pgf_and_branching_practice_problems

pgf_and_branching_practice_problems - Stat 333 Summer 2007...

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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 )= Var ( X )= λ . Solution :S in c e 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 Var ( X )= G ±± X (1) + G ± X (1) [ G ± X (1)] 2 = λ 2 + λ λ 2 = λ. 2. Suppose
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This note was uploaded on 01/19/2011 for the course STATISTICS STAT 333 taught by Professor Menzhongxian during the Fall '10 term at Waterloo.

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pgf_and_branching_practice_problems - Stat 333 Summer 2007...

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