07_03 - Some Notes. The mgf is a function of the variable...

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p. 7-21 Some Notes. The mgf is a function of the variable t . The mgf may only exist for some particular values of t . If X ~ Poisson( ), then for < t < , M X ( t )= x =0 e tx × e λ λ x x ! = e λ e λ e t x =0 e ( λ e t ) ( λ e t ) x x ! = e λ e λ e t = e λ ( e t 1) . If X ~ Exponential( ), then for t < , M X ( t 0 e tx × λ e λ x dx = λ 1 λ t 0 ( λ t ) e ( λ t ) x dx = λ λ t , and M X ( t ) does not exist for t . A list of some mgfs ( exercise ) If X ~ Binomial( n , p ), M X ( t )=(1 p + pe t ) n , for t< log(1 p ). If X is a discrete r.v. taking on values x i with probability p i , i =1, 2, 3, …, then Example. p. 7-22 If X ~ Negative Binomial( r , p ), M X ( t pe t 1 (1 p ) e t r , for log(1 p ) . M X ( t e β t e α t t ( β α ) . • Theorem (Uniqueness Theorem). Suppose that the mgfs M X ( t ) and M Y ( t ) of random variables X and Y exist for all | t |< h for some h >0. If M X ( t ) = M Y ( t ), for | t |< h , then F X ( z ) = F Y ( z ) for all z R , where F X and F Y are the cdfs of X and Y , respectively.
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This note was uploaded on 02/15/2012 for the course MATH 2810 taught by Professor Shao-weicheng during the Fall '11 term at National Tsing Hua University, China.

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07_03 - Some Notes. The mgf is a function of the variable...

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