# A Chebyshev tail for fair Bernoulli averages [← Guide index](https://ocme.evemisslab.com/en/derivations/) · [Original MKO](https://ocme.evemisslab.com/en/mko/mko-weak-law-large-numbers/) This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented. derivation-mko-weak-law-large-numbers-01 · ocme-new142-reader-adoption-20261021 ## Goal and scope Derive the WLLN from a whole-sequence IID premise. Finite-variance weak law; the conclusion is convergence in probability. ## Givens and symbols - work - G1 — 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. - bar X_n — n-term sample mean - N — epsilon-delta tail index ## Rules and sources - mko_definition · cited — 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. · mko-weak-law-large-numbers - work · prerequisite — Assume Chebyshev, linearity of expectation and variance expansion with covariances; independent finite-second-moment variables have zero covariance. Convergence in probability means that every epsilon>0 and delta>0 have an N with P(|Y_n-Y|>epsilon)0 After P(|\bar X_n-1/2|>\varepsilon)\le\frac1{4n\varepsilon^2} Conditions: none Rule: work · Uses: G1, S05, work Leaf basis: work Chebyshev's≥ event contains the strict> event, so the bound applies. S07 · impliesBefore \varepsilon,\delta>0,\quad N=\lfloor1/(4\varepsilon^2\delta)\rfloor+1 After n\ge N\implies P(|\bar X_n-1/2|>\varepsilon)<\delta Conditions: none Rule: work · Uses: G1, S06, work Leaf basis: work Explicit N strictly exceeds the critical value and controls the entire tail. S08 · impliesBefore \forall\varepsilon,\delta>0\ \exists N\ \forall n\ge N After \bar X_n\xrightarrow{P}1/2 Conditions: none Rule: work · Uses: G1, S07, work Leaf basis: work This is the definition of convergence in probability, completing the finite-variance WLLN instance. ## Substeps and lemmas No substeps in this guide. ## Reverse audits - S08 · premise_audit · G1, S07, work — It is neither an a.s. proof nor a finite-n normal equality. - S07 · premise_audit · G1, S06, work — Epsilon and delta are arbitrary first; N may depend on them. - S06 · premise_audit · G1, S05, work — Epsilon is positive and the mean and variance concern the same n. - S05 · premise_audit · G1, S04, work — Do not lose the squared scaling denominator. - S04 · premise_audit · G1, S03, work — Identical distribution alone does not ensure this equation. - S03 · premise_audit · G1, S02, work — The denominator requires n≥1. - S02 · premise_audit · G1, S01, work — This theorem permits zero variance; the present one is 1/4. - S01 · premise_audit · G1, work — The index i is arbitrary, not just1. ## Error controls and negative controls Use of independenceUse independence to eliminate cross covariances; do not replace it with identical marginals. Strict tailChoose floor of the critical value plus one to ensure