841

variance parameter1 outcome1 weighted centered square := mul(3/29, 161604/841)

variance parameter1 outcome1 weighted centered square = 484812/24389

variance parameter1 outcome2 T minus SAME mean := sub(17, 431/29)

variance parameter1 outcome2 T minus SAME mean = 62/29

variance parameter1 outcome2 centered square := mul(62/29, 62/29)

variance parameter1 outcome2 centered square = 3844/841

variance parameter1 outcome2 weighted centered square := mul(4/29, 3844/841)

variance parameter1 outcome2 weighted centered square = 15376/24389

variance parameter1 outcome3 T minus SAME mean := sub(18, 431/29)

variance parameter1 outcome3 T minus SAME mean = 91/29

variance parameter1 outcome3 centered square := mul(91/29, 91/29)

variance parameter1 outcome3 centered square = 8281/841

variance parameter1 outcome3 weighted centered square := mul(20/29, 8281/841)

variance parameter1 outcome3 weighted centered square = 165620/24389

variance parameter1 full centered variance := sum(371522/24389, 484812/24389, 15376/24389, 165620/24389)

variance parameter1 full centered variance = 35770/841

FIXED parameter SLOT=1;original same law=[2/29, 3/29, 4/29, 20/29];same statistic=[0, 1, 17, 18];its own mean=431/29;ALL centered deviations=[-431/29, -402/29, 62/29, 91/29]

ALL nonnegative weighted centered squares=[371522/24389, 484812/24389, 15376/24389, 165620/24389];variance=35770/841;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(2/29, 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(3/29, 1)

variance second moment outcome1 weighted T squared = 3/29

variance second moment outcome2 T squared := mul(17, 17)

variance second moment outcome2 T squared = 289

variance second moment outcome2 weighted T squared := mul(4/29, 289)

variance second moment outcome2 weighted T squared = 1156/29

variance second moment outcome3 T squared := mul(18, 18)

variance second moment outcome3 T squared = 324

variance second moment outcome3 weighted T squared := mul(20/29, 324)

variance second moment outcome3 weighted T squared = 6480/29

variance full SAME-law second moment := sum(0, 3/29, 1156/29, 6480/29)

variance full SAME-law second moment = 7639/29

variance square of SAME-law mean := mul(431/29, 431/29)

variance square of SAME-law mean = 185761/841

variance second moment minus mean squared := sub(7639/29, 185761/841)

variance second moment minus mean squared = 35770/841

FIXED original law at SLOT=1;mass sum1;E[T]=431/29;E[T^2]=7639/29;mean squared=185761/841

expand sum mass*(T-mean)^2=E[T^2]-2*mean*E[T]+mean^2*1=35770/841=centered variance=35770/841

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, 7639/29, 431/29, 35770/841, 35770/841, 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。