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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 ) = 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 (
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