第 20729 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-markov-square-derivation；資料={"masses":[[1,9],[2,9],[6,9]],"values":[0,4,7],"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-markov-square-derivation；資料={"masses":[[1,9],[2,9],[6,9]],"values":[0,4,7],"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-markov-square-derivation;original snapshot and mathematics retained

explicit ordered source output fields=[squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, 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

Chebyshev original atom0 absolute-event probability mass := mul(1/9, 1)

Chebyshev original atom0 absolute-event probability mass = 1/9

ORIGINAL atom=0;SIGNED centered deviation=-50/9;absolute deviation=50/9;positive epsilon=1

event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=1/9

Chebyshev original atom1 absolute-event probability mass := mul(2/9, 1)

Chebyshev original atom1 absolute-event probability mass = 2/9

ORIGINAL atom=1;SIGNED centered deviation=-14/9;absolute deviation=14/9;positive epsilon=1

event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=2/9

Chebyshev original atom2 absolute-event probability mass := mul(2/3, 1)

Chebyshev original atom2 absolute-event probability mass = 2/3

ORIGINAL atom=2;SIGNED centered deviation=13/9;absolute deviation=13/9;positive epsilon=1

event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=2/3

Chebyshev COMPLETE absolute-deviation tail probability := sum(1/9, 2/9, 2/3)

Chebyshev COMPLETE absolute-deviation tail probability = 1

Chebyshev POSITIVE epsilon SQUARED threshold := mul(1, 1)

Chebyshev POSITIVE epsilon SQUARED threshold = 1

positive original epsilon=1;consider ANY real centered deviation d;let r=abs(d)>=0

r+epsilon>0,so r>=epsilon iff(r-epsilon)*(r+epsilon)>=0 iff d^2>=epsilon^2;POINTWISE event identity,not just accidental probability equality

ORIGINAL atom=0;deviation=-50/9;its SQUARE=2500/81;EPSILON SQUARED=1

absolute event=是;squared event=是;same pointwise membership=是

ORIGINAL atom=1;deviation=-14/9;its SQUARE=196/81;EPSILON SQUARED=1

absolute event=是;squared event=是;same pointwise membership=是

ORIGINAL atom=2;deviation=13/9;its SQUARE=169/81;EPSILON SQUARED=1

absolute event=是;squared event=是;same pointwise membership=是

Chebyshev RAW variance over epsilon SQUARED := div(434/81, 1)

Chebyshev RAW variance over epsilon SQUARED = 434/81

Chebyshev Markov atom0 squared-threshold indicator := mul(1, 1)

Chebyshev Markov atom0 squared-threshold indicator = 1

Chebyshev Markov atom0 Y minus threshold indicator := sub(2500/81, 1)

Chebyshev Markov atom0 Y minus threshold indicator = 2419/81

Chebyshev Markov atom0 weighted nonnegative slack := mul(1/9, 2419/81)

Chebyshev Markov atom0 weighted nonnegative slack = 2419/729

Chebyshev Markov atom1 squared-threshold indicator := mul(1, 1)

Chebyshev Markov atom1 squared-threshold indicator = 1

Chebyshev Markov atom1 Y minus threshold indicator := sub(196/81, 1)

Chebyshev Markov atom1 Y minus threshold indicator = 115/81

Chebyshev Markov atom1 weighted nonnegative slack := mul(2/9, 115/81)

Chebyshev Markov atom1 weighted nonnegative slack = 230/729

Chebyshev Markov atom2 squared-threshold indicator := mul(1, 1)

Chebyshev Markov atom2 squared-threshold indicator = 1

Chebyshev Markov atom2 Y minus threshold indicator := sub(169/81, 1)

Chebyshev Markov atom2 Y minus threshold indicator = 88/81

Chebyshev Markov atom2 weighted nonnegative slack := mul(2/3, 88/81)

Chebyshev Markov atom2 weighted nonnegative slack = 176/243

Chebyshev Markov ALL weighted slacks sum := sum(2419/729, 230/729, 176/243)

Chebyshev Markov ALL weighted slacks sum = 353/81

Chebyshev epsilon squared times SAME event probability := mul(1, 1)

Chebyshev epsilon squared times SAME event probability = 1

Chebyshev variance minus threshold-event mass := sub(434/81, 1)

Chebyshev variance minus threshold-event mass = 353/81

Y=ORIGINAL squared deviations=[2500/81, 196/81, 169/81] nonnegative at EVERY atom;E[Y]=variance=434/81;positive threshold epsilon^2=1;complete weighted slacks=[2419/729, 230/729, 176/243]

variance-epsilon^2*tail=353/81>=0;divide POSITIVE epsilon^2 to obtain actual tail=1<=RAW bound=434/81;no independence and no equality/clipping claim

Chebyshev ORIGINAL atom0 SQUARED-event probability mass := mul(1/9, 1)

Chebyshev ORIGINAL atom0 SQUARED-event probability mass = 1/9

Chebyshev ORIGINAL atom1 SQUARED-event probability mass := mul(2/9, 1)

Chebyshev ORIGINAL atom1 SQUARED-event probability mass = 2/9

Chebyshev ORIGINAL atom2 SQUARED-event probability mass := mul(2/3, 1)

Chebyshev ORIGINAL atom2 SQUARED-event probability mass = 2/3

Chebyshev COMPLETE squared-event probability := sum(1/9, 2/9, 2/3)

Chebyshev COMPLETE squared-event probability = 1

ALL original outcome-order squared values=[2500/81, 196/81, 169/81];positive square threshold=1;ALL original squared-event weighted terms=[1/9, 2/9, 2/3]

complete squared-tail probability=1=absolute-tail probability=1;equality follows from POINTWISE event identity;RAW bound=434/81 remains separate

逐步反向核對

• S048 · 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。
• S047 · premise_and_result_audit — 以精確有理數重算 Chebyshev COMPLETE squared-event probability，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Chebyshev ORIGINAL atom2 SQUARED-event probability mass，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Chebyshev ORIGINAL atom1 SQUARED-event probability mass，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Chebyshev ORIGINAL atom0 SQUARED-event probability mass，不以小數近似替代。
• 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 Chebyshev variance minus threshold-event mass，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Chebyshev epsilon squared times SAME event probability，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Chebyshev Markov ALL weighted slacks sum，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Chebyshev Markov atom2 weighted nonnegative slack，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Chebyshev Markov atom2 Y minus threshold indicator，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Chebyshev Markov atom2 squared-threshold indicator，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Chebyshev Markov atom1 weighted nonnegative slack，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Chebyshev Markov atom1 Y minus threshold indicator，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Chebyshev Markov atom1 squared-threshold indicator，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Chebyshev Markov atom0 weighted nonnegative slack，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Chebyshev Markov atom0 Y minus threshold indicator，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Chebyshev Markov atom0 squared-threshold indicator，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Chebyshev RAW variance over epsilon SQUARED，不以小數近似替代。
• S029 · 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。
• S028 · 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。
• S027 · 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。
• S026 · 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。
• S025 · premise_and_result_audit — 以精確有理數重算 Chebyshev POSITIVE epsilon SQUARED threshold，不以小數近似替代。
• 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

([2500/81, 196/81, 169/81], 1, 1, 1, 是, 434/81, Markov_applied_to_nonnegative_squared_deviation)

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。