第 19227 題：弱大數律

原題與已知條件

IID通用證書、共同有限mean與variance及exact Chebyshev tail。操作=wlln-epsilon-delta-witness；資料={"mean":[1,2],"variance":[2,3],"epsilon":[1,3],"delta":[1,11],"iid_certificate":"symbolic_all_finite_subfamilies"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — IID通用證書、共同有限mean與variance及exact Chebyshev tail。操作=wlln-epsilon-delta-witness；資料={"mean":[1,2],"variance":[2,3],"epsilon":[1,3],"delta":[1,11],"iid_certificate":"symbolic_all_finite_subfamilies"}。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-weak-law-large-numbers

逐步推導

source operation=wlln-epsilon-delta-witness;original snapshot and mathematics retained

explicit ordered source output fields=[epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate]

original common finite mean=1/2;POSITIVE finite variance=2/3;whole-sequence iid_certificate=symbolic_all_finite_subfamilies

ASSUME for EVERY finite subfamily the declared independence and identical law/common moments;marker declares a premise,NOT independent evidence that a real sequence satisfies it

for ANY positive integer n:sample_mean_n=(X1+...+Xn)/n;every Xi has the SAME mean and variance;distinct Xi,Xj independent

linearity gives E(sample_mean_n)=mean;centered products have zero expectation for i!=j;Var(sample_mean_n)=(n*variance)/n^2=variance/n

WLLN strict-tail epsilon squared := mul(1/3, 1/3)

WLLN strict-tail epsilon squared = 1/9

WLLN strict-tail delta times epsilon squared := mul(1/11, 1/9)

WLLN strict-tail delta times epsilon squared = 1/99

WLLN strict-tail rational threshold := div(2/3, 1/99)

WLLN strict-tail rational threshold = 66

WLLN first strict bound-based index := sum(66, 1)

WLLN first strict bound-based index = 67

WLLN cutoff evaluation epsilon squared := mul(1/3, 1/3)

WLLN cutoff evaluation epsilon squared = 1/9

WLLN cutoff evaluation positive n times epsilon squared := mul(67, 1/9)

WLLN cutoff evaluation positive n times epsilon squared = 67/9

WLLN cutoff evaluation raw Chebyshev upper bound := div(2/3, 67/9)

WLLN cutoff evaluation raw Chebyshev upper bound = 6/67

WLLN cutoff evaluation:same whole-sequence IID premise;positive n=67;positive epsilon=1/3;common variance=2/3

P(abs(sample_mean_n-mean)>=epsilon)<=RAW bound=6/67;NOT event probability,NOT equality,NOT clipped to1

WLLN previous positive index := sub(67, 1)

WLLN previous positive index = 66

WLLN previous bound evaluation epsilon squared := mul(1/3, 1/3)

WLLN previous bound evaluation epsilon squared = 1/9

WLLN previous bound evaluation positive n times epsilon squared := mul(66, 1/9)

WLLN previous bound evaluation positive n times epsilon squared = 22/3

WLLN previous bound evaluation raw Chebyshev upper bound := div(2/3, 22/3)

WLLN previous bound evaluation raw Chebyshev upper bound = 1/11

WLLN previous bound evaluation:same whole-sequence IID premise;positive n=66;positive epsilon=1/3;common variance=2/3

P(abs(sample_mean_n-mean)>=epsilon)<=RAW bound=1/11;NOT event probability,NOT equality,NOT clipped to1

positive epsilon=1/3;delta=1/11;EXACT threshold=66;floor=66;N=67;bound_at_N=6/67;previous bound=1/11

strict Chebyshev comparison holds at N and fails at preceding positive index;least BOUND-BASED N,not necessarily least true-probability cutoff

SAME declared whole IID premise;positive epsilon=1/3;positive delta=1/11;chosen positive N=67;bound_at_N=6/67<delta

for EVERY n>=N,Chebyshev bound variance/(n*epsilon^2)<=bound_at_N<delta;therefore convergence in PROBABILITY under the declared premise

逐步反向核對

• S018 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S015 · premise_and_result_audit — 以精確有理數重算 WLLN previous bound evaluation raw Chebyshev upper bound，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 WLLN previous bound evaluation positive n times epsilon squared，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 WLLN previous bound evaluation epsilon squared，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 WLLN previous positive index，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 WLLN cutoff evaluation raw Chebyshev upper bound，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 WLLN cutoff evaluation positive n times epsilon squared，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 WLLN cutoff evaluation epsilon squared，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 WLLN first strict bound-based index，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 WLLN strict-tail rational threshold，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 WLLN strict-tail delta times epsilon squared，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 WLLN strict-tail epsilon squared，不以小數近似替代。
• S003 · 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。
• S002 · 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。
• 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

(1/3, 1/11, 67, 6/67, 是, forall_n_ge_N_monotone_Chebyshev_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。