TITLE: Product of two Gaussian PDFs is a Gaussian PDF, but Product of two Gaussian Variables is not Gaussian
QUESTION [65 upvotes]: The Product of Two Gaussian Random Variables is not Gaussian distributed:

Is the product of two Gaussian random variables also a Gaussian?
Also Wolfram Mathworld
So this is saying $X \sim N(\mu_1, \sigma_1^2)$, $Y \sim N(\mu_2, \sigma_2^2)$ then $XY \sim W$ where W is some other distribution, that is not Gaussian 

But the product of two Gaussian PDFs is a Gaussian PDF:

Calculate the product of two Gaussian PDF's
Full Proof
This tutorial which I am trying to understand Writes: $N(\mu_1, \sigma_1^2)\times N(\mu_2, \sigma_2^2) = N(\frac{\sigma_1^2 \mu_2 + \sigma_2^2 \mu_1}{\sigma_1^2 + \sigma_2^2},\frac{1}{\frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2}})$

What is going on here?
What am I doing when I take the product of two pdfs
vs. when I take the product of two variables from the pdfs?
When (what physical situation) is described by one,
and what by the other?
(I think a few real world examples would clear things up for me)

REPLY [3 votes]: In brief, you were confused by two totally different concepts.

For the first, you are calculating the distribution of transformed random variables. here,  you specify that as the product of XY.
For the second, you just calculate the product of two functions $\phi(x)\phi(y)$, which happen to be the PDF of two normal random variables. 

For some details, check here. In general, if you want to calculate the PDF of XY, you need to figure out $F(t) = P(XY < t)$, then $f(t) = F'(t)$. As for the product of two functions, that's easy as you can see.