in my probability class I was given this question on which I am stuck concerning a sum of random number of Poisson random variables:
Let us define the countable set of independent random variables $ X_i \sim \mathrm{Pois}(\lambda _i) $ and the random variable $N$ independent from the rest of the $ X_i $ which also has a Poisson distribution $ N \sim \mathrm{Pois}(\lambda) $. We are asked to check if the following sum $ Y = \sum_{i=1}^N X_i $ also has a Poisson distribution and if so, with what parameter? As a hint we are asked to look at the characteristic function of the variable to check.
I know the deterministic finite sum of Poisson random variables is again a Poisson random variable with the sum of parameters, but I cannot solve it for a random number of summands that is also Poisson-ly distributed, I know the characteristic function of a Poisson distributed random variable $ \Phi(t) = e^{\lambda (e^{it}-1)} $. I thank all helpers who can show me a way out of this.