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Stat 5101 notes - STAT5101 Lecture Notes 8 Moment...

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Unformatted text preview: STAT5101 Lecture Notes 8 Moment generating function (MGF): defi- nition, basic properties. Use MGF to compute the moments of a random variable. Use MGF to identify the distribution of a random variable. MGF and the central limit theorem. 1 Moment Generating Functions Definition: The moment generating function (MGF) of X , written as ( t ), is ( t ) = E [ e tX ] . When the expectation is not well-defined for some t = t , then we do not talk about ( t ). If X is a continuous random variable with pdf f ( x ), then ( t ) = e tx f ( x ) dx. If X is a discrete random variable with proba- bility function f , then ( t ) = x e tx f ( x ) . Remark: When t = 0, MGF is always well defined and equals 1. (0) = 1, because (0) = E [ e X ] = E [1] = 1 . 2 Ex1 Let X Ber ( p ), which means X is either 1 or 0, and P ( X = 1) = p , P ( X = 0) = 1 p. The moment generating function (MGF) of X is ( t ) = E [ e tX ] = e t 1 P ( X = 1) + e t P ( X = 0) = e t p + e (1 p ) = 1 p + pe t The above equation holds for all < t < . 3 Ex2 Let X Bin ( n,p ), which means X takes values from 0 , 1 , 2 ,...,n , and P ( X = k ) = ( n k ) p k (1 p ) n k . The moment generating function (MGF) of X is ( t ) = E [ e tX ] = n k =0 e t k P ( X = k ) = n k =0 e t k ( n k ) p k (1 p ) n k = n k =0 ( n k ) ( e t ) k p k (1 p ) n k = n k =0 ( n k ) ( e t p ) k (1 p ) n k = ( e t p + 1 p ) n where in the last step we applied the Binomial theorem. The above equation holds for all < t < . 4 Ex3 Let X has the pdf f ( x ) = e x I ( x > 0) . ( t ) = E [ e tX ] = e tx e x dx = e ( t 1) x dx If t 1, then e ( t 1) x 1 for all x 0. Thus e ( t 1) x dx = . If t < 1, ( t ) = e ( t 1) x dx = 1 t 1 e ( t 1) x | = 0 1 t 1 = 1 1 t 5 Ex4 Let X N (0 , 1). Find ( t ). ( t ) = E [ e tX ] = e tx 1 2 e x 2 2 dx = 1 2 e x 2 2 tx 2 dx = 1 2 e ( x t ) 2 t 2 2 dx = e 1 2 t 2 [ 1 2 e ( x t ) 2 2 dx ] = e 1 2 t 2 [ 1 2 e s 2 2 ds ] [ s = x t ] = e 1 2 t 2 1 = e 1 2 t 2 6 Compute Moments by MGF The k-th moment of X is defined as E [ X k ] for k = 1 , 2 , 3 , 4 ,... whenever the expectation is well defined.well defined....
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Stat 5101 notes - STAT5101 Lecture Notes 8 Moment...

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