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Unformatted text preview: 1 MOMENT GENERATING FUNCTION Lecture 13 ORIE3500/5500 Summer2009 Chen Class Today • Moment Generating Function • Characteristics Function • Inequalities 1 Moment Generating Function For a random variable X , the moment generating function of X is defined as φ X ( t ) = E ( e tX ) . There is a reason for this name. If one is given the moment generating function of a random variable then one can get all the moments of the random variable. If we evaluate the n th derivative of the moment generating function at 0, we will get the n th moment of X , that is, d n φ X ( t ) dt n fl fl fl fl t =0 = E ( X n ) . This is because d n φ X ( t ) dt n = d n E ( e tX ) dt n = E ( d n e tX ) dt n = E ( X n e tX ) . Example Suppose that X is continuous with pdf f X ( x ) = e- x ,x > 0. Then the moment generating function of X is given by φ X ( t ) = Z ∞ e tx e- x dx = Z ∞ e- (1- t ) x dx = 1 1- t ,t < 1 . Then φ X ( t ) = 1 (1- t ) 2 ,φ 00 X ( t ) = 2 (1- t ) 3 . 1 1 MOMENT GENERATING FUNCTION By induction we get that d n dt n φ X ( t ) = n ! (1- t ) n +1 , and hence E ( X n ) = d n dt n φ X ( t ) fl fl fl fl t =0 = n ! . Example Suppose X is discrete with pmf p X ( k ) = 2- k ,k = 1 , 2 , 3 ,.... Then the mgf of X is φ X ( t ) = ∞ X k =1 e tk 2- k = ∞ X k =1 ( e t / 2) k = 1 1- e t / 2- 1 ....
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This note was uploaded on 09/22/2009 for the course ORIE 3500 taught by Professor Weber during the Summer '08 term at Cornell.
- Summer '08