Lecture 10-2007 - Moment Generating Functions Lecture X I....

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Moment Generating Functions Lecture X I. Moment Generating Functions A. Definition 2.3.3. Let X be a random variable with cumulative distribution function FX . The moment generating function (mgf) of X (or ), denoted X Mt , is tX X M t E e provided that the expectation exists for t in some neighborhood of 0. That is, there is an 0 h such that, for all t in h t h , tX Ee 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 defined as: for continuous random variables, and for discrete random variables tx X tx X x M t e f x dx M t e P X x B. Theorem 2.3.2: If X has mgf X , then 0 n n X E X M where we define () 0 0 n n XX n t d M M t dt 1. First note that tX e can be approximated around zero using a Taylor series expansion: 23 0 0 2 0 3 0 11 0 0 0 26 1 tx t t t X M t E e E e te x t e x t e x tt E x t E x E x Note for any moment n : 1 2 2 n n n n n n d M M t E x E x t E x t dt Thus, as 0 t
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AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture X Professor Charles B. Moss Fall 2007
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Lecture 10-2007 - Moment Generating Functions Lecture X I....

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