第 19739 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-family-readback；資料={"laws":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],"statistic":[0,1,15,16]}。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-family-readback；資料={"laws":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],"statistic":[0,1,15,16]}。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-family-readback;original snapshot and mathematics retained

explicit ordered source output fields=[parameter_variance_rows, certificate]

variance parameter0 complete law mass := sum(1/23, 2/23, 3/23, 17/23)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/23, 2/23, 3/23, 17/23];ONE fixed statistic=[0, 1, 15, 16]

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

variance parameter1 complete law mass := sum(2/27, 1/9, 4/27, 2/3)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[2/27, 1/9, 4/27, 2/3];ONE fixed statistic=[0, 1, 15, 16]

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

variance parameter0 outcome0 weighted T := mul(1/23, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(2/23, 1)

variance parameter0 outcome1 weighted T = 2/23

variance parameter0 outcome2 weighted T := mul(3/23, 15)

variance parameter0 outcome2 weighted T = 45/23

variance parameter0 outcome3 weighted T := mul(17/23, 16)

variance parameter0 outcome3 weighted T = 272/23

variance parameter0 SAME-law mean := sum(0, 2/23, 45/23, 272/23)

variance parameter0 SAME-law mean = 319/23

variance parameter0 outcome0 T minus SAME mean := sub(0, 319/23)

variance parameter0 outcome0 T minus SAME mean = -319/23

variance parameter0 outcome0 centered square := mul(-319/23, -319/23)

variance parameter0 outcome0 centered square = 101761/529

variance parameter0 outcome0 weighted centered square := mul(1/23, 101761/529)

variance parameter0 outcome0 weighted centered square = 101761/12167

variance parameter0 outcome1 T minus SAME mean := sub(1, 319/23)

variance parameter0 outcome1 T minus SAME mean = -296/23

variance parameter0 outcome1 centered square := mul(-296/23, -296/23)

variance parameter0 outcome1 centered square = 87616/529

variance parameter0 outcome1 weighted centered square := mul(2/23, 87616/529)

variance parameter0 outcome1 weighted centered square = 175232/12167

variance parameter0 outcome2 T minus SAME mean := sub(15, 319/23)

variance parameter0 outcome2 T minus SAME mean = 26/23

variance parameter0 outcome2 centered square := mul(26/23, 26/23)

variance parameter0 outcome2 centered square = 676/529

variance parameter0 outcome2 weighted centered square := mul(3/23, 676/529)

variance parameter0 outcome2 weighted centered square = 2028/12167

variance parameter0 outcome3 T minus SAME mean := sub(16, 319/23)

variance parameter0 outcome3 T minus SAME mean = 49/23

variance parameter0 outcome3 centered square := mul(49/23, 49/23)

variance parameter0 outcome3 centered square = 2401/529

variance parameter0 outcome3 weighted centered square := mul(17/23, 2401/529)

variance parameter0 outcome3 weighted centered square = 40817/12167

variance parameter0 full centered variance := sum(101761/12167, 175232/12167, 2028/12167, 40817/12167)

variance parameter0 full centered variance = 13906/529

FIXED parameter SLOT=0;original same law=[1/23, 2/23, 3/23, 17/23];same statistic=[0, 1, 15, 16];its own mean=319/23;ALL centered deviations=[-319/23, -296/23, 26/23, 49/23]

ALL nonnegative weighted centered squares=[101761/12167, 175232/12167, 2028/12167, 40817/12167];variance=13906/529;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(2/27, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(1/9, 1)

variance parameter1 outcome1 weighted T = 1/9

variance parameter1 outcome2 weighted T := mul(4/27, 15)

variance parameter1 outcome2 weighted T = 20/9

variance parameter1 outcome3 weighted T := mul(2/3, 16)

variance parameter1 outcome3 weighted T = 32/3

variance parameter1 SAME-law mean := sum(0, 1/9, 20/9, 32/3)

variance parameter1 SAME-law mean = 13

variance parameter1 outcome0 T minus SAME mean := sub(0, 13)

variance parameter1 outcome0 T minus SAME mean = -13

variance parameter1 outcome0 centered square := mul(-13, -13)

variance parameter1 outcome0 centered square = 169

variance parameter1 outcome0 weighted centered square := mul(2/27, 169)

variance parameter1 outcome0 weighted centered square = 338/27

variance parameter1 outcome1 T minus SAME mean := sub(1, 13)

variance parameter1 outcome1 T minus SAME mean = -12

variance parameter1 outcome1 centered square := mul(-12, -12)

variance parameter1 outcome1 centered square = 144

variance parameter1 outcome1 weighted centered square := mul(1/9, 144)

variance parameter1 outcome1 weighted centered square = 16

variance parameter1 outcome2 T minus SAME mean := sub(15, 13)

variance parameter1 outcome2 T minus SAME mean = 2

variance parameter1 outcome2 centered square := mul(2, 2)

variance parameter1 outcome2 centered square = 4

variance parameter1 outcome2 weighted centered square := mul(4/27, 4)

variance parameter1 outcome2 weighted centered square = 16/27

variance parameter1 outcome3 T minus SAME mean := sub(16, 13)

variance parameter1 outcome3 T minus SAME mean = 3

variance parameter1 outcome3 centered square := mul(3, 3)

variance parameter1 outcome3 centered square = 9

variance parameter1 outcome3 weighted centered square := mul(2/3, 9)

variance parameter1 outcome3 weighted centered square = 6

variance parameter1 full centered variance := sum(338/27, 16, 16/27, 6)

variance parameter1 full centered variance = 316/9

FIXED parameter SLOT=1;original same law=[2/27, 1/9, 4/27, 2/3];same statistic=[0, 1, 15, 16];its own mean=13;ALL centered deviations=[-13, -12, 2, 3]

ALL nonnegative weighted centered squares=[338/27, 16, 16/27, 6];variance=316/9;variance0 iff EVERY positive-mass outcome has T=THIS mean

逐步反向核對

• S043 · 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。
• S042 · 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。
• S041 · 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。
• S040 · 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。
• S039 · 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。
• S038 · 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。
• S037 · 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。
• S036 · 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。
• S035 · 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。
• S034 · 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。
• S033 · 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。
• S032 · 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。
• S031 · 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。
• S030 · 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。
• S029 · 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。
• S028 · 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。
• S027 · 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。
• S026 · 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。
• S025 · 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。
• 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, 319/23, 13906/529], [1, 13, 316/9]], variance_is_parameter_indexed_family)

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。