第 20711 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":[[1,16],[2,16],[13,16]],"values":[0,11,14],"epsilon":[2,1]}。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":[[1,16],[2,16],[13,16]],"values":[0,11,14],"epsilon":[2,1]}。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(1/16, 1/8, 13/16)

Chebyshev complete probability law mass = 1

Chebyshev original atom0 weighted SIGNED X := mul(1/16, 0)

Chebyshev original atom0 weighted SIGNED X = 0

Chebyshev original atom1 weighted SIGNED X := mul(1/8, 11)

Chebyshev original atom1 weighted SIGNED X = 11/8

Chebyshev original atom2 weighted SIGNED X := mul(13/16, 14)

Chebyshev original atom2 weighted SIGNED X = 91/8

Chebyshev complete SAME-law mean := sum(0, 11/8, 91/8)

Chebyshev complete SAME-law mean = 51/4

Chebyshev original atom0 X minus SAME mean := sub(0, 51/4)

Chebyshev original atom0 X minus SAME mean = -51/4

Chebyshev original atom0 centered SQUARE := mul(-51/4, -51/4)

Chebyshev original atom0 centered SQUARE = 2601/16

Chebyshev original atom0 weighted centered square := mul(1/16, 2601/16)

Chebyshev original atom0 weighted centered square = 2601/256

Chebyshev original atom1 X minus SAME mean := sub(11, 51/4)

Chebyshev original atom1 X minus SAME mean = -7/4

Chebyshev original atom1 centered SQUARE := mul(-7/4, -7/4)

Chebyshev original atom1 centered SQUARE = 49/16

Chebyshev original atom1 weighted centered square := mul(1/8, 49/16)

Chebyshev original atom1 weighted centered square = 49/128

Chebyshev original atom2 X minus SAME mean := sub(14, 51/4)

Chebyshev original atom2 X minus SAME mean = 5/4

Chebyshev original atom2 centered SQUARE := mul(5/4, 5/4)

Chebyshev original atom2 centered SQUARE = 25/16

Chebyshev original atom2 weighted centered square := mul(13/16, 25/16)

Chebyshev original atom2 weighted centered square = 325/256

Chebyshev full variance E centered square := sum(2601/256, 49/128, 325/256)

Chebyshev full variance E centered square = 189/16

ORIGINAL complete normalized masses=[1/16, 1/8, 13/16];ALL aligned SIGNED values X=[0, 11, 14];SAME-law mean=51/4;all deviations=[-51/4, -7/4, 5/4]

ALL nonnegative squared deviations Y=[2601/16, 49/16, 25/16];variance=E[Y]=189/16;finite moments;variance0 iff X=mean on EVERY POSITIVE-mass atom,not necessarily null atoms

Chebyshev original atom0 absolute-event probability mass := mul(1/16, 1)

Chebyshev original atom0 absolute-event probability mass = 1/16

ORIGINAL atom=0;SIGNED centered deviation=-51/4;absolute deviation=51/4;positive epsilon=2

event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=1/16

Chebyshev original atom1 absolute-event probability mass := mul(1/8, 0)

Chebyshev original atom1 absolute-event probability mass = 0

ORIGINAL atom=1;SIGNED centered deviation=-7/4;absolute deviation=7/4;positive epsilon=2

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

Chebyshev original atom2 absolute-event probability mass := mul(13/16, 0)

Chebyshev original atom2 absolute-event probability mass = 0

ORIGINAL atom=2;SIGNED centered deviation=5/4;absolute deviation=5/4;positive epsilon=2

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

Chebyshev COMPLETE absolute-deviation tail probability := sum(1/16, 0, 0)

Chebyshev COMPLETE absolute-deviation tail probability = 1/16

Chebyshev POSITIVE epsilon SQUARED threshold := mul(2, 2)

Chebyshev POSITIVE epsilon SQUARED threshold = 4

positive original epsilon=2;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=-51/4;its SQUARE=2601/16;EPSILON SQUARED=4

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

ORIGINAL atom=1;deviation=-7/4;its SQUARE=49/16;EPSILON SQUARED=4

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

ORIGINAL atom=2;deviation=5/4;its SQUARE=25/16;EPSILON SQUARED=4

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

Chebyshev RAW variance over epsilon SQUARED := div(189/16, 4)

Chebyshev RAW variance over epsilon SQUARED = 189/64

Chebyshev Markov atom0 squared-threshold indicator := mul(4, 1)

Chebyshev Markov atom0 squared-threshold indicator = 4

Chebyshev Markov atom0 Y minus threshold indicator := sub(2601/16, 4)

Chebyshev Markov atom0 Y minus threshold indicator = 2537/16

Chebyshev Markov atom0 weighted nonnegative slack := mul(1/16, 2537/16)

Chebyshev Markov atom0 weighted nonnegative slack = 2537/256

Chebyshev Markov atom1 squared-threshold indicator := mul(4, 0)

Chebyshev Markov atom1 squared-threshold indicator = 0

Chebyshev Markov atom1 Y minus threshold indicator := sub(49/16, 0)

Chebyshev Markov atom1 Y minus threshold indicator = 49/16

Chebyshev Markov atom1 weighted nonnegative slack := mul(1/8, 49/16)

Chebyshev Markov atom1 weighted nonnegative slack = 49/128

Chebyshev Markov atom2 squared-threshold indicator := mul(4, 0)

Chebyshev Markov atom2 squared-threshold indicator = 0

Chebyshev Markov atom2 Y minus threshold indicator := sub(25/16, 0)

Chebyshev Markov atom2 Y minus threshold indicator = 25/16

Chebyshev Markov atom2 weighted nonnegative slack := mul(13/16, 25/16)

Chebyshev Markov atom2 weighted nonnegative slack = 325/256

Chebyshev Markov ALL weighted slacks sum := sum(2537/256, 49/128, 325/256)

Chebyshev Markov ALL weighted slacks sum = 185/16

Chebyshev epsilon squared times SAME event probability := mul(4, 1/16)

Chebyshev epsilon squared times SAME event probability = 1/4

Chebyshev variance minus threshold-event mass := sub(189/16, 1/4)

Chebyshev variance minus threshold-event mass = 185/16

Y=ORIGINAL squared deviations=[2601/16, 49/16, 25/16] nonnegative at EVERY atom;E[Y]=variance=189/16;positive threshold epsilon^2=4;complete weighted slacks=[2537/256, 49/128, 325/256]

variance-epsilon^2*tail=185/16>=0;divide POSITIVE epsilon^2 to obtain actual tail=1/16<=RAW bound=189/64;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

(1/16, 189/16, 189/64, 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。