第 20791 題：Chebyshev不等式

原題與已知條件

完整有限law、finite mean與variance及positive epsilon。操作=cheb-tail-probability；資料={"masses":[[1,21],[2,21],[18,21]],"values":[0,16,19],"epsilon":[1,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 完整有限law、finite mean與variance及positive epsilon。操作=cheb-tail-probability；資料={"masses":[[1,21],[2,21],[18,21]],"values":[0,16,19],"epsilon":[1,1]}。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-chebyshev-inequality

逐步推導

source operation=cheb-tail-probability;original snapshot and mathematics retained

explicit ordered source output fields=[mean, epsilon, tail_probability, certificate]

Chebyshev complete probability law mass := sum(1/21, 2/21, 6/7)

Chebyshev complete probability law mass = 1

Chebyshev original atom0 weighted SIGNED X := mul(1/21, 0)

Chebyshev original atom0 weighted SIGNED X = 0

Chebyshev original atom1 weighted SIGNED X := mul(2/21, 16)

Chebyshev original atom1 weighted SIGNED X = 32/21

Chebyshev original atom2 weighted SIGNED X := mul(6/7, 19)

Chebyshev original atom2 weighted SIGNED X = 114/7

Chebyshev complete SAME-law mean := sum(0, 32/21, 114/7)

Chebyshev complete SAME-law mean = 374/21

Chebyshev original atom0 X minus SAME mean := sub(0, 374/21)

Chebyshev original atom0 X minus SAME mean = -374/21

Chebyshev original atom0 centered SQUARE := mul(-374/21, -374/21)

Chebyshev original atom0 centered SQUARE = 139876/441

Chebyshev original atom0 weighted centered square := mul(1/21, 139876/441)

Chebyshev original atom0 weighted centered square = 139876/9261

Chebyshev original atom1 X minus SAME mean := sub(16, 374/21)

Chebyshev original atom1 X minus SAME mean = -38/21

Chebyshev original atom1 centered SQUARE := mul(-38/21, -38/21)

Chebyshev original atom1 centered SQUARE = 1444/441

Chebyshev original atom1 weighted centered square := mul(2/21, 1444/441)

Chebyshev original atom1 weighted centered square = 2888/9261

Chebyshev original atom2 X minus SAME mean := sub(19, 374/21)

Chebyshev original atom2 X minus SAME mean = 25/21

Chebyshev original atom2 centered SQUARE := mul(25/21, 25/21)

Chebyshev original atom2 centered SQUARE = 625/441

Chebyshev original atom2 weighted centered square := mul(6/7, 625/441)

Chebyshev original atom2 weighted centered square = 1250/1029

Chebyshev full variance E centered square := sum(139876/9261, 2888/9261, 1250/1029)

Chebyshev full variance E centered square = 7334/441

ORIGINAL complete normalized masses=[1/21, 2/21, 6/7];ALL aligned SIGNED values X=[0, 16, 19];SAME-law mean=374/21;all deviations=[-374/21, -38/21, 25/21]

ALL nonnegative squared deviations Y=[139876/441, 1444/441, 625/441];variance=E[Y]=7334/441;finite moments;variance0 iff X=mean on EVERY POSITIVE-mass atom,not necessarily null atoms

Chebyshev original atom0 absolute-event probability mass := mul(1/21, 1)

Chebyshev original atom0 absolute-event probability mass = 1/21

ORIGINAL atom=0;SIGNED centered deviation=-374/21;absolute deviation=374/21;positive epsilon=1

event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=1/21

Chebyshev original atom1 absolute-event probability mass := mul(2/21, 1)

Chebyshev original atom1 absolute-event probability mass = 2/21

ORIGINAL atom=1;SIGNED centered deviation=-38/21;absolute deviation=38/21;positive epsilon=1

event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=2/21

Chebyshev original atom2 absolute-event probability mass := mul(6/7, 1)

Chebyshev original atom2 absolute-event probability mass = 6/7

ORIGINAL atom=2;SIGNED centered deviation=25/21;absolute deviation=25/21;positive epsilon=1

event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=6/7

Chebyshev COMPLETE absolute-deviation tail probability := sum(1/21, 2/21, 6/7)

Chebyshev COMPLETE absolute-deviation tail probability = 1

逐步反向核對

• S024 · premise_and_result_audit — 以精確有理數重算 Chebyshev COMPLETE absolute-deviation tail probability，不以小數近似替代。
• S023 · 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。
• S022 · premise_and_result_audit — 以精確有理數重算 Chebyshev original atom2 absolute-event probability mass，不以小數近似替代。
• S021 · 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。
• S020 · premise_and_result_audit — 以精確有理數重算 Chebyshev original atom1 absolute-event probability mass，不以小數近似替代。
• S019 · 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。
• S018 · premise_and_result_audit — 以精確有理數重算 Chebyshev original atom0 absolute-event probability mass，不以小數近似替代。
• S017 · 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。
• S016 · premise_and_result_audit — 以精確有理數重算 Chebyshev full variance E centered square，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Chebyshev original atom2 weighted centered square，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Chebyshev original atom2 centered SQUARE，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Chebyshev original atom2 X minus SAME mean，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Chebyshev original atom1 weighted centered square，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Chebyshev original atom1 centered SQUARE，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Chebyshev original atom1 X minus SAME mean，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Chebyshev original atom0 weighted centered square，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Chebyshev original atom0 centered SQUARE，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Chebyshev original atom0 X minus SAME mean，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Chebyshev complete SAME-law mean，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Chebyshev original atom2 weighted SIGNED X，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Chebyshev original atom1 weighted SIGNED X，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Chebyshev original atom0 weighted SIGNED X，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Chebyshev complete probability 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

(374/21, 1, 1, exact_absolute_deviation_tail)

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。