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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.

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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 $\lambda_1 \lambda_2(\lambda_2 +1)$.

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