第 20722 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":[[1,27],[2,27],[24,27]],"values":[0,22,25],"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-bound-comparison；資料={"masses":[[1,27],[2,27],[24,27]],"values":[0,22,25],"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-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(1/27, 2/27, 8/9)

Chebyshev complete probability law mass = 1

Chebyshev original atom0 weighted SIGNED X := mul(1/27, 0)

Chebyshev original atom0 weighted SIGNED X = 0

Chebyshev original atom1 weighted SIGNED X := mul(2/27, 22)

Chebyshev original atom1 weighted SIGNED X = 44/27

Chebyshev original atom2 weighted SIGNED X := mul(8/9, 25)

Chebyshev original atom2 weighted SIGNED X = 200/9

Chebyshev complete SAME-law mean := sum(0, 44/27, 200/9)

Chebyshev complete SAME-law mean = 644/27

Chebyshev original atom0 X minus SAME mean := sub(0, 644/27)

Chebyshev original atom0 X minus SAME mean = -644/27

Chebyshev original atom0 centered SQUARE := mul(-644/27, -644/27)

Chebyshev original atom0 centered SQUARE = 414736/729

Chebyshev original atom0 weighted centered square := mul(1/27, 414736/729)

Chebyshev original atom0 weighted centered square = 414736/19683

Chebyshev original atom1 X minus SAME mean := sub(22, 644/27)

Chebyshev original atom1 X minus SAME mean = -50/27

Chebyshev original atom1 centered SQUARE := mul(-50/27, -50/27)

Chebyshev original atom1 centered SQUARE = 2500/729

Chebyshev original atom1 weighted centered square := mul(2/27, 2500/729)

Chebyshev original atom1 weighted centered square = 5000/19683

Chebyshev original atom2 X minus SAME mean := sub(25, 644/27)

Chebyshev original atom2 X minus SAME mean = 31/27

Chebyshev original atom2 centered SQUARE := mul(31/27, 31/27)

Chebyshev original atom2 centered SQUARE = 961/729

Chebyshev original atom2 weighted centered square := mul(8/9, 961/729)

Chebyshev original atom2 weighted centered square = 7688/6561

Chebyshev full variance E centered square := sum(414736/19683, 5000/19683, 7688/6561)

Chebyshev full variance E centered square = 16400/729

ORIGINAL complete normalized masses=[1/27, 2/27, 8/9];ALL aligned SIGNED values X=[0, 22, 25];SAME-law mean=644/27;all deviations=[-644/27, -50/27, 31/27]

ALL nonnegative squared deviations Y=[414736/729, 2500/729, 961/729];variance=E[Y]=16400/729;finite moments;variance0 iff X=mean on EVERY POSITIVE-mass atom,not necessarily null atoms

Chebyshev original atom0 absolute-event probability mass := mul(1/27, 1)

Chebyshev original atom0 absolute-event probability mass = 1/27

ORIGINAL atom=0;SIGNED centered deviation=-644/27;absolute deviation=644/27;positive epsilon=1

event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=1/27

Chebyshev original atom1 absolute-event probability mass := mul(2/27, 1)

Chebyshev original atom1 absolute-event probability mass = 2/27

ORIGINAL atom=1;SIGNED centered deviation=-50/27;absolute deviation=50/27;positive epsilon=1

event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=2/27

Chebyshev original atom2 absolute-event probability mass := mul(8/9, 1)

Chebyshev original atom2 absolute-event probability mass = 8/9

ORIGINAL atom=2;SIGNED centered deviation=31/27;absolute deviation=31/27;positive epsilon=1

event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=8/9

Chebyshev COMPLETE absolute-deviation tail probability := sum(1/27, 2/27, 8/9)

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=-644/27;its SQUARE=414736/729;EPSILON SQUARED=1

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

ORIGINAL atom=1;deviation=-50/27;its SQUARE=2500/729;EPSILON SQUARED=1

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

ORIGINAL atom=2;deviation=31/27;its SQUARE=961/729;EPSILON SQUARED=1

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

Chebyshev RAW variance over epsilon SQUARED := div(16400/729, 1)

Chebyshev RAW variance over epsilon SQUARED = 16400/729

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

Chebyshev Markov atom0 squared-threshold indicator = 1

Chebyshev Markov atom0 Y minus threshold indicator := sub(414736/729, 1)

Chebyshev Markov atom0 Y minus threshold indicator = 414007/729

Chebyshev Markov atom0 weighted nonnegative slack := mul(1/27, 414007/729)

Chebyshev Markov atom0 weighted nonnegative slack = 414007/19683

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

Chebyshev Markov atom1 squared-threshold indicator = 1

Chebyshev Markov atom1 Y minus threshold indicator := sub(2500/729, 1)

Chebyshev Markov atom1 Y minus threshold indicator = 1771/729

Chebyshev Markov atom1 weighted nonnegative slack := mul(2/27, 1771/729)

Chebyshev Markov atom1 weighted nonnegative slack = 3542/19683

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

Chebyshev Markov atom2 squared-threshold indicator = 1

Chebyshev Markov atom2 Y minus threshold indicator := sub(961/729, 1)

Chebyshev Markov atom2 Y minus threshold indicator = 232/729

Chebyshev Markov atom2 weighted nonnegative slack := mul(8/9, 232/729)

Chebyshev Markov atom2 weighted nonnegative slack = 1856/6561

Chebyshev Markov ALL weighted slacks sum := sum(414007/19683, 3542/19683, 1856/6561)

Chebyshev Markov ALL weighted slacks sum = 15671/729

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(16400/729, 1)

Chebyshev variance minus threshold-event mass = 15671/729

Y=ORIGINAL squared deviations=[414736/729, 2500/729, 961/729] nonnegative at EVERY atom;E[Y]=variance=16400/729;positive threshold epsilon^2=1;complete weighted slacks=[414007/19683, 3542/19683, 1856/6561]

variance-epsilon^2*tail=15671/729>=0;divide POSITIVE epsilon^2 to obtain actual tail=1<=RAW bound=16400/729;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, 16400/729, 16400/729, 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。