Section6Notes - SECTION NOTES YANKUN WANG 1. Moment...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
YANKUN WANG 1. Moment Generating Functions The Moment Generating Function (MGF) Φ( t ) of the random vari- able X is de±ned 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 di²erentiating 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) } Di²erentiation 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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/22/2009 for the course ORIE 3500 taught by Professor Weber during the Summer '08 term at Cornell University (Engineering School).

Page1 / 3

Section6Notes - SECTION NOTES YANKUN WANG 1. Moment...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online