NotesNov10

NotesNov10 - σ 2 = Var X i is finite then the...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Further properties of the bivariate Gaussian distribution Let X and Y be jointly normal distribution with the means μ X and μ Y , variances σ 2 X and σ 2 Y , and correlation ρ . Suppose that the correlation ρ = 0. Then the joint pdf f X,Y ( x,y ) = f X ( x ) f Y ( y ) and so X and Y are independent. Conclusion : For jointly normal random vari- ables zero correlations imply independence. This is NOT true without the assumption of joint normality .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
If μ X = μ Y = 0 and σ 2 X = σ 2 Y = 1, then X and Y are jointly normal with standard normal marginals. In this case the joint density takes a simpler form f X,Y ( x,y ) = 1 2 π q 1 - ρ 2 exp ( - x 2 - 2 ρxy + y 2 2(1 - ρ 2 ) ) , -∞ < x,y < .
Background image of page 2
All level curves of the density n ( x,y ) : f X,Y ( x,y ) = c o are ellipses.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Central Limit Theorem The Central Limit Theorem (CLT) describes how well the sample mean approximates the true mean. Let X 1 ,X 2 ,... be iid random variables with a common mean EX i = μ . The Law of Large Numbers (LLN) says that, as n → ∞ , the sample mean converges to the true mean: ¯ X n = X 1 + X 2 + ... + X n n μ = EX 1 .
Background image of page 4
The Central Limit Theorem : If the variance
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This note was uploaded on 12/03/2010 for the course OR 3500 at Cornell.

Page1 / 8

NotesNov10 - σ 2 = Var X i is finite then the...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online