第 19748 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":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],"statistic":[0,1,24,25]}。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":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],"statistic":[0,1,24,25]}。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(1/32, 1/16, 3/32, 13/16)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/32, 1/16, 3/32, 13/16];ONE fixed statistic=[0, 1, 24, 25]

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

variance parameter1 complete law mass := sum(1/18, 1/12, 1/9, 3/4)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[1/18, 1/12, 1/9, 3/4];ONE fixed statistic=[0, 1, 24, 25]

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

variance parameter0 outcome0 weighted T := mul(1/32, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(1/16, 1)

variance parameter0 outcome1 weighted T = 1/16

variance parameter0 outcome2 weighted T := mul(3/32, 24)

variance parameter0 outcome2 weighted T = 9/4

variance parameter0 outcome3 weighted T := mul(13/16, 25)

variance parameter0 outcome3 weighted T = 325/16

variance parameter0 SAME-law mean := sum(0, 1/16, 9/4, 325/16)

variance parameter0 SAME-law mean = 181/8

variance parameter0 outcome0 T minus SAME mean := sub(0, 181/8)

variance parameter0 outcome0 T minus SAME mean = -181/8

variance parameter0 outcome0 centered square := mul(-181/8, -181/8)

variance parameter0 outcome0 centered square = 32761/64

variance parameter0 outcome0 weighted centered square := mul(1/32, 32761/64)

variance parameter0 outcome0 weighted centered square = 32761/2048

variance parameter0 outcome1 T minus SAME mean := sub(1, 181/8)

variance parameter0 outcome1 T minus SAME mean = -173/8

variance parameter0 outcome1 centered square := mul(-173/8, -173/8)

variance parameter0 outcome1 centered square = 29929/64

variance parameter0 outcome1 weighted centered square := mul(1/16, 29929/64)

variance parameter0 outcome1 weighted centered square = 29929/1024

variance parameter0 outcome2 T minus SAME mean := sub(24, 181/8)

variance parameter0 outcome2 T minus SAME mean = 11/8

variance parameter0 outcome2 centered square := mul(11/8, 11/8)

variance parameter0 outcome2 centered square = 121/64

variance parameter0 outcome2 weighted centered square := mul(3/32, 121/64)

variance parameter0 outcome2 weighted centered square = 363/2048

variance parameter0 outcome3 T minus SAME mean := sub(25, 181/8)

variance parameter0 outcome3 T minus SAME mean = 19/8

variance parameter0 outcome3 centered square := mul(19/8, 19/8)

variance parameter0 outcome3 centered square = 361/64

variance parameter0 outcome3 weighted centered square := mul(13/16, 361/64)

variance parameter0 outcome3 weighted centered square = 4693/1024

variance parameter0 full centered variance := sum(32761/2048, 29929/1024, 363/2048, 4693/1024)

variance parameter0 full centered variance = 3199/64

FIXED parameter SLOT=0;original same law=[1/32, 1/16, 3/32, 13/16];same statistic=[0, 1, 24, 25];its own mean=181/8;ALL centered deviations=[-181/8, -173/8, 11/8, 19/8]

ALL nonnegative weighted centered squares=[32761/2048, 29929/1024, 363/2048, 4693/1024];variance=3199/64;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(1/18, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(1/12, 1)

variance parameter1 outcome1 weighted T = 1/12

variance parameter1 outcome2 weighted T := mul(1/9, 24)

variance parameter1 outcome2 weighted T = 8/3

variance parameter1 outcome3 weighted T := mul(3/4, 25)

variance parameter1 outcome3 weighted T = 75/4

variance parameter1 SAME-law mean := sum(0, 1/12, 8/3, 75/4)

variance parameter1 SAME-law mean = 43/2

variance parameter1 outcome0 T minus SAME mean := sub(0, 43/2)

variance parameter1 outcome0 T minus SAME mean = -43/2

variance parameter1 outcome0 centered square := mul(-43/2, -43/2)

variance parameter1 outcome0 centered square = 1849/4

variance parameter1 outcome0 weighted centered square := mul(1/18, 1849/4)

variance parameter1 outcome0 weighted centered square = 1849/72

variance parameter1 outcome1 T minus SAME mean := sub(1, 43/2)

variance parameter1 outcome1 T minus SAME mean = -41/2

variance parameter1 outcome1 centered square := mul(-41/2, -41/2)

variance parameter1 outcome1 centered square = 1681/4

variance parameter1 outcome1 weighted centered square := mul(1/12, 1681/4)

variance parameter1 outcome1 weighted centered square = 1681/48

variance parameter1 outcome2 T minus SAME mean := sub(24, 43/2)

variance parameter1 outcome2 T minus SAME mean = 5/2

variance parameter1 outcome2 centered square := mul(5/2, 5/2)

variance parameter1 outcome2 centered square = 25/4

variance parameter1 outcome2 weighted centered square := mul(1/9, 25/4)

variance parameter1 outcome2 weighted centered square = 25/36

variance parameter1 outcome3 T minus SAME mean := sub(25, 43/2)

variance parameter1 outcome3 T minus SAME mean = 7/2

variance parameter1 outcome3 centered square := mul(7/2, 7/2)

variance parameter1 outcome3 centered square = 49/4

variance parameter1 outcome3 weighted centered square := mul(3/4, 49/4)

variance parameter1 outcome3 weighted centered square = 147/16

variance parameter1 full centered variance := sum(1849/72, 1681/48, 25/36, 147/16)

variance parameter1 full centered variance = 847/12

FIXED parameter SLOT=1;original same law=[1/18, 1/12, 1/9, 3/4];same statistic=[0, 1, 24, 25];its own mean=43/2;ALL centered deviations=[-43/2, -41/2, 5/2, 7/2]

ALL nonnegative weighted centered squares=[1849/72, 1681/48, 25/36, 147/16];variance=847/12;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, 181/8, 3199/64], [1, 43/2, 847/12]], 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。