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. I needed to calculate this constant, so I’m recording the result here for my future reference and for anyone else who might find it useful.
Denote the normal PDF by
Then the product of two normal PDFs is given by the equation
Note that the product of two normal random variables is not normal, but the product of their PDFs is proportional to the PDF of another normal.