View source for ECE 600 Exams Addition of two independent Poisson random variables - Rhea

Example. Addition of two independent Poisson random variables
Let  where  and  are independent Poisson random variables with means λ and μ , respectively. 

(a) 

Find the pmf of  . 

According to the characteristic function of Poisson random variable 

. 

 and  are independent  and  are uncorrelated  and  are uncorrelated. 

 

Now, we know that \mathbf{Z} is a Poisson random variable with mean λ + μ . 

 

(b) 

Show that the conditional pmf of  conditioned on the event  is binomially distributed, and determine the parameters of binomial distribution (n and p ). 

 

This is a binomial pmf b(n,p) with parameters n and 

 λ 
p =  
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 λ + μ