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Slides16-2010

# Slides16-2010 - Empirical Examples of the Central Limit...

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Empirical Examples of the Central Limit Theorem: Lecture XVI Charles B. Moss October 7, 2010 Charles B. Moss () Empirical Examples of the Central Limit Theorem October 7, 2010 1 / 14

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1 Asymptotic Normality 2 Wrapping up Loose Ends 3 Normal Approximation of the Binomial Charles B. Moss () Empirical Examples of the Central Limit Theorem October 7, 2010 2 / 14
Back to Asymptotic Normality The characteristic function of a random variable X is deFned as φ X ( t )= E h e itX i = E [cos ( tX )+ i sin ( tX )] = E [cos ( tX )] + i E [sin ( tX )] . (1) Note that this deFnition parallels the deFnition of the moment-generating function M X ( t E h e tX i (2) Charles B. Moss () Empirical Examples of the Central Limit Theorem October 7, 2010 3 / 14

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Like the moment-generating function there is a one-to-one correspondence between the characteristic function and the distribution of random variable. Two random variables with the same characteristic function are distributed the same. The characteristic function of the uniform distribution function of the uniform distribution function is φ X ( t )= e it 1( 3 ) The characteristic function of the Normal distribution function is φ X ( t )=exp ± it μ σ 2 t 2 2 ² (4) Charles B. Moss () Empirical Examples of the Central Limit Theorem October 7, 2010 4 / 14
The Gamma distribution function f ( X )= a r x r 1 e α X Γ( r ) X (0 , )( 5 )

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Slides16-2010 - Empirical Examples of the Central Limit...

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