Lecture10-2010

# 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. Defnition 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 h e tX i (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 h e tX i 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- ±ned as M X ( t )= Z −∞ e tx f ( x ) dx for continuous random variables , and M x ( t X 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) wherewede±ne 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 h e tx i =E h 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 h x 2 i t 2 2 h x 3 i t 3 6 + (5) Note for any moment n M ( n ) x ( t )= d n dt n M X ( t )=E[ x n ]+E h x n +1 i h x n +2 i + (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 di±erentiable with respect to θ ,then d Z b ( θ ) a ( θ ) f ( x, θ ) dx = f ( b ( θ ) ) d a ( θ )+ f ( a ( θ )) d b ( θ ) Z b ( θ ) a ( θ )
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## This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.

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

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