This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: σ 2 = Var X i is ﬁnite then the distribution of √ n ( ¯ X nμ ) converges, as n → ∞ , to the normal N (0 ,σ 2 ) distribution. Conclusion : the diﬀerence  ¯ X nμ  ≈ 1 √ n . The CLT is used in diﬀerent forms: • For large n X 1 + X 2 + ... + X nnμ σ √ n has, approximately, the standard normal N (0 , 1) distribution. • P X 1 + X 2 + ... + X nnμ 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 nμ  ≈ 1 n 11 /α for some 1 < α < 2. • for large n , X 1 + ... + X n has approximately the αstable stable distribution , 1 < α < 2....
View
Full Document
 Fall '09
 SAMORODNITSKY
 Central Limit Theorem, Normal Distribution

Click to edit the document details