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MomGen1 - Moment Generating Functions a rst look Math431...

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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 * 0 * P { X = 0 } + e t * 1 * P { X = 1 } = (1 - p ) e t * 0 + 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 ) x q n - x = ( pe t + q ) n . Example 3. Let X Exp ( λ ). Then, M ( t ) = E [ e tX ] = Z 0 e tx λe - λx dx = λ Z 0 e - ( λ - t ) x dx = λ λ - t , for t < λ.
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