Product of two independent poisson variables

Suppose we have two independent poisson variables $X_1$ and $X_2$ such that $X_1 ∼ \operatorname{Poisson}(\lambda_1)$ and $X_2 ∼ \operatorname{Poisson}(\lambda_2)$. 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.

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Not a standard distribution. But independence guarantees that the mean of the product is the product $\lambda_1\lambda_2$ of the means. – Did Jan 7 '13 at 12:18
Can you provide the reference for that? That would be helpful. – anuj919 Jan 7 '13 at 12:21
– Did Jan 7 '13 at 12:30

Since $X_1,X_2$ are independent you get:
$$E[X_1X_2]=E[X_1]E[X_2]=\lambda_1\lambda_2$$