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Unformatted text preview: Moment Generating Functions: a first look Math431, Spring 2011 Instructor: David F. Anderson Section 7.7: Moment generating functions – a first look Moment generating functions have two major properties: 1. They allow us to calculate the moments of random variable. 2. No two different RVs have same moment generating function . Thus, to prove a RV has a certain distribution, you really only need moment generating function. Definition: For a random variable X , the moment generating function of X is M X ( t ) = E e tX . Therefore, for discrete, continuous RV we have M X ( t ) = X x ∈ R ( X ) e tx p X ( x ) discrete case M X ( t ) = Z ∞∞ e tx f ( x ) dx continuous case . Example 1. Let X be a Bernoulli RV with parameter p . Then: M X ( t ) = E e tX = e t * * P { X = 0 } + e t * 1 * P { X = 1 } = (1 p ) e t * + pe t * 1 = (1 p ) + pe t . Example 2. Let X be binomial( n,p ). Then, letting q = 1 p , we have M X ( t ) = E e tX = n X x =0 e tx n x p x q n x = n X x =0 n x ( pe t )...
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This note was uploaded on 04/17/2011 for the course MATH 431 taught by Professor Balazs during the Spring '05 term at University of Wisconsin.
 Spring '05
 BALAZS
 Math, Probability

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