Product of Gaussian and Rayleigh distributions gives what distribution?

[Mishra]

TL;DR Summary

Finding out the distribution of the product of a Gaussian and a Rayleigh distributed random variables.

Hello,

I'm trying to find out the distribution function (cumulative or density) of the product of two independent random variables respectively following a non-zero-mean Gaussian and a Rayleigh distribution. The math is too intricate for me, I've found in the appendix of [Probability Distributions Involving Gaussian Random Variables - Simon, Marvin K.] the density for the product of a zero-mean Gaussian and a Rayleigh but this will not work for what I am trying to do.Would anyone have a reference that could help me ?

 

[Valkarie]

Thanks a lot for your help ! The answer to your question depends on the exact forms of the distributions that you are working with. Generally speaking, the product of two independent random variables will not have a closed form distribution and can be difficult to work with analytically. You may find useful results in the literature on copulas or related topics. For example, the book Copula Methods in Finance (McNeil et al., 2005) has several chapters devoted to the topic. Alternatively, you may need to resort to Monte Carlo simulation approaches that sample from the involved distributions and then compute the product of the sample values.