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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 ψ ( t ), is ψ ( t ) = E [ e tX ] . When the expectation is not welldefined for some t = t , then we do not talk about ψ ( t ). If X is a continuous random variable with pdf f ( x ), then ψ ( t ) = ∫ ∞ −∞ e tx f ( x ) dx. If X is a discrete random variable with proba bility function f , then ψ ( t ) = ∑ x e tx f ( x ) . Remark: When t = 0, MGF is always well defined and equals 1. ψ (0) = 1, because ψ (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 ψ ( 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 ψ ( 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) . ψ ( 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, ψ ( 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 ψ ( t ). ψ ( t ) = E [ e tX ] = ∫ ∞ −∞ e tx 1 √ 2 π e − x 2 2 dx = ∫ ∞ −∞ 1 √ 2 π e − x 2 − 2 tx 2 dx = ∫ ∞ −∞ 1 √ 2 π e − ( x − t ) 2 − t 2 2 dx = e 1 2 t 2 [ ∫ ∞ −∞ 1 √ 2 π e − ( x − t ) 2 2 dx ] = e 1 2 t 2 [ ∫ ∞ −∞ 1 √ 2 π 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 kth 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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This note was uploaded on 02/24/2011 for the course STAT 5101 taught by Professor Staff during the Fall '02 term at Minnesota.
 Fall '02
 Staff
 Statistics, Central Limit Theorem

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