第 20281 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-tail-probability；資料={"masses":[[1,11],[2,11],[8,11]],"values":[0,6,8],"threshold":7}。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-tail-probability；資料={"masses":[[1,11],[2,11],[8,11]],"values":[0,6,8],"threshold":7}。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-tail-probability;original snapshot and mathematics retained

explicit ordered source output fields=[threshold, tail_probability, certificate]

Markov complete probability mass := sum(1/11, 2/11, 8/11)

Markov complete probability mass = 1

Markov original atom0 weighted X := mul(1/11, 0)

Markov original atom0 weighted X = 0

Markov original atom1 weighted X := mul(2/11, 6)

Markov original atom1 weighted X = 12/11

Markov original atom2 weighted X := mul(8/11, 8)

Markov original atom2 weighted X = 64/11

Markov complete SAME-law expectation := sum(0, 12/11, 64/11)

Markov complete SAME-law expectation = 76/11

ORIGINAL complete nonempty masses=[1/11, 2/11, 8/11];ALL aligned values X=[0, 6, 8];full mass=1;ALL weighted X terms=[0, 12/11, 64/11]

finite exact expectation=76/11;prototype EVERY-ENTRY nonnegative=是;actual positive-mass nonnegativity=是;null-negative atoms=[];positive-mass negative atoms=[]

Markov original atom0 threshold-event mass := mul(1/11, 0)

Markov original atom0 threshold-event mass = 0

ORIGINAL atom=0;value X=0;positive threshold=7;original mass=1/11

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

Markov original atom1 threshold-event mass := mul(2/11, 0)

Markov original atom1 threshold-event mass = 0

ORIGINAL atom=1;value X=6;positive threshold=7;original mass=2/11

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

Markov original atom2 threshold-event mass := mul(8/11, 1)

Markov original atom2 threshold-event mass = 8/11

ORIGINAL atom=2;value X=8;positive threshold=7;original mass=8/11

event X>=threshold=是;EQUALITY INCLUDED;event mass contribution=8/11

Markov complete exact tail probability := sum(0, 0, 8/11)

Markov complete exact tail probability = 8/11

ALL original threshold-event flags=[否, 否, 是];ALL law-weighted indicator terms=[0, 0, 8/11]

exact complete-law tail probability=8/11;this event calculation ALONE does not assert Markov applicability

逐步反向核對

• 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

(7, 8/11, exact_complete_finite_tail)

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。