第 20706 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-bound-comparison；資料={"masses":[[1,11],[2,11],[8,11]],"values":[0,6,9],"epsilon":[3,1]}。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-bound-comparison；資料={"masses":[[1,11],[2,11],[8,11]],"values":[0,6,9],"epsilon":[3,1]}。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-bound-comparison;original snapshot and mathematics retained

explicit ordered source output fields=[tail_probability, variance, raw_upper_bound, bound_holds, certificate]

Chebyshev complete probability law mass := sum(1/11, 2/11, 8/11)

Chebyshev complete probability law mass = 1

Chebyshev original atom0 weighted SIGNED X := mul(1/11, 0)

Chebyshev original atom0 weighted SIGNED X = 0

Chebyshev original atom1 weighted SIGNED X := mul(2/11, 6)

Chebyshev original atom1 weighted SIGNED X = 12/11

Chebyshev original atom2 weighted SIGNED X := mul(8/11, 9)

Chebyshev original atom2 weighted SIGNED X = 72/11

Chebyshev complete SAME-law mean := sum(0, 12/11, 72/11)

Chebyshev complete SAME-law mean = 84/11

Chebyshev original atom0 X minus SAME mean := sub(0, 84/11)

Chebyshev original atom0 X minus SAME mean = -84/11

Chebyshev original atom0 centered SQUARE := mul(-84/11, -84/11)

Chebyshev original atom0 centered SQUARE = 7056/121

Chebyshev original atom0 weighted centered square := mul(1/11, 7056/121)

Chebyshev original atom0 weighted centered square = 7056/1331

Chebyshev original atom1 X minus SAME mean := sub(6, 84/11)

Chebyshev original atom1 X minus SAME mean = -18/11

Chebyshev original atom1 centered SQUARE := mul(-18/11, -18/11)

Chebyshev original atom1 centered SQUARE = 324/121

Chebyshev original atom1 weighted centered square := mul(2/11, 324/121)

Chebyshev original atom1 weighted centered square = 648/1331

Chebyshev original atom2 X minus SAME mean := sub(9, 84/11)

Chebyshev original atom2 X minus SAME mean = 15/11

Chebyshev original atom2 centered SQUARE := mul(15/11, 15/11)

Chebyshev original atom2 centered SQUARE = 225/121

Chebyshev original atom2 weighted centered square := mul(8/11, 225/121)

Chebyshev original atom2 weighted centered square = 1800/1331

Chebyshev full variance E centered square := sum(7056/1331, 648/1331, 1800/1331)

Chebyshev full variance E centered square = 864/121

ORIGINAL complete normalized masses=[1/11, 2/11, 8/11];ALL aligned SIGNED values X=[0, 6, 9];SAME-law mean=84/11;all deviations=[-84/11, -18/11, 15/11]

ALL nonnegative squared deviations Y=[7056/121, 324/121, 225/121];variance=E[Y]=864/121;finite moments;variance0 iff X=mean on EVERY POSITIVE-mass atom,not necessarily null atoms

Chebyshev original atom0 absolute-event probability mass := mul(1/11, 1)

Chebyshev original atom0 absolute-event probability mass = 1/11

ORIGINAL atom=0;SIGNED centered deviation=-84/11;absolute deviation=84/11;positive epsilon=3

event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=1/11

Chebyshev original atom1 absolute-event probability mass := mul(2/11, 0)

Chebyshev original atom1 absolute-event probability mass = 0

ORIGINAL atom=1;SIGNED centered deviation=-18/11;absolute deviation=18/11;positive epsilon=3

event abs(deviation)>=epsilon=否;EQUALITY INCLUDED;probability contribution=0

Chebyshev original atom2 absolute-event probability mass := mul(8/11, 0)

Chebyshev original atom2 absolute-event probability mass = 0

ORIGINAL atom=2;SIGNED centered deviation=15/11;absolute deviation=15/11;positive epsilon=3

event abs(deviation)>=epsilon=否;EQUALITY INCLUDED;probability contribution=0

Chebyshev COMPLETE absolute-deviation tail probability := sum(1/11, 0, 0)

Chebyshev COMPLETE absolute-deviation tail probability = 1/11

Chebyshev POSITIVE epsilon SQUARED threshold := mul(3, 3)

Chebyshev POSITIVE epsilon SQUARED threshold = 9

positive original epsilon=3;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=-84/11;its SQUARE=7056/121;EPSILON SQUARED=9

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

ORIGINAL atom=1;deviation=-18/11;its SQUARE=324/121;EPSILON SQUARED=9

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

ORIGINAL atom=2;deviation=15/11;its SQUARE=225/121;EPSILON SQUARED=9

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

Chebyshev RAW variance over epsilon SQUARED := div(864/121, 9)

Chebyshev RAW variance over epsilon SQUARED = 96/121

Chebyshev Markov atom0 squared-threshold indicator := mul(9, 1)

Chebyshev Markov atom0 squared-threshold indicator = 9

Chebyshev Markov atom0 Y minus threshold indicator := sub(7056/121, 9)

Chebyshev Markov atom0 Y minus threshold indicator = 5967/121

Chebyshev Markov atom0 weighted nonnegative slack := mul(1/11, 5967/121)

Chebyshev Markov atom0 weighted nonnegative slack = 5967/1331

Chebyshev Markov atom1 squared-threshold indicator := mul(9, 0)

Chebyshev Markov atom1 squared-threshold indicator = 0

Chebyshev Markov atom1 Y minus threshold indicator := sub(324/121, 0)

Chebyshev Markov atom1 Y minus threshold indicator = 324/121

Chebyshev Markov atom1 weighted nonnegative slack := mul(2/11, 324/121)

Chebyshev Markov atom1 weighted nonnegative slack = 648/1331

Chebyshev Markov atom2 squared-threshold indicator := mul(9, 0)

Chebyshev Markov atom2 squared-threshold indicator = 0

Chebyshev Markov atom2 Y minus threshold indicator := sub(225/121, 0)

Chebyshev Markov atom2 Y minus threshold indicator = 225/121

Chebyshev Markov atom2 weighted nonnegative slack := mul(8/11, 225/121)

Chebyshev Markov atom2 weighted nonnegative slack = 1800/1331

Chebyshev Markov ALL weighted slacks sum := sum(5967/1331, 648/1331, 1800/1331)

Chebyshev Markov ALL weighted slacks sum = 765/121

Chebyshev epsilon squared times SAME event probability := mul(9, 1/11)

Chebyshev epsilon squared times SAME event probability = 9/11

Chebyshev variance minus threshold-event mass := sub(864/121, 9/11)

Chebyshev variance minus threshold-event mass = 765/121

Y=ORIGINAL squared deviations=[7056/121, 324/121, 225/121] nonnegative at EVERY atom;E[Y]=variance=864/121;positive threshold epsilon^2=9;complete weighted slacks=[5967/1331, 648/1331, 1800/1331]

variance-epsilon^2*tail=765/121>=0;divide POSITIVE epsilon^2 to obtain actual tail=1/11<=RAW bound=96/121;no independence and no equality/clipping claim

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 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

(1/11, 864/121, 96/121, 是, upper_bound_not_tail_equality_and_no_independence_needed)

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。