第 20210 題：Markov不等式

原題與已知條件

完整正規化有限有理law、明示隨機變數與正閾值。操作=markov-bound-comparison；資料={"masses":[[1,15],[2,15],[12,15]],"values":[0,10,12],"threshold":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 完整正規化有限有理law、明示隨機變數與正閾值。操作=markov-bound-comparison；資料={"masses":[[1,15],[2,15],[12,15]],"values":[0,10,12],"threshold":11}。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-markov-inequality

逐步推導

source operation=markov-bound-comparison;original snapshot and mathematics retained

explicit ordered source output fields=[tail_probability, expectation, raw_upper_bound, bound_holds, certificate]

Markov complete probability mass := sum(1/15, 2/15, 4/5)

Markov complete probability mass = 1

Markov original atom0 weighted X := mul(1/15, 0)

Markov original atom0 weighted X = 0

Markov original atom1 weighted X := mul(2/15, 10)

Markov original atom1 weighted X = 4/3

Markov original atom2 weighted X := mul(4/5, 12)

Markov original atom2 weighted X = 48/5

Markov complete SAME-law expectation := sum(0, 4/3, 48/5)

Markov complete SAME-law expectation = 164/15

ORIGINAL complete nonempty masses=[1/15, 2/15, 4/5];ALL aligned values X=[0, 10, 12];full mass=1;ALL weighted X terms=[0, 4/3, 48/5]

finite exact expectation=164/15;prototype EVERY-ENTRY nonnegative=是;actual positive-mass nonnegativity=是;null-negative atoms=[];positive-mass negative atoms=[]

Markov original atom0 threshold-event mass := mul(1/15, 0)

Markov original atom0 threshold-event mass = 0

ORIGINAL atom=0;value X=0;positive threshold=11;original mass=1/15

event X>=threshold=否;EQUALITY INCLUDED;event mass contribution=0

Markov original atom1 threshold-event mass := mul(2/15, 0)

Markov original atom1 threshold-event mass = 0

ORIGINAL atom=1;value X=10;positive threshold=11;original mass=2/15

event X>=threshold=否;EQUALITY INCLUDED;event mass contribution=0

Markov original atom2 threshold-event mass := mul(4/5, 1)

Markov original atom2 threshold-event mass = 4/5

ORIGINAL atom=2;value X=12;positive threshold=11;original mass=4/5

event X>=threshold=是;EQUALITY INCLUDED;event mass contribution=4/5

Markov complete exact tail probability := sum(0, 0, 4/5)

Markov complete exact tail probability = 4/5

ALL original threshold-event flags=[否, 否, 是];ALL law-weighted indicator terms=[0, 0, 4/5]

exact complete-law tail probability=4/5;this event calculation ALONE does not assert Markov applicability

Markov RAW expectation over positive threshold := div(164/15, 11)

Markov RAW expectation over positive threshold = 164/165

Markov proof atom0 threshold times indicator := mul(11, 0)

Markov proof atom0 threshold times indicator = 0

Markov proof atom0 pointwise X minus threshold indicator := sub(0, 0)

Markov proof atom0 pointwise X minus threshold indicator = 0

Markov proof atom0 law-weighted nonnegative slack := mul(1/15, 0)

Markov proof atom0 law-weighted nonnegative slack = 0

Markov proof atom1 threshold times indicator := mul(11, 0)

Markov proof atom1 threshold times indicator = 0

Markov proof atom1 pointwise X minus threshold indicator := sub(10, 0)

Markov proof atom1 pointwise X minus threshold indicator = 10

Markov proof atom1 law-weighted nonnegative slack := mul(2/15, 10)

Markov proof atom1 law-weighted nonnegative slack = 4/3

Markov proof atom2 threshold times indicator := mul(11, 1)

Markov proof atom2 threshold times indicator = 11

Markov proof atom2 pointwise X minus threshold indicator := sub(12, 11)

Markov proof atom2 pointwise X minus threshold indicator = 1

Markov proof atom2 law-weighted nonnegative slack := mul(4/5, 1)

Markov proof atom2 law-weighted nonnegative slack = 4/5

Markov complete nonnegative weighted slack sum := sum(0, 4/3, 4/5)

Markov complete nonnegative weighted slack sum = 32/15

Markov threshold times exact tail := mul(11, 4/5)

Markov threshold times exact tail = 44/5

Markov expectation minus threshold tail := sub(164/15, 44/5)

Markov expectation minus threshold tail = 32/15

ALL qualified original values nonnegative;for each atom threshold*1_event<=X;weighted slack terms=[0, 4/3, 4/5];threshold=11>0

E[X]-threshold*P(event)=32/15>=0;divide POSITIVE threshold to obtain actual tail=4/5<=RAW bound=164/165;NO equality claim and NO clipping to1

逐步反向核對

• 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 — 以精確有理數重算 Markov expectation minus threshold tail，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Markov threshold times exact tail，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Markov complete nonnegative weighted slack sum，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Markov proof atom2 law-weighted nonnegative slack，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Markov proof atom2 pointwise X minus threshold indicator，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Markov proof atom2 threshold times indicator，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Markov proof atom1 law-weighted nonnegative slack，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Markov proof atom1 pointwise X minus threshold indicator，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Markov proof atom1 threshold times indicator，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Markov proof atom0 law-weighted nonnegative slack，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Markov proof atom0 pointwise X minus threshold indicator，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Markov proof atom0 threshold times indicator，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Markov RAW expectation over positive threshold，不以小數近似替代。
• S015 · 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。
• S014 · premise_and_result_audit — 以精確有理數重算 Markov complete exact tail probability，不以小數近似替代。
• S013 · 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。
• S012 · premise_and_result_audit — 以精確有理數重算 Markov original atom2 threshold-event mass，不以小數近似替代。
• S011 · 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。
• S010 · premise_and_result_audit — 以精確有理數重算 Markov original atom1 threshold-event mass，不以小數近似替代。
• S009 · 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。
• S008 · premise_and_result_audit — 以精確有理數重算 Markov original atom0 threshold-event mass，不以小數近似替代。
• S007 · 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。
• S006 · premise_and_result_audit — 以精確有理數重算 Markov complete SAME-law expectation，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Markov original atom2 weighted X，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Markov original atom1 weighted X，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Markov original atom0 weighted X，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Markov complete probability 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

(4/5, 164/15, 164/165, 是, upper_bound_not_tail_equality)

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。