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Lecture16-2010 - 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 I. Back to Asymptotic Normality A. The characteristic function of a random variable X is defined as φ X ( t ) = E e itX = E [cos ( tX ) + i sin ( tX )] = E [cos ( tX )] + i E [sin ( tX )] . (1) Note that this definition parallels the definition of the moment- generating function M X ( t ) = E e tX (2) 1. 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. 2. The characteristic function of the uniform distribution func- tion of the uniform distribution function is φ X ( t ) = e it - 1 (3) The characteristic function of the Normal distribution func- tion is φ X ( t ) = exp itμ - σ 2 t 2 2 (4) The Gamma distribution function 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XVI Fall 2010 f ( X ) = a r x r - 1 e - α X Γ ( r ) X (0 , ) (5) which implies the characteristic function φ X ( t ) = 1 1 - it α r (6) B. Taking a Taylor series expansion of around the point
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