# Compound Poisson Distribution with Sum of Poisson Random Variables

I am trying to find the distribution of a compound Poisson distribution with sum of Poisson random variables.

Suppose that $$N \sim \text{Poisson}(\lambda_1)$$,

and that $$X_1, X_2, ...$$ are i.i.d. r.v.s. that are again $$\sim \text{Poisson}(\lambda_2)$$.

What is the distribution of $$Y = \displaystyle \sum_{n=1}^{N}X_n$$ ?

Is there any analytic solution for this distribution?

Any kind of help is thanked in advance.

## 1 Answer

The characteristic function is derived here.
$$\exp(\lambda_1 ( \exp(\lambda_2 (e^{it}-1))-1))$$ You could differentiate this to find the mean, variance, and other moments- see here.
Mean is $$\lambda_1 \lambda_2$$, variance is $$\lambda_1 \lambda_2(\lambda_2 +1)$$.