{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

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

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.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 3

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

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

View Full Document
Ask a homework question - tutors are online