第 19795 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":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],"statistic":[0,1,21,22],"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":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],"statistic":[0,1,21,22],"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(1/29, 2/29, 3/29, 23/29)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/29, 2/29, 3/29, 23/29];ONE fixed statistic=[0, 1, 21, 22]

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

variance parameter1 complete law mass := sum(2/33, 1/11, 4/33, 8/11)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[2/33, 1/11, 4/33, 8/11];ONE fixed statistic=[0, 1, 21, 22]

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

variance parameter1 outcome0 weighted T := mul(2/33, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(1/11, 1)

variance parameter1 outcome1 weighted T = 1/11

variance parameter1 outcome2 weighted T := mul(4/33, 21)

variance parameter1 outcome2 weighted T = 28/11

variance parameter1 outcome3 weighted T := mul(8/11, 22)

variance parameter1 outcome3 weighted T = 16

variance parameter1 SAME-law mean := sum(0, 1/11, 28/11, 16)

variance parameter1 SAME-law mean = 205/11

variance parameter1 outcome0 T minus SAME mean := sub(0, 205/11)

variance parameter1 outcome0 T minus SAME mean = -205/11

variance parameter1 outcome0 centered square := mul(-205/11, -205/11)

variance parameter1 outcome0 centered square = 42025/121

variance parameter1 outcome0 weighted centered square := mul(2/33, 42025/121)

variance parameter1 outcome0 weighted centered square = 84050/3993

variance parameter1 outcome1 T minus SAME mean := sub(1, 205/11)

variance parameter1 outcome1 T minus SAME mean = -194/11

variance parameter1 outcome1 centered square := mul(-194/11, -194/11)

variance parameter1 outcome1 centered square = 37636/121

variance parameter1 outcome1 weighted centered square := mul(1/11, 37636/121)

variance parameter1 outcome1 weighted centered square = 37636/1331

variance parameter1 outcome2 T minus SAME mean := sub(21, 205/11)

variance parameter1 outcome2 T minus SAME mean = 26/11

variance parameter1 outcome2 centered square := mul(26/11, 26/11)

variance parameter1 outcome2 centered square = 676/121

variance parameter1 outcome2 weighted centered square := mul(4/33, 676/121)

variance parameter1 outcome2 weighted centered square = 2704/3993

variance parameter1 outcome3 T minus SAME mean := sub(22, 205/11)

variance parameter1 outcome3 T minus SAME mean = 37/11

variance parameter1 outcome3 centered square := mul(37/11, 37/11)

variance parameter1 outcome3 centered square = 1369/121

variance parameter1 outcome3 weighted centered square := mul(8/11, 1369/121)

variance parameter1 outcome3 weighted centered square = 10952/1331

variance parameter1 full centered variance := sum(84050/3993, 37636/1331, 2704/3993, 10952/1331)

variance parameter1 full centered variance = 7046/121

FIXED parameter SLOT=1;original same law=[2/33, 1/11, 4/33, 8/11];same statistic=[0, 1, 21, 22];its own mean=205/11;ALL centered deviations=[-205/11, -194/11, 26/11, 37/11]

ALL nonnegative weighted centered squares=[84050/3993, 37636/1331, 2704/3993, 10952/1331];variance=7046/121;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(2/33, 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(1/11, 1)

variance second moment outcome1 weighted T squared = 1/11

variance second moment outcome2 T squared := mul(21, 21)

variance second moment outcome2 T squared = 441

variance second moment outcome2 weighted T squared := mul(4/33, 441)

variance second moment outcome2 weighted T squared = 588/11

variance second moment outcome3 T squared := mul(22, 22)

variance second moment outcome3 T squared = 484

variance second moment outcome3 weighted T squared := mul(8/11, 484)

variance second moment outcome3 weighted T squared = 352

variance full SAME-law second moment := sum(0, 1/11, 588/11, 352)

variance full SAME-law second moment = 4461/11

variance square of SAME-law mean := mul(205/11, 205/11)

variance square of SAME-law mean = 42025/121

variance second moment minus mean squared := sub(4461/11, 42025/121)

variance second moment minus mean squared = 7046/121

FIXED original law at SLOT=1;mass sum1;E[T]=205/11;E[T^2]=4461/11;mean squared=42025/121

expand sum mass*(T-mean)^2=E[T^2]-2*mean*E[T]+mean^2*1=7046/121=centered variance=7046/121

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, 4461/11, 205/11, 7046/121, 7046/121, 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。