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Lecture10-2010 - Moment Generating Functions Lecture X...

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Moment Generating Functions: Lecture X Charles B. Moss September 14, 2010 I. Moment Generating Function A. Definition 2.3.3. Let X be a random variable with cumula- tive distribution function F ( X ). The moment generating function (mgf) of X (or F ( X ) ), denoted M X ( t ), is M X ( t ) = E e tX (1) provided that the expectation exists for t in some neighborhood of 0. That is, there is an h > 0 such that, for all t in h < t < h , E e tX exists. 1. If the expectation does not exist in a neighborhood of 0, we say that the moment generating function does not exist. 2. More explicitly, the moment generating function can be de- fined as M X ( t ) = −∞ e tx f ( x ) dx for continuous random variables , and M x ( t ) = x e tx P [ X = x ] for discrete random variables (2) B. Theorem 2.3.2 If X has mgf M X ( t ) , then E [ X n ] = M ( n ) x (0) (3) where we define 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture X Fall 2010 M ( n ) X (0) = d n dt n M X ( t ) | t 0 (4) 1. First note that e tX can be approximated around zero using a Taylor series expansion M X ( t ) = E e tx = E e 0 + te t 0 ( x 0) + 1 2 t 2 e t 0 ( x 0) 2 + 1 6 t 3 e t 0 ( x 0) 3 + · · · = 1 + E [ x ] t + E x 2 t 2 2 + E x 3 t 3 6 + · · · (5) Note for any moment n M ( n ) x ( t ) = d n dt n M X ( t ) = E [ x n ]+E x n +1 +E x n +2 + · · · (6) Thus, as t 0 M ( n ) x (0) = E [ x n ] (7) 2. Leibnitzs Rule: If f ( x, θ ) , a ( θ ) , and b ( θ ) are differentiable with respect to θ , then d b ( θ ) a ( θ ) f (
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