Product of two normal distributions

1. What is the distribution of the product of two normal distributions?
2. Is the product of two normal distributions normal?
3. What happens when you combine two normal distributions?
4. Is the product of two Gaussian random variables also a Gaussian?

What is the distribution of the product of two normal distributions?

The product of two normal PDFs is proportional to a normal PDF. This is well known in Bayesian statistics because a normal likelihood times a normal prior gives a normal posterior. But because Bayesian applications don't usually need to know the proportionality constant, it's a little hard to find.

Is the product of two normal distributions normal?

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

What happens when you combine two normal distributions?

Independent random variables

This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations).

Is the product of two Gaussian random variables also a Gaussian?

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.