Product of gaussian functions

1. What is the product of two Gaussian distributions?
2. Is the product of Gaussians Gaussian?
3. Is the product of normal distributions normal?
4. What is the distribution of product of two Gaussian random variables?

What is the product of two Gaussian distributions?

The product of two Gaussian PDFs is proportional to a Gaussian PDF with a mean that is half the coefficient of x in Eq. 5 and a standard deviation that is the square root of half of the denominator i.e. as, due to the presence of the scaling factor, it will not have the correct normalisation.

Is the product of Gaussians Gaussian?

Since the product of two Gaussians is a Gaussian, the posterior probability is Gaussian.

Is the product of normal distributions normal?

It is clear the product of normal distributed variables is not normal distributed.

What is the distribution of product of two Gaussian random variables?

A random variable product of two independent gaussian random variables is not gaussian except in some degenerate cases such as one random variable in the product being constant. A product of two gaussian PDFs is proportional to a gaussian PDF, always, trivially.