Could we find two random variables $X$ and $Y$ which $XY \sim N(\mu, \sigma^2)$?
I found the ratio of two normal distributed random variables is distributed Cauchy distribution.
However, on the other way, the product ($U=XY$) of $X \sim Cauchy()$ and $Y \sim Normal()$ is not Normal distribution.
My first question is that I could not understand why the reverse statement does not hold.
My second question is how to find $X$ and $Y$ such the product is normal distribution. (Assume both $X$ and $Y$ are continuous.)