Suppose we have two independent poisson variables $X_1$ and $X_2$ such that $X_1 ∼ Poisson(\lambda_1)$ and $X_2 ∼ Poisson(\lambda_2)$. What will be the probability distribution of $X_1 \times X_2$? Is it some standard distribution?
I am particularly interested in the mean value of the distribution.
Additional question: If I have chain of N poisson variables, can we say anything about mean value of the multiplication of these variables?
I could not find any online resource discussing this.