第 20721 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":[[1,26],[2,26],[23,26]],"values":[0,21,24],"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":[[1,26],[2,26],[23,26]],"values":[0,21,24],"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(1/26, 1/13, 23/26)

Chebyshev complete probability law mass = 1

Chebyshev original atom0 weighted SIGNED X := mul(1/26, 0)

Chebyshev original atom0 weighted SIGNED X = 0

Chebyshev original atom1 weighted SIGNED X := mul(1/13, 21)

Chebyshev original atom1 weighted SIGNED X = 21/13

Chebyshev original atom2 weighted SIGNED X := mul(23/26, 24)

Chebyshev original atom2 weighted SIGNED X = 276/13

Chebyshev complete SAME-law mean := sum(0, 21/13, 276/13)

Chebyshev complete SAME-law mean = 297/13

Chebyshev original atom0 X minus SAME mean := sub(0, 297/13)

Chebyshev original atom0 X minus SAME mean = -297/13

Chebyshev original atom0 centered SQUARE := mul(-297/13, -297/13)

Chebyshev original atom0 centered SQUARE = 88209/169

Chebyshev original atom0 weighted centered square := mul(1/26, 88209/169)

Chebyshev original atom0 weighted centered square = 88209/4394

Chebyshev original atom1 X minus SAME mean := sub(21, 297/13)

Chebyshev original atom1 X minus SAME mean = -24/13

Chebyshev original atom1 centered SQUARE := mul(-24/13, -24/13)

Chebyshev original atom1 centered SQUARE = 576/169

Chebyshev original atom1 weighted centered square := mul(1/13, 576/169)

Chebyshev original atom1 weighted centered square = 576/2197

Chebyshev original atom2 X minus SAME mean := sub(24, 297/13)

Chebyshev original atom2 X minus SAME mean = 15/13

Chebyshev original atom2 centered SQUARE := mul(15/13, 15/13)

Chebyshev original atom2 centered SQUARE = 225/169

Chebyshev original atom2 weighted centered square := mul(23/26, 225/169)

Chebyshev original atom2 weighted centered square = 5175/4394

Chebyshev full variance E centered square := sum(88209/4394, 576/2197, 5175/4394)

Chebyshev full variance E centered square = 3636/169

ORIGINAL complete normalized masses=[1/26, 1/13, 23/26];ALL aligned SIGNED values X=[0, 21, 24];SAME-law mean=297/13;all deviations=[-297/13, -24/13, 15/13]

ALL nonnegative squared deviations Y=[88209/169, 576/169, 225/169];variance=E[Y]=3636/169;finite moments;variance0 iff X=mean on EVERY POSITIVE-mass atom,not necessarily null atoms

Chebyshev original atom0 absolute-event probability mass := mul(1/26, 1)

Chebyshev original atom0 absolute-event probability mass = 1/26

ORIGINAL atom=0;SIGNED centered deviation=-297/13;absolute deviation=297/13;positive epsilon=3

event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=1/26

Chebyshev original atom1 absolute-event probability mass := mul(1/13, 0)

Chebyshev original atom1 absolute-event probability mass = 0

ORIGINAL atom=1;SIGNED centered deviation=-24/13;absolute deviation=24/13;positive epsilon=3

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

Chebyshev original atom2 absolute-event probability mass := mul(23/26, 0)

Chebyshev original atom2 absolute-event probability mass = 0

ORIGINAL atom=2;SIGNED centered deviation=15/13;absolute deviation=15/13;positive epsilon=3

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

Chebyshev COMPLETE absolute-deviation tail probability := sum(1/26, 0, 0)

Chebyshev COMPLETE absolute-deviation tail probability = 1/26

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=-297/13;its SQUARE=88209/169;EPSILON SQUARED=9

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

ORIGINAL atom=1;deviation=-24/13;its SQUARE=576/169;EPSILON SQUARED=9

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

ORIGINAL atom=2;deviation=15/13;its SQUARE=225/169;EPSILON SQUARED=9

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

Chebyshev RAW variance over epsilon SQUARED := div(3636/169, 9)

Chebyshev RAW variance over epsilon SQUARED = 404/169

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

Chebyshev Markov atom0 squared-threshold indicator = 9

Chebyshev Markov atom0 Y minus threshold indicator := sub(88209/169, 9)

Chebyshev Markov atom0 Y minus threshold indicator = 86688/169

Chebyshev Markov atom0 weighted nonnegative slack := mul(1/26, 86688/169)

Chebyshev Markov atom0 weighted nonnegative slack = 43344/2197

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

Chebyshev Markov atom1 squared-threshold indicator = 0

Chebyshev Markov atom1 Y minus threshold indicator := sub(576/169, 0)

Chebyshev Markov atom1 Y minus threshold indicator = 576/169

Chebyshev Markov atom1 weighted nonnegative slack := mul(1/13, 576/169)

Chebyshev Markov atom1 weighted nonnegative slack = 576/2197

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

Chebyshev Markov atom2 squared-threshold indicator = 0

Chebyshev Markov atom2 Y minus threshold indicator := sub(225/169, 0)

Chebyshev Markov atom2 Y minus threshold indicator = 225/169

Chebyshev Markov atom2 weighted nonnegative slack := mul(23/26, 225/169)

Chebyshev Markov atom2 weighted nonnegative slack = 5175/4394

Chebyshev Markov ALL weighted slacks sum := sum(43344/2197, 576/2197, 5175/4394)

Chebyshev Markov ALL weighted slacks sum = 7155/338

Chebyshev epsilon squared times SAME event probability := mul(9, 1/26)

Chebyshev epsilon squared times SAME event probability = 9/26

Chebyshev variance minus threshold-event mass := sub(3636/169, 9/26)

Chebyshev variance minus threshold-event mass = 7155/338

Y=ORIGINAL squared deviations=[88209/169, 576/169, 225/169] nonnegative at EVERY atom;E[Y]=variance=3636/169;positive threshold epsilon^2=9;complete weighted slacks=[43344/2197, 576/2197, 5175/4394]

variance-epsilon^2*tail=7155/338>=0;divide POSITIVE epsilon^2 to obtain actual tail=1/26<=RAW bound=404/169;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

(1/26, 3636/169, 404/169, 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。