第 20733 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":[[1,13],[2,13],[10,13]],"values":[0,8,11],"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-markov-square-derivation；資料={"masses":[[1,13],[2,13],[10,13]],"values":[0,8,11],"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-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(1/13, 2/13, 10/13)

Chebyshev complete probability law mass = 1

Chebyshev original atom0 weighted SIGNED X := mul(1/13, 0)

Chebyshev original atom0 weighted SIGNED X = 0

Chebyshev original atom1 weighted SIGNED X := mul(2/13, 8)

Chebyshev original atom1 weighted SIGNED X = 16/13

Chebyshev original atom2 weighted SIGNED X := mul(10/13, 11)

Chebyshev original atom2 weighted SIGNED X = 110/13

Chebyshev complete SAME-law mean := sum(0, 16/13, 110/13)

Chebyshev complete SAME-law mean = 126/13

Chebyshev original atom0 X minus SAME mean := sub(0, 126/13)

Chebyshev original atom0 X minus SAME mean = -126/13

Chebyshev original atom0 centered SQUARE := mul(-126/13, -126/13)

Chebyshev original atom0 centered SQUARE = 15876/169

Chebyshev original atom0 weighted centered square := mul(1/13, 15876/169)

Chebyshev original atom0 weighted centered square = 15876/2197

Chebyshev original atom1 X minus SAME mean := sub(8, 126/13)

Chebyshev original atom1 X minus SAME mean = -22/13

Chebyshev original atom1 centered SQUARE := mul(-22/13, -22/13)

Chebyshev original atom1 centered SQUARE = 484/169

Chebyshev original atom1 weighted centered square := mul(2/13, 484/169)

Chebyshev original atom1 weighted centered square = 968/2197

Chebyshev original atom2 X minus SAME mean := sub(11, 126/13)

Chebyshev original atom2 X minus SAME mean = 17/13

Chebyshev original atom2 centered SQUARE := mul(17/13, 17/13)

Chebyshev original atom2 centered SQUARE = 289/169

Chebyshev original atom2 weighted centered square := mul(10/13, 289/169)

Chebyshev original atom2 weighted centered square = 2890/2197

Chebyshev full variance E centered square := sum(15876/2197, 968/2197, 2890/2197)

Chebyshev full variance E centered square = 1518/169

ORIGINAL complete normalized masses=[1/13, 2/13, 10/13];ALL aligned SIGNED values X=[0, 8, 11];SAME-law mean=126/13;all deviations=[-126/13, -22/13, 17/13]

ALL nonnegative squared deviations Y=[15876/169, 484/169, 289/169];variance=E[Y]=1518/169;finite moments;variance0 iff X=mean on EVERY POSITIVE-mass atom,not necessarily null atoms

Chebyshev original atom0 absolute-event probability mass := mul(1/13, 1)

Chebyshev original atom0 absolute-event probability mass = 1/13

ORIGINAL atom=0;SIGNED centered deviation=-126/13;absolute deviation=126/13;positive epsilon=2

event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=1/13

Chebyshev original atom1 absolute-event probability mass := mul(2/13, 0)

Chebyshev original atom1 absolute-event probability mass = 0

ORIGINAL atom=1;SIGNED centered deviation=-22/13;absolute deviation=22/13;positive epsilon=2

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

Chebyshev original atom2 absolute-event probability mass := mul(10/13, 0)

Chebyshev original atom2 absolute-event probability mass = 0

ORIGINAL atom=2;SIGNED centered deviation=17/13;absolute deviation=17/13;positive epsilon=2

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

Chebyshev COMPLETE absolute-deviation tail probability := sum(1/13, 0, 0)

Chebyshev COMPLETE absolute-deviation tail probability = 1/13

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=-126/13;its SQUARE=15876/169;EPSILON SQUARED=4

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

ORIGINAL atom=1;deviation=-22/13;its SQUARE=484/169;EPSILON SQUARED=4

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

ORIGINAL atom=2;deviation=17/13;its SQUARE=289/169;EPSILON SQUARED=4

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

Chebyshev RAW variance over epsilon SQUARED := div(1518/169, 4)

Chebyshev RAW variance over epsilon SQUARED = 759/338

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

Chebyshev Markov atom0 squared-threshold indicator = 4

Chebyshev Markov atom0 Y minus threshold indicator := sub(15876/169, 4)

Chebyshev Markov atom0 Y minus threshold indicator = 15200/169

Chebyshev Markov atom0 weighted nonnegative slack := mul(1/13, 15200/169)

Chebyshev Markov atom0 weighted nonnegative slack = 15200/2197

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

Chebyshev Markov atom1 squared-threshold indicator = 0

Chebyshev Markov atom1 Y minus threshold indicator := sub(484/169, 0)

Chebyshev Markov atom1 Y minus threshold indicator = 484/169

Chebyshev Markov atom1 weighted nonnegative slack := mul(2/13, 484/169)

Chebyshev Markov atom1 weighted nonnegative slack = 968/2197

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

Chebyshev Markov atom2 squared-threshold indicator = 0

Chebyshev Markov atom2 Y minus threshold indicator := sub(289/169, 0)

Chebyshev Markov atom2 Y minus threshold indicator = 289/169

Chebyshev Markov atom2 weighted nonnegative slack := mul(10/13, 289/169)

Chebyshev Markov atom2 weighted nonnegative slack = 2890/2197

Chebyshev Markov ALL weighted slacks sum := sum(15200/2197, 968/2197, 2890/2197)

Chebyshev Markov ALL weighted slacks sum = 1466/169

Chebyshev epsilon squared times SAME event probability := mul(4, 1/13)

Chebyshev epsilon squared times SAME event probability = 4/13

Chebyshev variance minus threshold-event mass := sub(1518/169, 4/13)

Chebyshev variance minus threshold-event mass = 1466/169

Y=ORIGINAL squared deviations=[15876/169, 484/169, 289/169] nonnegative at EVERY atom;E[Y]=variance=1518/169;positive threshold epsilon^2=4;complete weighted slacks=[15200/2197, 968/2197, 2890/2197]

variance-epsilon^2*tail=1466/169>=0;divide POSITIVE epsilon^2 to obtain actual tail=1/13<=RAW bound=759/338;no independence and no equality/clipping claim

Chebyshev ORIGINAL atom0 SQUARED-event probability mass := mul(1/13, 1)

Chebyshev ORIGINAL atom0 SQUARED-event probability mass = 1/13

Chebyshev ORIGINAL atom1 SQUARED-event probability mass := mul(2/13, 0)

Chebyshev ORIGINAL atom1 SQUARED-event probability mass = 0

Chebyshev ORIGINAL atom2 SQUARED-event probability mass := mul(10/13, 0)

Chebyshev ORIGINAL atom2 SQUARED-event probability mass = 0

Chebyshev COMPLETE squared-event probability := sum(1/13, 0, 0)

Chebyshev COMPLETE squared-event probability = 1/13

ALL original outcome-order squared values=[15876/169, 484/169, 289/169];positive square threshold=4;ALL original squared-event weighted terms=[1/13, 0, 0]

complete squared-tail probability=1/13=absolute-tail probability=1/13;equality follows from POINTWISE event identity;RAW bound=759/338 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

([15876/169, 484/169, 289/169], 4, 1/13, 1/13, 是, 759/338, 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。