第 19232 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":[6,7],"variance":[7,8],"epsilon":[1,8],"delta":[1,16],"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":[6,7],"variance":[7,8],"epsilon":[1,8],"delta":[1,16],"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=6/7;POSITIVE finite variance=7/8;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(1/8, 1/8)

WLLN strict-tail epsilon squared = 1/64

WLLN strict-tail delta times epsilon squared := mul(1/16, 1/64)

WLLN strict-tail delta times epsilon squared = 1/1024

WLLN strict-tail rational threshold := div(7/8, 1/1024)

WLLN strict-tail rational threshold = 896

WLLN first strict bound-based index := sum(896, 1)

WLLN first strict bound-based index = 897

WLLN cutoff evaluation epsilon squared := mul(1/8, 1/8)

WLLN cutoff evaluation epsilon squared = 1/64

WLLN cutoff evaluation positive n times epsilon squared := mul(897, 1/64)

WLLN cutoff evaluation positive n times epsilon squared = 897/64

WLLN cutoff evaluation raw Chebyshev upper bound := div(7/8, 897/64)

WLLN cutoff evaluation raw Chebyshev upper bound = 56/897

WLLN cutoff evaluation:same whole-sequence IID premise;positive n=897;positive epsilon=1/8;common variance=7/8

P(abs(sample_mean_n-mean)>=epsilon)<=RAW bound=56/897;NOT event probability,NOT equality,NOT clipped to1

WLLN previous positive index := sub(897, 1)

WLLN previous positive index = 896

WLLN previous bound evaluation epsilon squared := mul(1/8, 1/8)

WLLN previous bound evaluation epsilon squared = 1/64

WLLN previous bound evaluation positive n times epsilon squared := mul(896, 1/64)

WLLN previous bound evaluation positive n times epsilon squared = 14

WLLN previous bound evaluation raw Chebyshev upper bound := div(7/8, 14)

WLLN previous bound evaluation raw Chebyshev upper bound = 1/16

WLLN previous bound evaluation:same whole-sequence IID premise;positive n=896;positive epsilon=1/8;common variance=7/8

P(abs(sample_mean_n-mean)>=epsilon)<=RAW bound=1/16;NOT event probability,NOT equality,NOT clipped to1

positive epsilon=1/8;delta=1/16;EXACT threshold=896;floor=896;N=897;bound_at_N=56/897;previous bound=1/16

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=1/8;positive delta=1/16;chosen positive N=897;bound_at_N=56/897<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

(1/8, 1/16, 897, 56/897, 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。