第 20201 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":[[1,6],[2,6],[3,6]],"values":[0,1,3],"threshold":2}。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":[[1,6],[2,6],[3,6]],"values":[0,1,3],"threshold":2}。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(1/6, 1/3, 1/2)

Markov complete probability mass = 1

Markov original atom0 weighted X := mul(1/6, 0)

Markov original atom0 weighted X = 0

Markov original atom1 weighted X := mul(1/3, 1)

Markov original atom1 weighted X = 1/3

Markov original atom2 weighted X := mul(1/2, 3)

Markov original atom2 weighted X = 3/2

Markov complete SAME-law expectation := sum(0, 1/3, 3/2)

Markov complete SAME-law expectation = 11/6

ORIGINAL complete nonempty masses=[1/6, 1/3, 1/2];ALL aligned values X=[0, 1, 3];full mass=1;ALL weighted X terms=[0, 1/3, 3/2]

finite exact expectation=11/6;prototype EVERY-ENTRY nonnegative=是;actual positive-mass nonnegativity=是;null-negative atoms=[];positive-mass negative atoms=[]

Markov original atom0 threshold-event mass := mul(1/6, 0)

Markov original atom0 threshold-event mass = 0

ORIGINAL atom=0;value X=0;positive threshold=2;original mass=1/6

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

Markov original atom1 threshold-event mass := mul(1/3, 0)

Markov original atom1 threshold-event mass = 0

ORIGINAL atom=1;value X=1;positive threshold=2;original mass=1/3

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

Markov original atom2 threshold-event mass := mul(1/2, 1)

Markov original atom2 threshold-event mass = 1/2

ORIGINAL atom=2;value X=3;positive threshold=2;original mass=1/2

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

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

Markov complete exact tail probability = 1/2

ALL original threshold-event flags=[否, 否, 是];ALL law-weighted indicator terms=[0, 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(11/6, 2)

Markov RAW expectation over positive threshold = 11/12

Markov proof atom0 threshold times indicator := mul(2, 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(1/6, 0)

Markov proof atom0 law-weighted nonnegative slack = 0

Markov proof atom1 threshold times indicator := mul(2, 0)

Markov proof atom1 threshold times indicator = 0

Markov proof atom1 pointwise X minus threshold indicator := sub(1, 0)

Markov proof atom1 pointwise X minus threshold indicator = 1

Markov proof atom1 law-weighted nonnegative slack := mul(1/3, 1)

Markov proof atom1 law-weighted nonnegative slack = 1/3

Markov proof atom2 threshold times indicator := mul(2, 1)

Markov proof atom2 threshold times indicator = 2

Markov proof atom2 pointwise X minus threshold indicator := sub(3, 2)

Markov proof atom2 pointwise X minus threshold indicator = 1

Markov proof atom2 law-weighted nonnegative slack := mul(1/2, 1)

Markov proof atom2 law-weighted nonnegative slack = 1/2

Markov complete nonnegative weighted slack sum := sum(0, 1/3, 1/2)

Markov complete nonnegative weighted slack sum = 5/6

Markov threshold times exact tail := mul(2, 1/2)

Markov threshold times exact tail = 1

Markov expectation minus threshold tail := sub(11/6, 1)

Markov expectation minus threshold tail = 5/6

ALL qualified original values nonnegative;for each atom threshold*1_event<=X;weighted slack terms=[0, 1/3, 1/2];threshold=2>0

E[X]-threshold*P(event)=5/6>=0;divide POSITIVE threshold to obtain actual tail=1/2<=RAW bound=11/12;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

(1/2, 11/6, 11/12, 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。