# 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. Commented Nov 4, 2013 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. Commented Nov 4, 2013 at 11:56
• 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. Commented Nov 11, 2022 at 15:29