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.

  • $\begingroup$ 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. $\endgroup$
    – M. Berk
    Nov 4, 2013 at 9:57
  • 1
    $\begingroup$ (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. $\endgroup$
    – Glen_b
    Nov 4, 2013 at 11:56

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.