第 19788 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":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],"statistic":[0,1,14,15],"parameter":0}。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":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],"statistic":[0,1,14,15],"parameter":0}。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(1/22, 1/11, 3/22, 8/11)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/22, 1/11, 3/22, 8/11];ONE fixed statistic=[0, 1, 14, 15]

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

variance parameter1 complete law mass := sum(1/13, 3/26, 2/13, 17/26)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[1/13, 3/26, 2/13, 17/26];ONE fixed statistic=[0, 1, 14, 15]

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

variance parameter0 outcome0 weighted T := mul(1/22, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(1/11, 1)

variance parameter0 outcome1 weighted T = 1/11

variance parameter0 outcome2 weighted T := mul(3/22, 14)

variance parameter0 outcome2 weighted T = 21/11

variance parameter0 outcome3 weighted T := mul(8/11, 15)

variance parameter0 outcome3 weighted T = 120/11

variance parameter0 SAME-law mean := sum(0, 1/11, 21/11, 120/11)

variance parameter0 SAME-law mean = 142/11

variance parameter0 outcome0 T minus SAME mean := sub(0, 142/11)

variance parameter0 outcome0 T minus SAME mean = -142/11

variance parameter0 outcome0 centered square := mul(-142/11, -142/11)

variance parameter0 outcome0 centered square = 20164/121

variance parameter0 outcome0 weighted centered square := mul(1/22, 20164/121)

variance parameter0 outcome0 weighted centered square = 10082/1331

variance parameter0 outcome1 T minus SAME mean := sub(1, 142/11)

variance parameter0 outcome1 T minus SAME mean = -131/11

variance parameter0 outcome1 centered square := mul(-131/11, -131/11)

variance parameter0 outcome1 centered square = 17161/121

variance parameter0 outcome1 weighted centered square := mul(1/11, 17161/121)

variance parameter0 outcome1 weighted centered square = 17161/1331

variance parameter0 outcome2 T minus SAME mean := sub(14, 142/11)

variance parameter0 outcome2 T minus SAME mean = 12/11

variance parameter0 outcome2 centered square := mul(12/11, 12/11)

variance parameter0 outcome2 centered square = 144/121

variance parameter0 outcome2 weighted centered square := mul(3/22, 144/121)

variance parameter0 outcome2 weighted centered square = 216/1331

variance parameter0 outcome3 T minus SAME mean := sub(15, 142/11)

variance parameter0 outcome3 T minus SAME mean = 23/11

variance parameter0 outcome3 centered square := mul(23/11, 23/11)

variance parameter0 outcome3 centered square = 529/121

variance parameter0 outcome3 weighted centered square := mul(8/11, 529/121)

variance parameter0 outcome3 weighted centered square = 4232/1331

variance parameter0 full centered variance := sum(10082/1331, 17161/1331, 216/1331, 4232/1331)

variance parameter0 full centered variance = 2881/121

FIXED parameter SLOT=0;original same law=[1/22, 1/11, 3/22, 8/11];same statistic=[0, 1, 14, 15];its own mean=142/11;ALL centered deviations=[-142/11, -131/11, 12/11, 23/11]

ALL nonnegative weighted centered squares=[10082/1331, 17161/1331, 216/1331, 4232/1331];variance=2881/121;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(1/22, 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(14, 14)

variance second moment outcome2 T squared = 196

variance second moment outcome2 weighted T squared := mul(3/22, 196)

variance second moment outcome2 weighted T squared = 294/11

variance second moment outcome3 T squared := mul(15, 15)

variance second moment outcome3 T squared = 225

variance second moment outcome3 weighted T squared := mul(8/11, 225)

variance second moment outcome3 weighted T squared = 1800/11

variance full SAME-law second moment := sum(0, 1/11, 294/11, 1800/11)

variance full SAME-law second moment = 2095/11

variance square of SAME-law mean := mul(142/11, 142/11)

variance square of SAME-law mean = 20164/121

variance second moment minus mean squared := sub(2095/11, 20164/121)

variance second moment minus mean squared = 2881/121

FIXED original law at SLOT=0;mass sum1;E[T]=142/11;E[T^2]=2095/11;mean squared=20164/121

expand sum mass*(T-mean)^2=E[T^2]-2*mean*E[T]+mean^2*1=2881/121=centered variance=2881/121

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 parameter0 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 parameter0 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 parameter0 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 parameter0 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 parameter0 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 parameter0 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 parameter0 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 parameter0 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 parameter0 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 parameter0 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 parameter0 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 parameter0 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 parameter0 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 parameter0 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 parameter0 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 parameter0 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 parameter0 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 parameter0 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

(0, 2095/11, 142/11, 2881/121, 2881/121, 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。