第 19707 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-at-parameter；資料={"laws":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],"statistic":[0,1,8,9],"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-at-parameter；資料={"laws":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],"statistic":[0,1,8,9],"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-at-parameter;original snapshot and mathematics retained

explicit ordered source output fields=[parameter_index, expectation, variance, variance_certificate, certificate]

variance parameter0 complete law mass := sum(1/16, 1/8, 3/16, 5/8)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/16, 1/8, 3/16, 5/8];ONE fixed statistic=[0, 1, 8, 9]

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

variance parameter1 complete law mass := sum(1/10, 3/20, 1/5, 11/20)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[1/10, 3/20, 1/5, 11/20];ONE fixed statistic=[0, 1, 8, 9]

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

variance parameter0 outcome0 weighted T := mul(1/16, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(1/8, 1)

variance parameter0 outcome1 weighted T = 1/8

variance parameter0 outcome2 weighted T := mul(3/16, 8)

variance parameter0 outcome2 weighted T = 3/2

variance parameter0 outcome3 weighted T := mul(5/8, 9)

variance parameter0 outcome3 weighted T = 45/8

variance parameter0 SAME-law mean := sum(0, 1/8, 3/2, 45/8)

variance parameter0 SAME-law mean = 29/4

variance parameter0 outcome0 T minus SAME mean := sub(0, 29/4)

variance parameter0 outcome0 T minus SAME mean = -29/4

variance parameter0 outcome0 centered square := mul(-29/4, -29/4)

variance parameter0 outcome0 centered square = 841/16

variance parameter0 outcome0 weighted centered square := mul(1/16, 841/16)

variance parameter0 outcome0 weighted centered square = 841/256

variance parameter0 outcome1 T minus SAME mean := sub(1, 29/4)

variance parameter0 outcome1 T minus SAME mean = -25/4

variance parameter0 outcome1 centered square := mul(-25/4, -25/4)

variance parameter0 outcome1 centered square = 625/16

variance parameter0 outcome1 weighted centered square := mul(1/8, 625/16)

variance parameter0 outcome1 weighted centered square = 625/128

variance parameter0 outcome2 T minus SAME mean := sub(8, 29/4)

variance parameter0 outcome2 T minus SAME mean = 3/4

variance parameter0 outcome2 centered square := mul(3/4, 3/4)

variance parameter0 outcome2 centered square = 9/16

variance parameter0 outcome2 weighted centered square := mul(3/16, 9/16)

variance parameter0 outcome2 weighted centered square = 27/256

variance parameter0 outcome3 T minus SAME mean := sub(9, 29/4)

variance parameter0 outcome3 T minus SAME mean = 7/4

variance parameter0 outcome3 centered square := mul(7/4, 7/4)

variance parameter0 outcome3 centered square = 49/16

variance parameter0 outcome3 weighted centered square := mul(5/8, 49/16)

variance parameter0 outcome3 weighted centered square = 245/128

variance parameter0 full centered variance := sum(841/256, 625/128, 27/256, 245/128)

variance parameter0 full centered variance = 163/16

FIXED parameter SLOT=0;original same law=[1/16, 1/8, 3/16, 5/8];same statistic=[0, 1, 8, 9];its own mean=29/4;ALL centered deviations=[-29/4, -25/4, 3/4, 7/4]

ALL nonnegative weighted centered squares=[841/256, 625/128, 27/256, 245/128];variance=163/16;variance0 iff EVERY positive-mass outcome has T=THIS mean

逐步反向核對

• 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, 29/4, 163/16, 是, same_theta_sampling_law_support_and_mean)

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。