Let 
where 
and 
are
independent Poisson random variables with
means λ and μ, respectively.
Find the pmf of 
and show that X + Y is itself a Poisson random variable with parameter
λ + μ
First, let’s find
the pmf of .
According to the characteristic
function of a Poisson random variable
and are
independent 
and are
uncorrelated 
and 
are
uncorrelated.
From the above, we know that is
a Poisson random variable with mean λ + μ.
