TITLE: PDF of the product of normal and Cauchy distributions
QUESTION [5 upvotes]: I am having trouble in finding out the resulting PDF of the product of normal and Cauchy distributions. It turns out that we have a general formula for calculating the PDF of product of two random distribution but the integral is not converging . Also, I tried using Mellin Transform method but it is getting too complicated. 
If anybody has any idea about how to approach the problem, please share it with me.
Thanks

REPLY [3 votes]: Since the normal distribution is a $ t $ distribution with $\infty $ degrees of freedom, perhaps you could find the pdf of the product of two $ t $ distributed random variables and then take a limit.