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Lecture 16-2007

# Lecture 16-2007 - Empirical Examples of the Central Limit...

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Empirical Examples of the Central Limit Theorem: Lecture XVI I. Back to Asymptotic Normality A. The characteristic function of a random variable X is defined as           cos sin cos sin itX X t E e E tX i tX E tX iE tX     Note that this definition parallels the definition of the moment-generating function   tX X M t E e 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 function of the uniform distribution function is   1 it X te  The characteristic function of the Normal distribution function is   22 2 t it X The Gamma distribution function       1 0, r r X xe f X X r  which implies the characteristic function     1 1 X r t it B. Taking a Taylor series expansion of around the point 0 t yields           11 0 0 0 1!

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Lecture 16-2007 - Empirical Examples of the Central Limit...

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