第 20754 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-qualification；資料={"masses":[[1,9],[2,9],[6,9]],"values":[0,4,7]}。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-qualification；資料={"masses":[[1,9],[2,9],[6,9]],"values":[0,4,7]}。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-chebyshev-inequality

逐步推導

source operation=cheb-qualification;original snapshot and mathematics retained

explicit ordered source output fields=[masses, values, mean, variance, finite_moment_certificate, certificate]

Chebyshev complete probability law mass := sum(1/9, 2/9, 2/3)

Chebyshev complete probability law mass = 1

Chebyshev original atom0 weighted SIGNED X := mul(1/9, 0)

Chebyshev original atom0 weighted SIGNED X = 0

Chebyshev original atom1 weighted SIGNED X := mul(2/9, 4)

Chebyshev original atom1 weighted SIGNED X = 8/9

Chebyshev original atom2 weighted SIGNED X := mul(2/3, 7)

Chebyshev original atom2 weighted SIGNED X = 14/3

Chebyshev complete SAME-law mean := sum(0, 8/9, 14/3)

Chebyshev complete SAME-law mean = 50/9

Chebyshev original atom0 X minus SAME mean := sub(0, 50/9)

Chebyshev original atom0 X minus SAME mean = -50/9

Chebyshev original atom0 centered SQUARE := mul(-50/9, -50/9)

Chebyshev original atom0 centered SQUARE = 2500/81

Chebyshev original atom0 weighted centered square := mul(1/9, 2500/81)

Chebyshev original atom0 weighted centered square = 2500/729

Chebyshev original atom1 X minus SAME mean := sub(4, 50/9)

Chebyshev original atom1 X minus SAME mean = -14/9

Chebyshev original atom1 centered SQUARE := mul(-14/9, -14/9)

Chebyshev original atom1 centered SQUARE = 196/81

Chebyshev original atom1 weighted centered square := mul(2/9, 196/81)

Chebyshev original atom1 weighted centered square = 392/729

Chebyshev original atom2 X minus SAME mean := sub(7, 50/9)

Chebyshev original atom2 X minus SAME mean = 13/9

Chebyshev original atom2 centered SQUARE := mul(13/9, 13/9)

Chebyshev original atom2 centered SQUARE = 169/81

Chebyshev original atom2 weighted centered square := mul(2/3, 169/81)

Chebyshev original atom2 weighted centered square = 338/243

Chebyshev full variance E centered square := sum(2500/729, 392/729, 338/243)

Chebyshev full variance E centered square = 434/81

ORIGINAL complete normalized masses=[1/9, 2/9, 2/3];ALL aligned SIGNED values X=[0, 4, 7];SAME-law mean=50/9;all deviations=[-50/9, -14/9, 13/9]

ALL nonnegative squared deviations Y=[2500/81, 196/81, 169/81];variance=E[Y]=434/81;finite moments;variance0 iff X=mean on EVERY POSITIVE-mass atom,not necessarily null atoms

逐步反向核對

• 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

([1/9, 2/9, 2/3], [0, 4, 7], 50/9, 434/81, 是, finite_mean_and_variance)

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。