I need direction to approximate the resultant probability distribution of the product of two independent distributions: $N(\mu, \sigma^2)$ and $lognormal(\mu_{N}, \sigma_{N}^2)$, where
$\mu_{N}$ is the normal mean and $\mu_{N} = 0$, and
$\sigma_{N}$ is the normal SD
Is the resultant distribution a log-normal distribution? If yes, what are the normal mean and normal SD? If no, is there a way to approximate the pdf for dummies? I am an R user.
See this method for calculating the PDF of the product of two independent continuous random variables, in terms of their PDFs. But no such calculation is needed to note that, since a Normally distributed $X$ can be positive or negative but a log-normally distributed $Y$ is non-negative, $XY$ can be positive or negative and hence isn't log-normal. As for the mean and variance, note that $$\Bbb EXY=\Bbb EX\Bbb EY=\mu\exp\left(\mu_N+\frac12\sigma_N^2\right),\,\\\operatorname{Var}XY=\Bbb EX^2\Bbb EY^2-(\Bbb EXY)^2\\=(\mu^2+\sigma^2)\exp\left(2\mu_N+2\sigma_N^2\right)-\mu^2\exp\left(2\mu_N+\sigma_N^2\right).$$