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**Unformatted text preview: **σ 2 = Var X i is ﬁnite then the distribution of √ n ( ¯ X n-μ ) converges, as n → ∞ , to the normal N (0 ,σ 2 ) distribution. Conclusion : the diﬀerence | ¯ X n-μ | ≈ 1 √ n . The CLT is used in diﬀerent forms: • For large n X 1 + X 2 + ... + X n-nμ σ √ n has, approximately, the standard normal N (0 , 1) distribution. • P X 1 + X 2 + ... + X n-nμ σ √ n ≤ x ! ≈ Φ( x ) if n is large. Rule of thumb: n ≥ 30. • A loose way to formulate the CLT: for large n , X 1 + ... + X n has approximately the normal N ( nμ,nσ 2 ) distribution . • If the variance of X 1 ,X 2 ,... is inﬁnite, then the Central Limit Theorem may still hold, but | ¯ X n-μ | ≈ 1 n 1-1 /α for some 1 < α < 2. • for large n , X 1 + ... + X n has approximately the α-stable stable distribution , 1 < α < 2....

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- Fall '09
- SAMORODNITSKY
- Central Limit Theorem, Normal Distribution