第 19781 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":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],"statistic":[0,1,7,8],"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":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],"statistic":[0,1,7,8],"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(1/15, 2/15, 1/5, 3/5)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/15, 2/15, 1/5, 3/5];ONE fixed statistic=[0, 1, 7, 8]

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

variance parameter1 complete law mass := sum(2/19, 3/19, 4/19, 10/19)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[2/19, 3/19, 4/19, 10/19];ONE fixed statistic=[0, 1, 7, 8]

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

variance parameter1 outcome0 weighted T := mul(2/19, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/19, 1)

variance parameter1 outcome1 weighted T = 3/19

variance parameter1 outcome2 weighted T := mul(4/19, 7)

variance parameter1 outcome2 weighted T = 28/19

variance parameter1 outcome3 weighted T := mul(10/19, 8)

variance parameter1 outcome3 weighted T = 80/19

variance parameter1 SAME-law mean := sum(0, 3/19, 28/19, 80/19)

variance parameter1 SAME-law mean = 111/19

variance parameter1 outcome0 T minus SAME mean := sub(0, 111/19)

variance parameter1 outcome0 T minus SAME mean = -111/19

variance parameter1 outcome0 centered square := mul(-111/19, -111/19)

variance parameter1 outcome0 centered square = 12321/361

variance parameter1 outcome0 weighted centered square := mul(2/19, 12321/361)

variance parameter1 outcome0 weighted centered square = 24642/6859

variance parameter1 outcome1 T minus SAME mean := sub(1, 111/19)

variance parameter1 outcome1 T minus SAME mean = -92/19

variance parameter1 outcome1 centered square := mul(-92/19, -92/19)

variance parameter1 outcome1 centered square = 8464/361

variance parameter1 outcome1 weighted centered square := mul(3/19, 8464/361)

variance parameter1 outcome1 weighted centered square = 25392/6859

variance parameter1 outcome2 T minus SAME mean := sub(7, 111/19)

variance parameter1 outcome2 T minus SAME mean = 22/19

variance parameter1 outcome2 centered square := mul(22/19, 22/19)

variance parameter1 outcome2 centered square = 484/361

variance parameter1 outcome2 weighted centered square := mul(4/19, 484/361)

variance parameter1 outcome2 weighted centered square = 1936/6859

variance parameter1 outcome3 T minus SAME mean := sub(8, 111/19)

variance parameter1 outcome3 T minus SAME mean = 41/19

variance parameter1 outcome3 centered square := mul(41/19, 41/19)

variance parameter1 outcome3 centered square = 1681/361

variance parameter1 outcome3 weighted centered square := mul(10/19, 1681/361)

variance parameter1 outcome3 weighted centered square = 16810/6859

variance parameter1 full centered variance := sum(24642/6859, 25392/6859, 1936/6859, 16810/6859)

variance parameter1 full centered variance = 3620/361

FIXED parameter SLOT=1;original same law=[2/19, 3/19, 4/19, 10/19];same statistic=[0, 1, 7, 8];its own mean=111/19;ALL centered deviations=[-111/19, -92/19, 22/19, 41/19]

ALL nonnegative weighted centered squares=[24642/6859, 25392/6859, 1936/6859, 16810/6859];variance=3620/361;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(2/19, 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(3/19, 1)

variance second moment outcome1 weighted T squared = 3/19

variance second moment outcome2 T squared := mul(7, 7)

variance second moment outcome2 T squared = 49

variance second moment outcome2 weighted T squared := mul(4/19, 49)

variance second moment outcome2 weighted T squared = 196/19

variance second moment outcome3 T squared := mul(8, 8)

variance second moment outcome3 T squared = 64

variance second moment outcome3 weighted T squared := mul(10/19, 64)

variance second moment outcome3 weighted T squared = 640/19

variance full SAME-law second moment := sum(0, 3/19, 196/19, 640/19)

variance full SAME-law second moment = 839/19

variance square of SAME-law mean := mul(111/19, 111/19)

variance square of SAME-law mean = 12321/361

variance second moment minus mean squared := sub(839/19, 12321/361)

variance second moment minus mean squared = 3620/361

FIXED original law at SLOT=1;mass sum1;E[T]=111/19;E[T^2]=839/19;mean squared=12321/361

expand sum mass*(T-mean)^2=E[T^2]-2*mean*E[T]+mean^2*1=3620/361=centered variance=3620/361

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, 839/19, 111/19, 3620/361, 3620/361, 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。