第 19793 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":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],"statistic":[0,1,19,20],"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":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],"statistic":[0,1,19,20],"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(1/27, 2/27, 1/9, 7/9)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/27, 2/27, 1/9, 7/9];ONE fixed statistic=[0, 1, 19, 20]

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

variance parameter1 complete law mass := sum(2/31, 3/31, 4/31, 22/31)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[2/31, 3/31, 4/31, 22/31];ONE fixed statistic=[0, 1, 19, 20]

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

variance parameter1 outcome0 weighted T := mul(2/31, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/31, 1)

variance parameter1 outcome1 weighted T = 3/31

variance parameter1 outcome2 weighted T := mul(4/31, 19)

variance parameter1 outcome2 weighted T = 76/31

variance parameter1 outcome3 weighted T := mul(22/31, 20)

variance parameter1 outcome3 weighted T = 440/31

variance parameter1 SAME-law mean := sum(0, 3/31, 76/31, 440/31)

variance parameter1 SAME-law mean = 519/31

variance parameter1 outcome0 T minus SAME mean := sub(0, 519/31)

variance parameter1 outcome0 T minus SAME mean = -519/31

variance parameter1 outcome0 centered square := mul(-519/31, -519/31)

variance parameter1 outcome0 centered square = 269361/961

variance parameter1 outcome0 weighted centered square := mul(2/31, 269361/961)

variance parameter1 outcome0 weighted centered square = 538722/29791

variance parameter1 outcome1 T minus SAME mean := sub(1, 519/31)

variance parameter1 outcome1 T minus SAME mean = -488/31

variance parameter1 outcome1 centered square := mul(-488/31, -488/31)

variance parameter1 outcome1 centered square = 238144/961

variance parameter1 outcome1 weighted centered square := mul(3/31, 238144/961)

variance parameter1 outcome1 weighted centered square = 714432/29791

variance parameter1 outcome2 T minus SAME mean := sub(19, 519/31)

variance parameter1 outcome2 T minus SAME mean = 70/31

variance parameter1 outcome2 centered square := mul(70/31, 70/31)

variance parameter1 outcome2 centered square = 4900/961

variance parameter1 outcome2 weighted centered square := mul(4/31, 4900/961)

variance parameter1 outcome2 weighted centered square = 19600/29791

variance parameter1 outcome3 T minus SAME mean := sub(20, 519/31)

variance parameter1 outcome3 T minus SAME mean = 101/31

variance parameter1 outcome3 centered square := mul(101/31, 101/31)

variance parameter1 outcome3 centered square = 10201/961

variance parameter1 outcome3 weighted centered square := mul(22/31, 10201/961)

variance parameter1 outcome3 weighted centered square = 224422/29791

variance parameter1 full centered variance := sum(538722/29791, 714432/29791, 19600/29791, 224422/29791)

variance parameter1 full centered variance = 48296/961

FIXED parameter SLOT=1;original same law=[2/31, 3/31, 4/31, 22/31];same statistic=[0, 1, 19, 20];its own mean=519/31;ALL centered deviations=[-519/31, -488/31, 70/31, 101/31]

ALL nonnegative weighted centered squares=[538722/29791, 714432/29791, 19600/29791, 224422/29791];variance=48296/961;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(2/31, 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(3/31, 1)

variance second moment outcome1 weighted T squared = 3/31

variance second moment outcome2 T squared := mul(19, 19)

variance second moment outcome2 T squared = 361

variance second moment outcome2 weighted T squared := mul(4/31, 361)

variance second moment outcome2 weighted T squared = 1444/31

variance second moment outcome3 T squared := mul(20, 20)

variance second moment outcome3 T squared = 400

variance second moment outcome3 weighted T squared := mul(22/31, 400)

variance second moment outcome3 weighted T squared = 8800/31

variance full SAME-law second moment := sum(0, 3/31, 1444/31, 8800/31)

variance full SAME-law second moment = 10247/31

variance square of SAME-law mean := mul(519/31, 519/31)

variance square of SAME-law mean = 269361/961

variance second moment minus mean squared := sub(10247/31, 269361/961)

variance second moment minus mean squared = 48296/961

FIXED original law at SLOT=1;mass sum1;E[T]=519/31;E[T^2]=10247/31;mean squared=269361/961

expand sum mass*(T-mean)^2=E[T^2]-2*mean*E[T]+mean^2*1=48296/961=centered variance=48296/961

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, 10247/31, 519/31, 48296/961, 48296/961, 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。