第 19753 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-property-boundary；資料={"laws":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],"statistic":[0,1,4,5]}。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-property-boundary；資料={"laws":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],"statistic":[0,1,4,5]}。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-property-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary]

variance parameter0 complete law mass := sum(1/12, 1/6, 1/4, 1/2)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/12, 1/6, 1/4, 1/2];ONE fixed statistic=[0, 1, 4, 5]

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

variance parameter1 complete law mass := sum(1/8, 3/16, 1/4, 7/16)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[1/8, 3/16, 1/4, 7/16];ONE fixed statistic=[0, 1, 4, 5]

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

variance parameter0 outcome0 weighted T := mul(1/12, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(1/6, 1)

variance parameter0 outcome1 weighted T = 1/6

variance parameter0 outcome2 weighted T := mul(1/4, 4)

variance parameter0 outcome2 weighted T = 1

variance parameter0 outcome3 weighted T := mul(1/2, 5)

variance parameter0 outcome3 weighted T = 5/2

variance parameter0 SAME-law mean := sum(0, 1/6, 1, 5/2)

variance parameter0 SAME-law mean = 11/3

variance parameter0 outcome0 T minus SAME mean := sub(0, 11/3)

variance parameter0 outcome0 T minus SAME mean = -11/3

variance parameter0 outcome0 centered square := mul(-11/3, -11/3)

variance parameter0 outcome0 centered square = 121/9

variance parameter0 outcome0 weighted centered square := mul(1/12, 121/9)

variance parameter0 outcome0 weighted centered square = 121/108

variance parameter0 outcome1 T minus SAME mean := sub(1, 11/3)

variance parameter0 outcome1 T minus SAME mean = -8/3

variance parameter0 outcome1 centered square := mul(-8/3, -8/3)

variance parameter0 outcome1 centered square = 64/9

variance parameter0 outcome1 weighted centered square := mul(1/6, 64/9)

variance parameter0 outcome1 weighted centered square = 32/27

variance parameter0 outcome2 T minus SAME mean := sub(4, 11/3)

variance parameter0 outcome2 T minus SAME mean = 1/3

variance parameter0 outcome2 centered square := mul(1/3, 1/3)

variance parameter0 outcome2 centered square = 1/9

variance parameter0 outcome2 weighted centered square := mul(1/4, 1/9)

variance parameter0 outcome2 weighted centered square = 1/36

variance parameter0 outcome3 T minus SAME mean := sub(5, 11/3)

variance parameter0 outcome3 T minus SAME mean = 4/3

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

variance parameter0 outcome3 centered square = 16/9

variance parameter0 outcome3 weighted centered square := mul(1/2, 16/9)

variance parameter0 outcome3 weighted centered square = 8/9

variance parameter0 full centered variance := sum(121/108, 32/27, 1/36, 8/9)

variance parameter0 full centered variance = 29/9

FIXED parameter SLOT=0;original same law=[1/12, 1/6, 1/4, 1/2];same statistic=[0, 1, 4, 5];its own mean=11/3;ALL centered deviations=[-11/3, -8/3, 1/3, 4/3]

ALL nonnegative weighted centered squares=[121/108, 32/27, 1/36, 8/9];variance=29/9;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(1/8, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/16, 1)

variance parameter1 outcome1 weighted T = 3/16

variance parameter1 outcome2 weighted T := mul(1/4, 4)

variance parameter1 outcome2 weighted T = 1

variance parameter1 outcome3 weighted T := mul(7/16, 5)

variance parameter1 outcome3 weighted T = 35/16

variance parameter1 SAME-law mean := sum(0, 3/16, 1, 35/16)

variance parameter1 SAME-law mean = 27/8

variance parameter1 outcome0 T minus SAME mean := sub(0, 27/8)

variance parameter1 outcome0 T minus SAME mean = -27/8

variance parameter1 outcome0 centered square := mul(-27/8, -27/8)

variance parameter1 outcome0 centered square = 729/64

variance parameter1 outcome0 weighted centered square := mul(1/8, 729/64)

variance parameter1 outcome0 weighted centered square = 729/512

variance parameter1 outcome1 T minus SAME mean := sub(1, 27/8)

variance parameter1 outcome1 T minus SAME mean = -19/8

variance parameter1 outcome1 centered square := mul(-19/8, -19/8)

variance parameter1 outcome1 centered square = 361/64

variance parameter1 outcome1 weighted centered square := mul(3/16, 361/64)

variance parameter1 outcome1 weighted centered square = 1083/1024

variance parameter1 outcome2 T minus SAME mean := sub(4, 27/8)

variance parameter1 outcome2 T minus SAME mean = 5/8

variance parameter1 outcome2 centered square := mul(5/8, 5/8)

variance parameter1 outcome2 centered square = 25/64

variance parameter1 outcome2 weighted centered square := mul(1/4, 25/64)

variance parameter1 outcome2 weighted centered square = 25/256

variance parameter1 outcome3 T minus SAME mean := sub(5, 27/8)

variance parameter1 outcome3 T minus SAME mean = 13/8

variance parameter1 outcome3 centered square := mul(13/8, 13/8)

variance parameter1 outcome3 centered square = 169/64

variance parameter1 outcome3 weighted centered square := mul(7/16, 169/64)

variance parameter1 outcome3 weighted centered square = 1183/1024

variance parameter1 full centered variance := sum(729/512, 1083/1024, 25/256, 1183/1024)

variance parameter1 full centered variance = 239/64

FIXED parameter SLOT=1;original same law=[1/8, 3/16, 1/4, 7/16];same statistic=[0, 1, 4, 5];its own mean=27/8;ALL centered deviations=[-27/8, -19/8, 5/8, 13/8]

ALL nonnegative weighted centered squares=[729/512, 1083/1024, 25/256, 1183/1024];variance=239/64;variance0 iff EVERY positive-mass outcome has T=THIS mean

ALL original parameter variance rows=[[0, 29/9], [1, 239/64]];no target,bias or sequence of statistics supplied

three literal No markers WITHHOLD unbiasedness,consistency or MSE claims;not proved negations and NOT an assumption all listed variances0

逐步反向核對

• S044 · 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。
• 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, 29/9], [1, 239/64]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)

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。