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 =
λ + μ
