# 第 19249 題：弱大數律 [返回所在分頁 97](https://ocme.evemisslab.com/zh-Hant/questions/page/97/) [← 逐題詳解目錄](https://ocme.evemisslab.com/zh-Hant/questions/) · [原題批次](https://ocme.evemisslab.com/zh-Hant/practice/batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations/#q-20261014-advance-20261105-r07-weak-law-large-numbers-0074) 本頁是已完成第二對話 AI 數理與中英語義覆核的教學詳解，只支持本題明示的模型與前提。沒有人工審查或新 Lean 證明，也不提升原題或 MKO 的 Canonical 真值狀態。 R0037-19249 · q-20261014-advance-20261105-r07-weak-law-large-numbers-0074 ## 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":[23,24],"variance":[24,25],"epsilon":[1,25],"delta":[1,33],"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":[23,24],"variance":[24,25],"epsilon":[1,25],"delta":[1,33],"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](https://ocme.evemisslab.com/zh-Hant/mko/mko-weak-law-large-numbers/) ## 逐步推導 S001 · implies操作前 source operation=wlln-epsilon-delta-witness;original snapshot and mathematics retained 操作後 explicit ordered source output fields=[epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate] 條件：無 原題快照保留歷史介面標記；本頁直接列原始欄位順序，不需JavaScript或查外部表。欄位名字本身不自動是數學證據，No依各題的聲稱範圍解讀。 步驟識別與依賴規則：source_ordered_interface_explicit_without_javascript · 依賴：G001 S002 · implies操作前 original common finite mean=23/24;POSITIVE finite variance=24/25;whole-sequence iid_certificate=symbolic_all_finite_subfamilies 操作後 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 條件：無 標記是本題前提而非靠字串或有限資料證IID。以下是若全序列真的具此前提才成立的條件式推導；不新增Lean證明或實際資料認證。 步驟識別與依賴規則：wlln_whole_IID_declared_premise_not_token_proof · 依賴：G001, S001 S003 · implies操作前 for ANY positive integer n:sample_mean_n=(X1+...+Xn)/n;every Xi has the SAME mean and variance;distinct Xi,Xj independent 操作後 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 條件：無 有限方差保乘積可積；獨立讓交叉中心化期望分解成0·0，只有n個對角variance項。再乘1/n平方得variance/n，這是任意n的代數證書，不從已觀察平均值等於mean推。 步驟識別與依賴規則：wlln_universal_variance_sum_independence_not_one_sample · 依賴：G001, S002 S004 · calculation操作前 WLLN strict-tail epsilon squared := mul(1/25, 1/25) 操作後 WLLN strict-tail epsilon squared = 1/625 條件：無 依原始角色與已核中間量精確代值，保留參數、樣本、目標與分母，不交換未授權的聲稱。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S003 S005 · calculation操作前 WLLN strict-tail delta times epsilon squared := mul(1/33, 1/625) 操作後 WLLN strict-tail delta times epsilon squared = 1/20625 條件：無 依原始角色與已核中間量精確代值，保留參數、樣本、目標與分母，不交換未授權的聲稱。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S004 S006 · calculation操作前 WLLN strict-tail rational threshold := div(24/25, 1/20625) 操作後 WLLN strict-tail rational threshold = 19800 條件：無 依原始角色與已核中間量精確代值，保留參數、樣本、目標與分母，不交換未授權的聲稱。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S005 S007 · calculation操作前 WLLN first strict bound-based index := sum(19800, 1) 操作後 WLLN first strict bound-based index = 19801 條件：無 依原始角色與已核中間量精確代值，保留參數、樣本、目標與分母，不交換未授權的聲稱。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S006 S008 · calculation操作前 WLLN cutoff evaluation epsilon squared := mul(1/25, 1/25) 操作後 WLLN cutoff evaluation epsilon squared = 1/625 條件：無 依原始角色與已核中間量精確代值，保留參數、樣本、目標與分母，不交換未授權的聲稱。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S007 S009 · calculation操作前 WLLN cutoff evaluation positive n times epsilon squared := mul(19801, 1/625) 操作後 WLLN cutoff evaluation positive n times epsilon squared = 19801/625 條件：無 依原始角色與已核中間量精確代值，保留參數、樣本、目標與分母，不交換未授權的聲稱。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S008 S010 · calculation操作前 WLLN cutoff evaluation raw Chebyshev upper bound := div(24/25, 19801/625) 操作後 WLLN cutoff evaluation raw Chebyshev upper bound = 600/19801 條件：無 依原始角色與已核中間量精確代值，保留參數、樣本、目標與分母，不交換未授權的聲稱。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S009 S011 · implies操作前 WLLN cutoff evaluation:same whole-sequence IID premise;positive n=19801;positive epsilon=1/25;common variance=24/25 操作後 P(abs(sample_mean_n-mean)>=epsilon)<=RAW bound=600/19801;NOT event probability,NOT equality,NOT clipped to1 條件：無 Chebyshev用同一平均值的variance除epsilon平方。上界大於1仍可合法但鬆，不把它當事件機率，也不擅自把來源字串截成1；此欄仍條件於明示全序列前提。 步驟識別與依賴規則：wlln_Chebyshev_raw_upper_bound_not_probability_value · 依賴：G001, S010 S012 · calculation操作前 WLLN previous positive index := sub(19801, 1) 操作後 WLLN previous positive index = 19800 條件：無 依原始角色與已核中間量精確代值，保留參數、樣本、目標與分母，不交換未授權的聲稱。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S011 S013 · calculation操作前 WLLN previous bound evaluation epsilon squared := mul(1/25, 1/25) 操作後 WLLN previous bound evaluation epsilon squared = 1/625 條件：無 依原始角色與已核中間量精確代值，保留參數、樣本、目標與分母，不交換未授權的聲稱。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S012 S014 · calculation操作前 WLLN previous bound evaluation positive n times epsilon squared := mul(19800, 1/625) 操作後 WLLN previous bound evaluation positive n times epsilon squared = 792/25 條件：無 依原始角色與已核中間量精確代值，保留參數、樣本、目標與分母，不交換未授權的聲稱。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S013 S015 · calculation操作前 WLLN previous bound evaluation raw Chebyshev upper bound := div(24/25, 792/25) 操作後 WLLN previous bound evaluation raw Chebyshev upper bound = 1/33 條件：無 依原始角色與已核中間量精確代值，保留參數、樣本、目標與分母，不交換未授權的聲稱。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S014 S016 · implies操作前 WLLN previous bound evaluation:same whole-sequence IID premise;positive n=19800;positive epsilon=1/25;common variance=24/25 操作後 P(abs(sample_mean_n-mean)>=epsilon)<=RAW bound=1/33;NOT event probability,NOT equality,NOT clipped to1 條件：無 Chebyshev用同一平均值的variance除epsilon平方。上界大於1仍可合法但鬆，不把它當事件機率，也不擅自把來源字串截成1；此欄仍條件於明示全序列前提。 步驟識別與依賴規則：wlln_Chebyshev_raw_upper_bound_not_probability_value · 依賴：G001, S015 S017 · implies操作前 positive epsilon=1/25;delta=1/33;EXACT threshold=19800;floor=19800;N=19801;bound_at_N=600/19801;previous bound=1/33 操作後 strict Chebyshev comparison holds at N and fails at preceding positive index;least BOUND-BASED N,not necessarily least true-probability cutoff 條件：無 界variance/(delta epsilon²)，所以floor加1，包括整數門檻都不能少加1。前一正索引失敗證此Chebyshev界的最小N，不宣稱實際機率沒有更早變小。 步驟識別與依賴規則：wlln_exact_strict_bound_cutoff_not_actual_probability_minimum · 依賴：G001, S016 S018 · implies操作前 SAME declared whole IID premise;positive epsilon=1/25;positive delta=1/33;chosen positive N=19801;bound_at_N=600/19801=N,Chebyshev bound variance/(n*epsilon^2)<=bound_at_N