# Distribution of Product of Normal and Poisson?

Suppose that X is distributed Poisson with a known rate and Y is a normal distributed with a know mean and variance. My goal is to approximate the distribution Z where P(Z) = P(X) * P(Y), where Z is a non-negative integer. I could get a good approximation by sampling, but I'd really like to have a fast solution, ideally closed-form.

• Do you have an idea of likely values for the parameters of each distribution? Specifically, if the rate parameter of the Poisson distribution is large then you could use a Normal approximation and the product of two Normal distributions appears to be well studied. – M. Berk Nov 4 '13 at 9:57
• (i) Did you mean to take the product of a pdf and a pmf there where you say P(X)*P(Y)? That doesn't seem to match your title which implies a product of random variables. (ii) You don't state what the bivariate distribution of $X$ and $Y$ is, only the margins. – Glen_b Nov 4 '13 at 11:56

## 1 Answer

There is one book dedicated to the problem of products of random variables: http://www.amazon.com/Products-Random-Variables-Applications-Arithmetical/dp/0824754026/ref=sr_1_1?s=books&ie=UTF8&qid=1383564424&sr=1-1&keywords=product+of+random+variables

Maybe you can find it in a library. (Or search google scholar with the author names)

There is a connection between products of independent random variables and the Mellin transform, see the paper: "Some Applications of the Mellin Transform in Statistics" by Benjamin Epstein, which is on JSTOR. There is a Wikipedia article on the Mellin Transform, and search google scholar for "Mellin transform product of random variables" gives some relevant papers.