Weak Law of Large Numbers

Probability · Axiom Academy

LESSON Weak Law of Large Numbers Understanding convergence in probability as sample size grows Let X₁, X₂, X₃, ... be independent and identically distributed (i.i.d.) random variables with mean μ and finite variance σ². Define the sample mean: The Weak Law states that for any ε > 0, no matter how small: This means the probability that the sample mean differs from μ by more than ε goes to zero as n increases. 2. Chebyshev's Inequality Foundation The proof relies on Chebyshev's inequality, which bounds the probability of deviating from the mean: For our sample mean X̄ₙ, we know: E[X̄ₙ] = μ (expected value is the population mean) Var(X̄ₙ) = σ²/n (variance decreases with sample size) Applying Chebyshev's inequality to X̄ₙ: As n → ∞, we have σ²/(nε²) → 0, therefore: This completes the proof of the Weak Law of Large Numbers.

This is the written version of the interactive lesson above. See the full Probability course.