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.
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$\begingroup$ 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. BerkCommented Nov 4, 2013 at 9:57
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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_bCommented Nov 4, 2013 at 11:56
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1$\begingroup$ To put it in slightly different terms what already pointed out by @Glen_b, Poisson is a discrete distribution, where your variable X assumes integer values 0...n, but a normal distribution is a continuous one, where your variable Y takes values -infinity to +infinity. $\endgroup$– gcirianiCommented Nov 11, 2022 at 15:29
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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.
answered Nov 4, 2013 at 12:06