Unformatted text preview: with mean Î» 1 and X 2 is an independent Poisson random variable with mean Î» 2 , then X 1 + X 2 is a Poisson random variable with mean Î» 1 + Î» 2 . 7) Suppose that X 1 , X 2 , . . . is a sequence of independent, identically distributed nonnegative integer valued random variables with common pgf Ï† , and suppose that N is an independent nonnegative integer valued random variable with pgf Ïˆ . Show that the random variable âˆ‘ N i =1 X i has pgf Î³ ( s ) = Ïˆ ( Ï† ( s )). (Hint: First condition on the value of N and use (5).) 8) In the setup of (7), suppose that X 1 , X 2 , . . . are Bernoulli with success probability p and N is Poisson with mean Î» . Show that âˆ‘ N i =1 X i is Poisson with mean Î»p . 1...
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 Spring '08
 Evans
 Probability, Probability theory, #, PGF

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