Product distribution of independent Normal and Exponential random variables

Question

Let $Z = XY$ be a random variable which is the product of two independent random variables $X\sim\mathcal{N}(0,\sigma^2)$ and $Y\sim \text{Exp}(\lambda)$. I'm wondering, is there some way to write the expression for the PDF of $Z$? (Side note: this question is very similar, but didn't get an answer)

Based on the standard definition of the product of two independent random variables, the PDF of $Z$, $f_Z(z)$, can be written as:

\begin{align} f_Z(z) &= \int_{0}^{\infty}f_Y(x)f_X(z/x)\frac{1}{x}dx\\ &= \int_{0}^{\infty}\lambda\exp(-\lambda x)\frac{1}{\sqrt{2\pi\sigma^2}}\exp\Big(\frac{-z^2}{2\sigma^2x^2}\Big)\frac{1}{x}dx \end{align}

I'm not too sure how to go about simplifying this. Any ideas? Approximations?

Answer 1

I don't think it can be expressed in an elementary way. Maple evaluates the integral as $$ \frac{1}{\pi z} G^{3, 0}_{0, 3}\left({\frac {{z}^{2}{\lambda}^{2}}{{8\sigma}^{2}}}\, \Big\vert\,^{}_{1, 1/2, 1/2}\right)$$ where $G$ is the Meijer G function.