第 19790 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":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],"statistic":[0,1,16,17],"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":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],"statistic":[0,1,16,17],"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(1/24, 1/12, 1/8, 3/4)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/24, 1/12, 1/8, 3/4];ONE fixed statistic=[0, 1, 16, 17]

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

variance parameter1 complete law mass := sum(1/14, 3/28, 1/7, 19/28)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[1/14, 3/28, 1/7, 19/28];ONE fixed statistic=[0, 1, 16, 17]

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

variance parameter0 outcome0 weighted T := mul(1/24, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(1/12, 1)

variance parameter0 outcome1 weighted T = 1/12

variance parameter0 outcome2 weighted T := mul(1/8, 16)

variance parameter0 outcome2 weighted T = 2

variance parameter0 outcome3 weighted T := mul(3/4, 17)

variance parameter0 outcome3 weighted T = 51/4

variance parameter0 SAME-law mean := sum(0, 1/12, 2, 51/4)

variance parameter0 SAME-law mean = 89/6

variance parameter0 outcome0 T minus SAME mean := sub(0, 89/6)

variance parameter0 outcome0 T minus SAME mean = -89/6

variance parameter0 outcome0 centered square := mul(-89/6, -89/6)

variance parameter0 outcome0 centered square = 7921/36

variance parameter0 outcome0 weighted centered square := mul(1/24, 7921/36)

variance parameter0 outcome0 weighted centered square = 7921/864

variance parameter0 outcome1 T minus SAME mean := sub(1, 89/6)

variance parameter0 outcome1 T minus SAME mean = -83/6

variance parameter0 outcome1 centered square := mul(-83/6, -83/6)

variance parameter0 outcome1 centered square = 6889/36

variance parameter0 outcome1 weighted centered square := mul(1/12, 6889/36)

variance parameter0 outcome1 weighted centered square = 6889/432

variance parameter0 outcome2 T minus SAME mean := sub(16, 89/6)

variance parameter0 outcome2 T minus SAME mean = 7/6

variance parameter0 outcome2 centered square := mul(7/6, 7/6)

variance parameter0 outcome2 centered square = 49/36

variance parameter0 outcome2 weighted centered square := mul(1/8, 49/36)

variance parameter0 outcome2 weighted centered square = 49/288

variance parameter0 outcome3 T minus SAME mean := sub(17, 89/6)

variance parameter0 outcome3 T minus SAME mean = 13/6

variance parameter0 outcome3 centered square := mul(13/6, 13/6)

variance parameter0 outcome3 centered square = 169/36

variance parameter0 outcome3 weighted centered square := mul(3/4, 169/36)

variance parameter0 outcome3 weighted centered square = 169/48

variance parameter0 full centered variance := sum(7921/864, 6889/432, 49/288, 169/48)

variance parameter0 full centered variance = 1037/36

FIXED parameter SLOT=0;original same law=[1/24, 1/12, 1/8, 3/4];same statistic=[0, 1, 16, 17];its own mean=89/6;ALL centered deviations=[-89/6, -83/6, 7/6, 13/6]

ALL nonnegative weighted centered squares=[7921/864, 6889/432, 49/288, 169/48];variance=1037/36;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(1/24, 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(1/12, 1)

variance second moment outcome1 weighted T squared = 1/12

variance second moment outcome2 T squared := mul(16, 16)

variance second moment outcome2 T squared = 256

variance second moment outcome2 weighted T squared := mul(1/8, 256)

variance second moment outcome2 weighted T squared = 32

variance second moment outcome3 T squared := mul(17, 17)

variance second moment outcome3 T squared = 289

variance second moment outcome3 weighted T squared := mul(3/4, 289)

variance second moment outcome3 weighted T squared = 867/4

variance full SAME-law second moment := sum(0, 1/12, 32, 867/4)

variance full SAME-law second moment = 1493/6

variance square of SAME-law mean := mul(89/6, 89/6)

variance square of SAME-law mean = 7921/36

variance second moment minus mean squared := sub(1493/6, 7921/36)

variance second moment minus mean squared = 1037/36

FIXED original law at SLOT=0;mass sum1;E[T]=89/6;E[T^2]=1493/6;mean squared=7921/36

expand sum mass*(T-mean)^2=E[T^2]-2*mean*E[T]+mean^2*1=1037/36=centered variance=1037/36

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, 1493/6, 89/6, 1037/36, 1037/36, 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。