第 20730 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":[[1,10],[2,10],[7,10]],"values":[0,5,8],"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":[[1,10],[2,10],[7,10]],"values":[0,5,8],"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(1/10, 1/5, 7/10)

Chebyshev complete probability law mass = 1

Chebyshev original atom0 weighted SIGNED X := mul(1/10, 0)

Chebyshev original atom0 weighted SIGNED X = 0

Chebyshev original atom1 weighted SIGNED X := mul(1/5, 5)

Chebyshev original atom1 weighted SIGNED X = 1

Chebyshev original atom2 weighted SIGNED X := mul(7/10, 8)

Chebyshev original atom2 weighted SIGNED X = 28/5

Chebyshev complete SAME-law mean := sum(0, 1, 28/5)

Chebyshev complete SAME-law mean = 33/5

Chebyshev original atom0 X minus SAME mean := sub(0, 33/5)

Chebyshev original atom0 X minus SAME mean = -33/5

Chebyshev original atom0 centered SQUARE := mul(-33/5, -33/5)

Chebyshev original atom0 centered SQUARE = 1089/25

Chebyshev original atom0 weighted centered square := mul(1/10, 1089/25)

Chebyshev original atom0 weighted centered square = 1089/250

Chebyshev original atom1 X minus SAME mean := sub(5, 33/5)

Chebyshev original atom1 X minus SAME mean = -8/5

Chebyshev original atom1 centered SQUARE := mul(-8/5, -8/5)

Chebyshev original atom1 centered SQUARE = 64/25

Chebyshev original atom1 weighted centered square := mul(1/5, 64/25)

Chebyshev original atom1 weighted centered square = 64/125

Chebyshev original atom2 X minus SAME mean := sub(8, 33/5)

Chebyshev original atom2 X minus SAME mean = 7/5

Chebyshev original atom2 centered SQUARE := mul(7/5, 7/5)

Chebyshev original atom2 centered SQUARE = 49/25

Chebyshev original atom2 weighted centered square := mul(7/10, 49/25)

Chebyshev original atom2 weighted centered square = 343/250

Chebyshev full variance E centered square := sum(1089/250, 64/125, 343/250)

Chebyshev full variance E centered square = 156/25

ORIGINAL complete normalized masses=[1/10, 1/5, 7/10];ALL aligned SIGNED values X=[0, 5, 8];SAME-law mean=33/5;all deviations=[-33/5, -8/5, 7/5]

ALL nonnegative squared deviations Y=[1089/25, 64/25, 49/25];variance=E[Y]=156/25;finite moments;variance0 iff X=mean on EVERY POSITIVE-mass atom,not necessarily null atoms

Chebyshev original atom0 absolute-event probability mass := mul(1/10, 1)

Chebyshev original atom0 absolute-event probability mass = 1/10

ORIGINAL atom=0;SIGNED centered deviation=-33/5;absolute deviation=33/5;positive epsilon=2

event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=1/10

Chebyshev original atom1 absolute-event probability mass := mul(1/5, 0)

Chebyshev original atom1 absolute-event probability mass = 0

ORIGINAL atom=1;SIGNED centered deviation=-8/5;absolute deviation=8/5;positive epsilon=2

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

Chebyshev original atom2 absolute-event probability mass := mul(7/10, 0)

Chebyshev original atom2 absolute-event probability mass = 0

ORIGINAL atom=2;SIGNED centered deviation=7/5;absolute deviation=7/5;positive epsilon=2

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

Chebyshev COMPLETE absolute-deviation tail probability := sum(1/10, 0, 0)

Chebyshev COMPLETE absolute-deviation tail probability = 1/10

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=-33/5;its SQUARE=1089/25;EPSILON SQUARED=4

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

ORIGINAL atom=1;deviation=-8/5;its SQUARE=64/25;EPSILON SQUARED=4

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

ORIGINAL atom=2;deviation=7/5;its SQUARE=49/25;EPSILON SQUARED=4

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

Chebyshev RAW variance over epsilon SQUARED := div(156/25, 4)

Chebyshev RAW variance over epsilon SQUARED = 39/25

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

Chebyshev Markov atom0 squared-threshold indicator = 4

Chebyshev Markov atom0 Y minus threshold indicator := sub(1089/25, 4)

Chebyshev Markov atom0 Y minus threshold indicator = 989/25

Chebyshev Markov atom0 weighted nonnegative slack := mul(1/10, 989/25)

Chebyshev Markov atom0 weighted nonnegative slack = 989/250

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

Chebyshev Markov atom1 squared-threshold indicator = 0

Chebyshev Markov atom1 Y minus threshold indicator := sub(64/25, 0)

Chebyshev Markov atom1 Y minus threshold indicator = 64/25

Chebyshev Markov atom1 weighted nonnegative slack := mul(1/5, 64/25)

Chebyshev Markov atom1 weighted nonnegative slack = 64/125

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

Chebyshev Markov atom2 squared-threshold indicator = 0

Chebyshev Markov atom2 Y minus threshold indicator := sub(49/25, 0)

Chebyshev Markov atom2 Y minus threshold indicator = 49/25

Chebyshev Markov atom2 weighted nonnegative slack := mul(7/10, 49/25)

Chebyshev Markov atom2 weighted nonnegative slack = 343/250

Chebyshev Markov ALL weighted slacks sum := sum(989/250, 64/125, 343/250)

Chebyshev Markov ALL weighted slacks sum = 146/25

Chebyshev epsilon squared times SAME event probability := mul(4, 1/10)

Chebyshev epsilon squared times SAME event probability = 2/5

Chebyshev variance minus threshold-event mass := sub(156/25, 2/5)

Chebyshev variance minus threshold-event mass = 146/25

Y=ORIGINAL squared deviations=[1089/25, 64/25, 49/25] nonnegative at EVERY atom;E[Y]=variance=156/25;positive threshold epsilon^2=4;complete weighted slacks=[989/250, 64/125, 343/250]

variance-epsilon^2*tail=146/25>=0;divide POSITIVE epsilon^2 to obtain actual tail=1/10<=RAW bound=39/25;no independence and no equality/clipping claim

Chebyshev ORIGINAL atom0 SQUARED-event probability mass := mul(1/10, 1)

Chebyshev ORIGINAL atom0 SQUARED-event probability mass = 1/10

Chebyshev ORIGINAL atom1 SQUARED-event probability mass := mul(1/5, 0)

Chebyshev ORIGINAL atom1 SQUARED-event probability mass = 0

Chebyshev ORIGINAL atom2 SQUARED-event probability mass := mul(7/10, 0)

Chebyshev ORIGINAL atom2 SQUARED-event probability mass = 0

Chebyshev COMPLETE squared-event probability := sum(1/10, 0, 0)

Chebyshev COMPLETE squared-event probability = 1/10

ALL original outcome-order squared values=[1089/25, 64/25, 49/25];positive square threshold=4;ALL original squared-event weighted terms=[1/10, 0, 0]

complete squared-tail probability=1/10=absolute-tail probability=1/10;equality follows from POINTWISE event identity;RAW bound=39/25 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

([1089/25, 64/25, 49/25], 4, 1/10, 1/10, 是, 39/25, 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。