第 20234 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-negative-variable-boundary；資料={"masses":[[1,2],[1,2]],"values":[-9,11],"threshold":10}。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-negative-variable-boundary；資料={"masses":[[1,2],[1,2]],"values":[-9,11],"threshold":10}。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-negative-variable-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary]

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

Markov complete probability mass = 1

Markov original atom0 weighted X := mul(1/2, -9)

Markov original atom0 weighted X = -9/2

Markov original atom1 weighted X := mul(1/2, 11)

Markov original atom1 weighted X = 11/2

Markov complete SAME-law expectation := sum(-9/2, 11/2)

Markov complete SAME-law expectation = 1

ORIGINAL complete nonempty masses=[1/2, 1/2];ALL aligned values X=[-9, 11];full mass=1;ALL weighted X terms=[-9/2, 11/2]

finite exact expectation=1;prototype EVERY-ENTRY nonnegative=否;actual positive-mass nonnegativity=否;null-negative atoms=[];positive-mass negative atoms=[0]

Markov original atom0 threshold-event mass := mul(1/2, 0)

Markov original atom0 threshold-event mass = 0

ORIGINAL atom=0;value X=-9;positive threshold=10;original mass=1/2

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

Markov original atom1 threshold-event mass := mul(1/2, 1)

Markov original atom1 threshold-event mass = 1/2

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

event X>=threshold=是;EQUALITY INCLUDED;event mass contribution=1/2

Markov complete exact tail probability := sum(0, 1/2)

Markov complete exact tail probability = 1/2

ALL original threshold-event flags=[否, 是];ALL law-weighted indicator terms=[0, 1/2]

exact complete-law tail probability=1/2;this event calculation ALONE does not assert Markov applicability

Markov RAW expectation over positive threshold := div(1, 10)

Markov RAW expectation over positive threshold = 1/10

ACTUAL negative X at positive mass exists;original values=[-9, 11];tail=1/2;finite expectation=1;positive threshold=10

SOURCE pointwise applicability=否;raw E[X]/threshold=1/10;actual numeric tail<=raw quotient happens=否;NO general Markov guarantee or reverse-inequality claim

逐步反向核對

• S014 · 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。
• S013 · 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。
• S012 · 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。
• S011 · 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。
• S010 · 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。
• S009 · 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。
• S008 · 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。
• S007 · 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。
• S006 · 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。
• S005 · 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。
• 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

([-9, 11], 否, 1/2, 1/10, Markov_not_applicable_to_signed_variable_without_transformation)

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。