{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261105-r07-weak-law-large-numbers-0065", "pilot_number": "R0037-19240", "number": 19240, "family_id": "qf-mod36-wlln-epsilon-delta-witness", "title": { "zh-Hant": "第 19240 題：弱大數律", "en": "Question 19240: Choose a Strict All-n Chebyshev Tail Witness" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261105-r07-weak-law-large-numbers-0065", "batch_id": "batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-weak-law-large-numbers" ], "title_zh": "弱大數律", "stem_zh": "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\":[14,15],\"variance\":[15,16],\"epsilon\":[1,16],\"delta\":[1,24],\"iid_certificate\":\"symbolic_all_finite_subfamilies\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(1/16, 1/24, 5761, 240/5761, 是, forall_n_ge_N_monotone_Chebyshev_tail)", "explanation_zh": "Wave5第五層核對=(1/16, 1/24, 5761, 240/5761, 是, forall_n_ge_N_monotone_Chebyshev_tail)；依完整明示資料獨立重算；independence給Var(sample mean)=sigma squared/n並以forall n大於等於N證依機率收斂；有限樣本或模擬不足，且不宣稱almost sure、CLT或有限n已等於極限。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "弱大數律" ] }, "variant": { "family_id": "qf-mod36-wlln-epsilon-delta-witness", "seed": 202611057065, "parameters": { "mean": [ 14, 15 ], "variance": [ 15, 16 ], "epsilon": [ 1, 16 ], "delta": [ 1, 24 ], "iid_certificate": "symbolic_all_finite_subfamilies", "domain": "Symbolic_iid_Chebyshev_WLLN", "operation": "wlln-epsilon-delta-witness", "semantic_template": "qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,24],\"epsilon\":[1,16],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[14,15],\"variance\":[15,16]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261105-r07-weak-law-large-numbers-0065", "batch_id": "batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "74b0cc8538a95848bb5bf9c39c453f85dc2217c7a9720a5a2206e0baf8a72932", "family_id": "qf-mod36-wlln-epsilon-delta-witness", "seed": 202611057065, "semantic_template": "qf-mod36-wlln-epsilon-delta-witness/{\"delta\":[1,24],\"epsilon\":[1,16],\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[14,15],\"variance\":[15,16]}" }, "fields": { "title": "Choose a Strict All-n Chebyshev Tail Witness", "stem": "Use declared mean=14/15, positive finite variance=15/16, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/16, delta=1/24. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).", "answer": { "kind": "expression", "value": "(1/16, 1/24, 5761, 240/5761, Yes, forall_n_ge_N_monotone_Chebyshev_tail)", "explanation": "Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/16, 1/24, 5761, 240/5761, Yes, forall_n_ge_N_monotone_Chebyshev_tail)." }, "skills": [ "Choose a Strict All-n Chebyshev Tail Witness" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "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\":[14,15],\"variance\":[15,16],\"epsilon\":[1,16],\"delta\":[1,24],\"iid_certificate\":\"symbolic_all_finite_subfamilies\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use declared mean=14/15, positive finite variance=15/16, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/16, delta=1/24. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=wlln-epsilon-delta-witness;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field names are not mathematical evidence,and No retains the operation's specific claim scope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "original common finite mean=14/15;POSITIVE finite variance=15/16;whole-sequence iid_certificate=symbolic_all_finite_subfamilies", "after": "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", "relation": "implies", "rule_id": "wlln_whole_IID_declared_premise_not_token_proof", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "The marker is a premise of this problem,not proof of IID by a string or finite data.What follows is conditional on a whole sequence satisfying that premise,not a Lean proof or certification of empirical data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "for ANY positive integer n:sample_mean_n=(X1+...+Xn)/n;every Xi has the SAME mean and variance;distinct Xi,Xj independent", "after": "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", "relation": "implies", "rule_id": "wlln_universal_variance_sum_independence_not_one_sample", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Finite variances ensure integrability of the products.Independence factors every distinct centered cross term into0*0,leaving n diagonal variance terms.Scaling by1/n squared gives variance/n for arbitrary n,not a deduction from one observed average matching the mean." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "WLLN strict-tail epsilon squared := mul(1/16, 1/16)", "after": "WLLN strict-tail epsilon squared = 1/256", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 WLLN strict-tail epsilon squared，不以小數近似替代。", "en": "Recompute WLLN strict-tail epsilon squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/16", "1/16" ], "output": "1/256", "label": "WLLN strict-tail epsilon squared" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "WLLN strict-tail delta times epsilon squared := mul(1/24, 1/256)", "after": "WLLN strict-tail delta times epsilon squared = 1/6144", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 WLLN strict-tail delta times epsilon squared，不以小數近似替代。", "en": "Recompute WLLN strict-tail delta times epsilon squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/24", "1/256" ], "output": "1/6144", "label": "WLLN strict-tail delta times epsilon squared" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "WLLN strict-tail rational threshold := div(15/16, 1/6144)", "after": "WLLN strict-tail rational threshold = 5760", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 WLLN strict-tail rational threshold，不以小數近似替代。", "en": "Recompute WLLN strict-tail rational threshold with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "15/16", "1/6144" ], "output": "5760", "label": "WLLN strict-tail rational threshold" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "WLLN first strict bound-based index := sum(5760, 1)", "after": "WLLN first strict bound-based index = 5761", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 WLLN first strict bound-based index，不以小數近似替代。", "en": "Recompute WLLN first strict bound-based index with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "5760", "1" ], "output": "5761", "label": "WLLN first strict bound-based index" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "WLLN cutoff evaluation epsilon squared := mul(1/16, 1/16)", "after": "WLLN cutoff evaluation epsilon squared = 1/256", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 WLLN cutoff evaluation epsilon squared，不以小數近似替代。", "en": "Recompute WLLN cutoff evaluation epsilon squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/16", "1/16" ], "output": "1/256", "label": "WLLN cutoff evaluation epsilon squared" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "WLLN cutoff evaluation positive n times epsilon squared := mul(5761, 1/256)", "after": "WLLN cutoff evaluation positive n times epsilon squared = 5761/256", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 WLLN cutoff evaluation positive n times epsilon squared，不以小數近似替代。", "en": "Recompute WLLN cutoff evaluation positive n times epsilon squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5761", "1/256" ], "output": "5761/256", "label": "WLLN cutoff evaluation positive n times epsilon squared" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "WLLN cutoff evaluation raw Chebyshev upper bound := div(15/16, 5761/256)", "after": "WLLN cutoff evaluation raw Chebyshev upper bound = 240/5761", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 WLLN cutoff evaluation raw Chebyshev upper bound，不以小數近似替代。", "en": "Recompute WLLN cutoff evaluation raw Chebyshev upper bound with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "15/16", "5761/256" ], "output": "240/5761", "label": "WLLN cutoff evaluation raw Chebyshev upper bound" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "WLLN cutoff evaluation:same whole-sequence IID premise;positive n=5761;positive epsilon=1/16;common variance=15/16", "after": "P(abs(sample_mean_n-mean)>=epsilon)<=RAW bound=240/5761;NOT event probability,NOT equality,NOT clipped to1", "relation": "implies", "rule_id": "wlln_Chebyshev_raw_upper_bound_not_probability_value", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Chebyshev divides the variance of this same average by epsilon squared.An upper bound above1 is valid but loose,not an event probability.Do not clip the source value to1;the statement remains conditional on the declared whole-sequence premise." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "WLLN previous positive index := sub(5761, 1)", "after": "WLLN previous positive index = 5760", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 WLLN previous positive index，不以小數近似替代。", "en": "Recompute WLLN previous positive index with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "5761", "1" ], "output": "5760", "label": "WLLN previous positive index" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "WLLN previous bound evaluation epsilon squared := mul(1/16, 1/16)", "after": "WLLN previous bound evaluation epsilon squared = 1/256", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 WLLN previous bound evaluation epsilon squared，不以小數近似替代。", "en": "Recompute WLLN previous bound evaluation epsilon squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/16", "1/16" ], "output": "1/256", "label": "WLLN previous bound evaluation epsilon squared" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "WLLN previous bound evaluation positive n times epsilon squared := mul(5760, 1/256)", "after": "WLLN previous bound evaluation positive n times epsilon squared = 45/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "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。", "en": "Recompute WLLN previous bound evaluation positive n times epsilon squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5760", "1/256" ], "output": "45/2", "label": "WLLN previous bound evaluation positive n times epsilon squared" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "WLLN previous bound evaluation raw Chebyshev upper bound := div(15/16, 45/2)", "after": "WLLN previous bound evaluation raw Chebyshev upper bound = 1/24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 WLLN previous bound evaluation raw Chebyshev upper bound，不以小數近似替代。", "en": "Recompute WLLN previous bound evaluation raw Chebyshev upper bound with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "15/16", "45/2" ], "output": "1/24", "label": "WLLN previous bound evaluation raw Chebyshev upper bound" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "WLLN previous bound evaluation:same whole-sequence IID premise;positive n=5760;positive epsilon=1/16;common variance=15/16", "after": "P(abs(sample_mean_n-mean)>=epsilon)<=RAW bound=1/24;NOT event probability,NOT equality,NOT clipped to1", "relation": "implies", "rule_id": "wlln_Chebyshev_raw_upper_bound_not_probability_value", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Chebyshev divides the variance of this same average by epsilon squared.An upper bound above1 is valid but loose,not an event probability.Do not clip the source value to1;the statement remains conditional on the declared whole-sequence premise." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "positive epsilon=1/16;delta=1/24;EXACT threshold=5760;floor=5760;N=5761;bound_at_N=240/5761;previous bound=1/24", "after": "strict Chebyshev comparison holds at N and fails at preceding positive index;least BOUND-BASED N,not necessarily least true-probability cutoff", "relation": "implies", "rule_id": "wlln_exact_strict_bound_cutoff_not_actual_probability_minimum", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "The bound is below delta iff n>variance/(delta epsilon squared),so floor+1 is required even at integral thresholds.Previous failure proves minimality for this Chebyshev bound,not for the actual unknown event probability." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "SAME declared whole IID premise;positive epsilon=1/16;positive delta=1/24;chosen positive N=5761;bound_at_N=240/5761=N,Chebyshev bound variance/(n*epsilon^2)<=bound_at_N