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Section6Notes

Section6Notes - SECTION NOTES YANKUN WANG 1 Moment...

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SECTION NOTES YANKUN WANG 1. Moment Generating Functions The Moment Generating Function (MGF) Φ( t ) of the random vari- able X is defined for all values t by: Φ( t ) = E [ e tx ] = x e tx p ( x ) if x is discrete R + -∞ e tx f ( x ) dx if x is continuous We call Φ( t ) the MGF because all of the moments of X can be obtained by successively differentiating Φ( t ). For example, Φ ( n ) (0) = E [ x n ]. Example 1 . The Poisson Distribution with Mean λ : Φ( t ) = E [ e tx ] = X n =0 e tn e - λ λ n n ! = e - λ X n =0 ( e t λ ) n n ! = e - λ e λe t = exp { λ ( e t - 1) } Differentiation yields: Φ 0 ( t ) = λe t exp { λ ( e t - 1) } , Φ 00 ( t ) = ( λe t ) 2 exp { λ ( e t - 1) } + λe t exp { λ ( e t - 1) } , and so E [ X ] = Φ 0 (0) = λ, E [ X 2 ] = Φ 00 (0) = λ 2 + λ, V ar [ X ] = E [ X 2 ] - ( E [ X ]) 2 = λ Property: The MGF uniquely determines the distribution; i.e., there exists a 1-1 correspondence between the MGF and the distribution function of a r.v. Example 2 . Suppose the MGF of a random variable X is given by: Φ( t ) = e 3( e t - 1) , what is P { x = 0 } ?

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Section6Notes - SECTION NOTES YANKUN WANG 1 Moment...

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