# 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.

• Not a standard distribution. But independence guarantees that the mean of the product is the product $\lambda_1\lambda_2$ of the means.
– Did
Commented Jan 7, 2013 at 12:18
• Can you provide the reference for that? That would be helpful. Commented Jan 7, 2013 at 12:21
• – Did
Commented Jan 7, 2013 at 12:30
• Would the Mellin transform be a step in obtaining the distribution? Commented Apr 23 at 16:30

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