第 19730 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":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],"statistic":[0,1,6,7]}。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":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],"statistic":[0,1,6,7]}。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(1/14, 1/7, 3/14, 4/7)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/14, 1/7, 3/14, 4/7];ONE fixed statistic=[0, 1, 6, 7]

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

variance parameter1 complete law mass := sum(1/9, 1/6, 2/9, 1/2)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[1/9, 1/6, 2/9, 1/2];ONE fixed statistic=[0, 1, 6, 7]

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

variance parameter0 outcome0 weighted T := mul(1/14, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(1/7, 1)

variance parameter0 outcome1 weighted T = 1/7

variance parameter0 outcome2 weighted T := mul(3/14, 6)

variance parameter0 outcome2 weighted T = 9/7

variance parameter0 outcome3 weighted T := mul(4/7, 7)

variance parameter0 outcome3 weighted T = 4

variance parameter0 SAME-law mean := sum(0, 1/7, 9/7, 4)

variance parameter0 SAME-law mean = 38/7

variance parameter0 outcome0 T minus SAME mean := sub(0, 38/7)

variance parameter0 outcome0 T minus SAME mean = -38/7

variance parameter0 outcome0 centered square := mul(-38/7, -38/7)

variance parameter0 outcome0 centered square = 1444/49

variance parameter0 outcome0 weighted centered square := mul(1/14, 1444/49)

variance parameter0 outcome0 weighted centered square = 722/343

variance parameter0 outcome1 T minus SAME mean := sub(1, 38/7)

variance parameter0 outcome1 T minus SAME mean = -31/7

variance parameter0 outcome1 centered square := mul(-31/7, -31/7)

variance parameter0 outcome1 centered square = 961/49

variance parameter0 outcome1 weighted centered square := mul(1/7, 961/49)

variance parameter0 outcome1 weighted centered square = 961/343

variance parameter0 outcome2 T minus SAME mean := sub(6, 38/7)

variance parameter0 outcome2 T minus SAME mean = 4/7

variance parameter0 outcome2 centered square := mul(4/7, 4/7)

variance parameter0 outcome2 centered square = 16/49

variance parameter0 outcome2 weighted centered square := mul(3/14, 16/49)

variance parameter0 outcome2 weighted centered square = 24/343

variance parameter0 outcome3 T minus SAME mean := sub(7, 38/7)

variance parameter0 outcome3 T minus SAME mean = 11/7

variance parameter0 outcome3 centered square := mul(11/7, 11/7)

variance parameter0 outcome3 centered square = 121/49

variance parameter0 outcome3 weighted centered square := mul(4/7, 121/49)

variance parameter0 outcome3 weighted centered square = 484/343

variance parameter0 full centered variance := sum(722/343, 961/343, 24/343, 484/343)

variance parameter0 full centered variance = 313/49

FIXED parameter SLOT=0;original same law=[1/14, 1/7, 3/14, 4/7];same statistic=[0, 1, 6, 7];its own mean=38/7;ALL centered deviations=[-38/7, -31/7, 4/7, 11/7]

ALL nonnegative weighted centered squares=[722/343, 961/343, 24/343, 484/343];variance=313/49;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(1/9, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(1/6, 1)

variance parameter1 outcome1 weighted T = 1/6

variance parameter1 outcome2 weighted T := mul(2/9, 6)

variance parameter1 outcome2 weighted T = 4/3

variance parameter1 outcome3 weighted T := mul(1/2, 7)

variance parameter1 outcome3 weighted T = 7/2

variance parameter1 SAME-law mean := sum(0, 1/6, 4/3, 7/2)

variance parameter1 SAME-law mean = 5

variance parameter1 outcome0 T minus SAME mean := sub(0, 5)

variance parameter1 outcome0 T minus SAME mean = -5

variance parameter1 outcome0 centered square := mul(-5, -5)

variance parameter1 outcome0 centered square = 25

variance parameter1 outcome0 weighted centered square := mul(1/9, 25)

variance parameter1 outcome0 weighted centered square = 25/9

variance parameter1 outcome1 T minus SAME mean := sub(1, 5)

variance parameter1 outcome1 T minus SAME mean = -4

variance parameter1 outcome1 centered square := mul(-4, -4)

variance parameter1 outcome1 centered square = 16

variance parameter1 outcome1 weighted centered square := mul(1/6, 16)

variance parameter1 outcome1 weighted centered square = 8/3

variance parameter1 outcome2 T minus SAME mean := sub(6, 5)

variance parameter1 outcome2 T minus SAME mean = 1

variance parameter1 outcome2 centered square := mul(1, 1)

variance parameter1 outcome2 centered square = 1

variance parameter1 outcome2 weighted centered square := mul(2/9, 1)

variance parameter1 outcome2 weighted centered square = 2/9

variance parameter1 outcome3 T minus SAME mean := sub(7, 5)

variance parameter1 outcome3 T minus SAME mean = 2

variance parameter1 outcome3 centered square := mul(2, 2)

variance parameter1 outcome3 centered square = 4

variance parameter1 outcome3 weighted centered square := mul(1/2, 4)

variance parameter1 outcome3 weighted centered square = 2

variance parameter1 full centered variance := sum(25/9, 8/3, 2/9, 2)

variance parameter1 full centered variance = 23/3

FIXED parameter SLOT=1;original same law=[1/9, 1/6, 2/9, 1/2];same statistic=[0, 1, 6, 7];its own mean=5;ALL centered deviations=[-5, -4, 1, 2]

ALL nonnegative weighted centered squares=[25/9, 8/3, 2/9, 2];variance=23/3;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, 38/7, 313/49], [1, 5, 23/3]], 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。