第 19783 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":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],"statistic":[0,1,9,10],"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":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],"statistic":[0,1,9,10],"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(1/17, 2/17, 3/17, 11/17)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/17, 2/17, 3/17, 11/17];ONE fixed statistic=[0, 1, 9, 10]

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

variance parameter1 complete law mass := sum(2/21, 1/7, 4/21, 4/7)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[2/21, 1/7, 4/21, 4/7];ONE fixed statistic=[0, 1, 9, 10]

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

variance parameter1 outcome0 weighted T := mul(2/21, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(1/7, 1)

variance parameter1 outcome1 weighted T = 1/7

variance parameter1 outcome2 weighted T := mul(4/21, 9)

variance parameter1 outcome2 weighted T = 12/7

variance parameter1 outcome3 weighted T := mul(4/7, 10)

variance parameter1 outcome3 weighted T = 40/7

variance parameter1 SAME-law mean := sum(0, 1/7, 12/7, 40/7)

variance parameter1 SAME-law mean = 53/7

variance parameter1 outcome0 T minus SAME mean := sub(0, 53/7)

variance parameter1 outcome0 T minus SAME mean = -53/7

variance parameter1 outcome0 centered square := mul(-53/7, -53/7)

variance parameter1 outcome0 centered square = 2809/49

variance parameter1 outcome0 weighted centered square := mul(2/21, 2809/49)

variance parameter1 outcome0 weighted centered square = 5618/1029

variance parameter1 outcome1 T minus SAME mean := sub(1, 53/7)

variance parameter1 outcome1 T minus SAME mean = -46/7

variance parameter1 outcome1 centered square := mul(-46/7, -46/7)

variance parameter1 outcome1 centered square = 2116/49

variance parameter1 outcome1 weighted centered square := mul(1/7, 2116/49)

variance parameter1 outcome1 weighted centered square = 2116/343

variance parameter1 outcome2 T minus SAME mean := sub(9, 53/7)

variance parameter1 outcome2 T minus SAME mean = 10/7

variance parameter1 outcome2 centered square := mul(10/7, 10/7)

variance parameter1 outcome2 centered square = 100/49

variance parameter1 outcome2 weighted centered square := mul(4/21, 100/49)

variance parameter1 outcome2 weighted centered square = 400/1029

variance parameter1 outcome3 T minus SAME mean := sub(10, 53/7)

variance parameter1 outcome3 T minus SAME mean = 17/7

variance parameter1 outcome3 centered square := mul(17/7, 17/7)

variance parameter1 outcome3 centered square = 289/49

variance parameter1 outcome3 weighted centered square := mul(4/7, 289/49)

variance parameter1 outcome3 weighted centered square = 1156/343

variance parameter1 full centered variance := sum(5618/1029, 2116/343, 400/1029, 1156/343)

variance parameter1 full centered variance = 754/49

FIXED parameter SLOT=1;original same law=[2/21, 1/7, 4/21, 4/7];same statistic=[0, 1, 9, 10];its own mean=53/7;ALL centered deviations=[-53/7, -46/7, 10/7, 17/7]

ALL nonnegative weighted centered squares=[5618/1029, 2116/343, 400/1029, 1156/343];variance=754/49;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(2/21, 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(1/7, 1)

variance second moment outcome1 weighted T squared = 1/7

variance second moment outcome2 T squared := mul(9, 9)

variance second moment outcome2 T squared = 81

variance second moment outcome2 weighted T squared := mul(4/21, 81)

variance second moment outcome2 weighted T squared = 108/7

variance second moment outcome3 T squared := mul(10, 10)

variance second moment outcome3 T squared = 100

variance second moment outcome3 weighted T squared := mul(4/7, 100)

variance second moment outcome3 weighted T squared = 400/7

variance full SAME-law second moment := sum(0, 1/7, 108/7, 400/7)

variance full SAME-law second moment = 509/7

variance square of SAME-law mean := mul(53/7, 53/7)

variance square of SAME-law mean = 2809/49

variance second moment minus mean squared := sub(509/7, 2809/49)

variance second moment minus mean squared = 754/49

FIXED original law at SLOT=1;mass sum1;E[T]=53/7;E[T^2]=509/7;mean squared=2809/49

expand sum mass*(T-mean)^2=E[T^2]-2*mean*E[T]+mean^2*1=754/49=centered variance=754/49

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, 509/7, 53/7, 754/49, 754/49, 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。