{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

pgf_and_branching_practice_problems

# pgf_and_branching_practice_problems - Stat 333 Summer 2007...

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

Stat 333 Summer 2007 Probability generating function, Branching process Practice Problems READING: Ross’s textbook, Page 233 to Page 236. The following are some problems regarding the generating function, the probability generating function and the branching process. 1. Let X be a Poisson random variable with parameter λ . Find the probability generating function G X ( s ) and indicate the interval of convergence. Use G X ( s ) to show that E ( X ) = V ar ( X ) = λ . Solution : Since X is a Poisson random variable with parameter λ , then P ( X = n ) = λ n e λ n ! for n = 0 , 1 , . . . . G X ( s ) = E [ s X ] = n =0 s n P ( X = n ) = n =0 ( ) n e λ n ! = e λ and the series converges for | | < i.e. the series converges for all values of s . From the formulas in course notes, we have E ( X ) = G X (1) = λ and V ar ( X ) = G X (1) + G X (1) [ G X (1)] 2 = λ 2 + λ λ 2 = λ. 2. Suppose U has a discrete uniform distribution over the integers 1 , 2 , ..., k 1 , k . That is, P ( U = i ) = 1 /k for each of i = 1 , ..., k . Find the probability generating function G U ( s ) and its interval of convergence. Use G U (

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 ]}