Slides10-2010 - Moment Generating Functions: Lecture X...

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Moment Generating Functions: Lecture X Charles B. Moss September 14, 2010 Charles B. Moss () Moment Generating Functions September 14, 2010 1 / 13
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1 Moment Generating Function Charles B. Moss () Moment Generating Functions September 14, 2010 2 / 13
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Moment Generating Function Defnition 2.3.3. Let X be a random variable with cumulative 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 ± e tX ² exists. I If the expectation does not exist in a neighborhood of 0, we say that the moment generating function does not exist. I More explicitly, the moment generating function can be deFned 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) Charles B. Moss () Moment Generating Functions September 14, 2010 3 / 13
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Theorem 2.3.2 If X has mgf M X ( t ),then E [ X n ]= M ( n ) x (0) (3) whe rewedeFne M ( n ) X (0) = d n dt n M X ( t ) | t 0 (4) 1. ±irst note that e tX can be approximated around zero using a Taylor series expansion M X ( t )= E ± e tx ² = 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 + ··· ³ =1+ E [ x ] t + E ± x 2 ² t 2 2 + E ± x 3 ² 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 ² + E ± x n +2 ² + (6) Thus, as t 0 Charles B. Moss () Moment Generating Functions September 14, 2010 4 / 13
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Continued I Leibnitzs Rule: If f ( x ), a ( θ ),and b ( 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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Slides10-2010 - Moment Generating Functions: Lecture X...

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