NotesNov10 - Further properties of the bivariate Gaussian...

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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 .
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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 < .
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All level curves of the density n ( x, y ) : f X,Y ( x, y ) = c o are ellipses.
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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 .
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The Central Limit Theorem : If the variance
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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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