{ "schema_version": "ocme-derivation-reader-v1", "guide_id": "derivation-mko-weak-law-large-numbers-01", "mko_id": "mko-weak-law-large-numbers", "title": { "zh-Hant": "公平Bernoulli平均的Chebyshev尾端", "en": "A Chebyshev tail for fair Bernoulli averages" }, "contract": { "schema_version": "ocme-derivation-guide-v0.1", "guide_id": "derivation-mko-weak-law-large-numbers-01", "source": { "mko_id": "mko-weak-law-large-numbers", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "sha256": "7eb5ea7326a95ec1926f661b129d1f6ed74f95c53574aea5f3878111f80bb55c", "english_sha256": "cbabd753c9362d2298518ae3dd4332e7dc4007f7b436746fdb53dd41565de8a3" }, "mode": "theorem_proof", "audience_prerequisites": [ "work" ], "goal": { "zh-Hant": "由全序列IID前提導出WLLN。", "en": "Derive the WLLN from a whole-sequence IID premise." }, "givens": [ { "id": "G1", "statement": { "zh-Hant": "在一個已給定機率空間上，(X_i)_{i≥1}為無限IID序列，對任意有限不同indices的joint law都因子分解為Bernoulli(1/2)的乘積。此為全序列前提，不從有限資料估出。", "en": "On a supplied probability space, (X_i)_{i≥1} is an infinite IID sequence: every finite distinct-index joint law factors into Bernoulli(1/2) marginals. This is a whole-process premise, not inferred from finite data." } } ], "steps": [ { "id": "S01", "uses": [ "G1", "work" ], "before_tex": "P(X_i=0)=P(X_i=1)=1/2", "after_tex": "\\mu=1/2,\\quad E[X_i^2]=1/2", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "共同邊際law完整給出一階與二階矩。", "en": "The complete common marginal gives the first and second moments." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "work" ], "explanation": { "zh-Hant": "i任意，非只對X1成立。", "en": "The index i is arbitrary, not just1." } }, "leaf_basis": "work" }, { "id": "S02", "uses": [ "G1", "S01", "work" ], "before_tex": "\\sigma^2=E[X_i^2]-\\mu^2", "after_tex": "\\sigma^2=1/2-1/4=1/4", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "共同variance嚴格有限。", "en": "The common variance is finite." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01", "work" ], "explanation": { "zh-Hant": "此定理可容許零variance，本例則為1/4。", "en": "This theorem permits zero variance; the present one is 1/4." } }, "leaf_basis": "work" }, { "id": "S03", "uses": [ "G1", "S02", "work" ], "before_tex": "\\bar X_n=\\frac1n\\sum_{i=1}^nX_i", "after_tex": "E[\\bar X_n]=\\frac1n\\sum_{i=1}^n\\mu=1/2", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "線性期望不需獨立即可保留mean。", "en": "Linearity preserves the mean without needing independence." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S02", "work" ], "explanation": { "zh-Hant": "分母n必須正，取n≥1。", "en": "The denominator requires n≥1." } }, "leaf_basis": "work" }, { "id": "S04", "uses": [ "G1", "S03", "work" ], "before_tex": "i\\ne j,\\quad E[X_iX_j]=E[X_i]E[X_j]", "after_tex": "\\operatorname{Cov}(X_i,X_j)=0", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "IID的獨立性在這一步消掉所有cross covariances。", "en": "IID independence eliminates every cross covariance here." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S03", "work" ], "explanation": { "zh-Hant": "同分布本身不保證此式。", "en": "Identical distribution alone does not ensure this equation." } }, "leaf_basis": "work" }, { "id": "S05", "uses": [ "G1", "S04", "work" ], "before_tex": "\\operatorname{Var}(\\bar X_n)=n^{-2}\\sum_{i=1}^n\\operatorname{Var}(X_i)", "after_tex": "\\operatorname{Var}(\\bar X_n)=1/(4n)", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "全部cross terms零，n項variance再除n平方。", "en": "All cross terms vanish; divide n variances by n squared." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S04", "work" ], "explanation": { "zh-Hant": "不能把分母n平方漏成n。", "en": "Do not lose the squared scaling denominator." } }, "leaf_basis": "work" }, { "id": "S06", "uses": [ "G1", "S05", "work" ], "before_tex": "\\varepsilon>0", "after_tex": "P(|\\bar X_n-1/2|>\\varepsilon)\\le\\frac1{4n\\varepsilon^2}", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "Chebyshev的≥事件包含嚴格>事件，故同一上界可用。", "en": "Chebyshev's≥ event contains the strict> event, so the bound applies." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S05", "work" ], "explanation": { "zh-Hant": "epsilon必須正且每個mean、variance來自同一n。", "en": "Epsilon is positive and the mean and variance concern the same n." } }, "leaf_basis": "work" }, { "id": "S07", "uses": [ "G1", "S06", "work" ], "before_tex": "\\varepsilon,\\delta>0,\\quad N=\\lfloor1/(4\\varepsilon^2\\delta)\\rfloor+1", "after_tex": "n\\ge N\\implies P(|\\bar X_n-1/2|>\\varepsilon)<\\delta", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "明示N嚴格大於臨界值，控制全部尾端。", "en": "Explicit N strictly exceeds the critical value and controls the entire tail." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S06", "work" ], "explanation": { "zh-Hant": "epsilon與delta先任取，N可依它們改變。", "en": "Epsilon and delta are arbitrary first; N may depend on them." } }, "leaf_basis": "work" }, { "id": "S08", "uses": [ "G1", "S07", "work" ], "before_tex": "\\forall\\varepsilon,\\delta>0\\ \\exists N\\ \\forall n\\ge N", "after_tex": "\\bar X_n\\xrightarrow{P}1/2", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "這是依機率收斂定義，完成有限variance WLLN例證。", "en": "This is the definition of convergence in probability, completing the finite-variance WLLN instance." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S07", "work" ], "explanation": { "zh-Hant": "不是a.s.收斂證明或有限n常態等式。", "en": "It is neither an a.s. proof nor a finite-n normal equality." } }, "leaf_basis": "work" } ], "result": { "zh-Hant": "在宣告無限IID前提下，樣本平均依機率收斂到1/2。", "en": "Under the declared infinite IID premise, the sample mean converges in probability to 1/2." }, "negative_controls": [ { "claim": { "zh-Hant": "相同邊際即可消covariance。", "en": "Equal marginals eliminate covariance." }, "expected": "reject", "reason": { "zh-Hant": "仍需要獨立；完全相同的重複變數不會平均掉variance。", "en": "Independence is needed; repeating one variable does not average away its variance." } }, { "claim": { "zh-Hant": "弱律證所有路徑收斂。", "en": "The weak law proves every path converges." }, "expected": "reject", "reason": { "zh-Hant": "此處結論是依機率而非逐路徑。", "en": "The conclusion here is probabilistic, not pathwise." } }, { "claim": { "zh-Hant": "有限樣本像IID就足夠。", "en": "A finite sample looking IID suffices." }, "expected": "reject", "reason": { "zh-Hant": "所用前提涵蓋整個無限序列。", "en": "The premise covers the whole infinite sequence." } } ], "status": "candidate", "gaps": [], "verification": { "structure": "checked_local_script", "mathematics": "candidate_argument", "formalization": "not_formalized" }, "rule_glossary": [ { "id": "mko_definition", "kind": "cited", "statement": { "zh-Hant": "限明示IID通用證書、共同有限mean mu與variance sigma squared。independence先給Var(sample mean)=sigma squared/n，再用exact Chebyshev epsilon-delta tail證依機率收斂；有限模擬不足。", "en": "Use a declared whole-sequence IID premise, common finite mean mu and variance sigma_squared. Independence gives Var(sample_mean_n)=sigma_squared/n; an exact Chebyshev epsilon-delta tail then gives convergence in probability. Finite simulation is insufficient." }, "source": { "mko_id": "mko-weak-law-large-numbers", "sha256": "7eb5ea7326a95ec1926f661b129d1f6ed74f95c53574aea5f3878111f80bb55c", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f" } }, { "id": "work", "kind": "prerequisite", "statement": { "zh-Hant": "假設Chebyshev、期望線性與variance加法含covariance；獨立的有限二次矩變數covariance零。依機率收斂表示任意epsilon>0、delta>0皆有N，使所有n≥N的偏差機率P(|Y_n-Y|>epsilon)0 and delta>0 have an N with P(|Y_n-Y|>epsilon)