Lecture 14 - Law of Large Numbers

# Lecture 14 - Law of Large Numbers - 1 LAW OF LARGE NUMBERS...

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1 LAW OF LARGE NUMBERS Lecture 14 ORIE3500/5500 Summer2011 Li 1 Law of Large Numbers The law of large numbers or l.l.n. is one of the most important theorems in probability and is the backbone of most statistical procedures. Theorem. If X 1 , . . . , X n are independent and identically distributed(iid) with mean μ , then the sample mean ¯ X n converges to the true mean μ as n in- creases, that is, ¯ X n -→ μ, n → ∞ . Before we try to see why we should expect this let us recall a few properties of the the sample mean, ¯ X n = X 1 + · · · + X n n . 1. Expected value of the sample mean, E ( ¯ X n ) = E X 1 + · · · + X n n · = 1 n E ( X 1 + · · · + X n ) = 1 n nE ( X 1 ) = μ. 2. If var ( X 1 ) = σ 2 , then variance of the sample mean, var ( ¯ X n ) = var X 1 + · · · + X n n · = 1 n 2 var ( X 1 + · · · + X n ) = 1 n 2 ( var ( X 1 ) + · · · + var ( X n )) (by independence) = 1 n 2 n · var ( X 1 ) = σ 2 n . 1

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2 NOTIONS OF CONVERGENCE This means that the variance of the sample mean decreases as the sample size increases. Recall that the variance of a random variable measures the dispersion of the random variable about its mean. So if the variance is decreasing to 0, then the random variable is slowly shrinking to its mean. It becomes more and more concentrated around the population mean. The Chebyshev’s inequality completes the argument. For any ² > 0 P [ | ¯ X n - μ | > ² ] var ( ¯ X n ) ² 2 = σ 2 /n ² 2 0 , n → ∞ . This shows that whatever small number positive ² we choose, the probability that the sample mean is more than ² distance away from the true mean goes to zero. So we proved the LLN in the case when the variance of X 1 is finite. It can be proved without this assumption as well. Note that the statement of LLN does not assume anything about the variance of X 1 .
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