Analytical solution of Product of Normal distribution and Exponential Distribution?

Question

Numerically (Monte Carlo) we can get the distribution of the product of normal distribution and Exponential distribution (Both are independent). Could we do analytically?

This question is linked with Meijer G function, but still not yet calculated

Answer 1

The normal distribution is $f_N(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$. The exponential distribution has probability distribution function $f_E(x)=\lambda e^{-\lambda x}$ for $x\geq 0$ and $f_E(x)=0$ for $x<0$.

The probability distribution function $f(z)$ for the product distribution is given by the integral $f(z)=\int_{-\infty}^{\infty}f_N(x)f_E(\frac{z}{x})\frac{1}{|x|}\mathrm{d}x$.

This comes out as $f(z)=\int_0^{\infty}\frac{\lambda}{x\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}-\frac{\lambda z}{x}}$.

Wolfram Alpha could not find an analytic solution so I doubt that an analytic solution exists.