Multiplying two univariate Gaussian PDFs
$$ X \sim \mathcal{N}(\mu_X,\sigma_X) \\ Y \sim \mathcal{N}(\mu_Y,\sigma_Y) \\ Z = XY $$
results in closed form equations for $\mu_Z$ and $\sigma_Z^2$:
$$ \mu_Z = \frac{\sigma^2_X*\mu_Y+\sigma^2_Y*\mu_X}{\sigma^2_X+\sigma^2_Y} $$ and $$ \sigma_Z^2 = \frac{\sigma_X^2*\sigma_Y^2}{\sigma_X^2+\sigma_Y^2} $$
some derivations of this here
This is a direct consequence of the exponential functions and the addition of exponents.
But, it is said by a lot on here that it is "obvious" that the multiplication of two Gaussian random variables is not a Gaussian random variable.
I just don't get why it's obvious. There seems to be a move to jump to the integral to combine two random variables. Why? Why is multiplying two PDFs not the same as multiplying two random variables?
My motivation: there is a literature within psychophysics to use the combinations of gaussian functions to approximate sensory cue combination in humans. An example. I want to extrapolate this to a similar case, but I'm running into this theoretical distinction people are making that I don't understand.
Links to other posts about similar things that don't quite answer this question:
Is the product of two Gaussian random variables also a Gaussian?
Texbook with proof that product of two Gaussian functions is also Gaussian
What's the densitiy of the product of two independent Gaussian random variables?
Product of two Gaussian PDFs is a Gaussain PDF, but Produt of two Gaussan Variables is not Gaussian
A random variable and a density function are two very different things.
Suppose $X$ and $Y$ are random variables and $f_X$ and $f_Y$ are their density functions.
The density function of $X+Y$ is not $f_X + f_Y$. The latter is not the density of anything because it integrates to 2!
Similarly, the density of $X*Y$ is not $f_X*f_Y$.
The density of $X+Y$ is the convolution of $f_X$ and $f_Y$. There is no way to relate the density of $X*Y$ to the densities of $X$ and $Y$ that I know of.