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∼N(μ1,σ21), Y∼N(μ2,σ22) then XY∼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(μ1,σ21)×N(μ2,σ22)=N(σ21μ2+σ22μ1σ21+σ22,11σ21+1σ22)
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)
The product of the PDFs of two random variables X and Y will give the joint distribution of the vector-valued random variable (X,Y) in the case that X and Y are independent. Therefore, if X and Y are normally distributed independent random variables, the product of their PDFs is bivariate normal with zero correlation.
On the other hand, even in the case that X and Y are IID standard normal random variables, their product is not itself normal, as the links you provide show. The product of X and Y is a scalar-valued random variable, not a vector-valued one as in the above case.