Laws that describe the way in which the sample mean approaches the population mean as the sample size increases. The phrase is due to Poisson, who, in 1835, referred to 'La loi des grands nombres'. The weak law of large numbers states that if X1, X2,..., Xn are a set of independent identically distributed random variables, each with expectation μ, and if
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X̄ → μ 'almost surely', as n → ∞.
The strong law of large numbers states that, for every positive δ, it is always possible to find positive values ε and N such that P(|X̄-μ|≥ε)≤δ, n=N, N+1,....
An equivalent statement is that X̄ → μ 'in probability', as n → ∞.
