*Posted by Danny Tarlow*

A ~ N(mu_A, sigma_A^2) B ~ N(mu_B, sigma_B^2)then you define random variable C to take on the value of the sum A + B, then C will be distributed according to a Gaussian distribution:

C ~ N(mu_A + mu_B, sigma_A^2 + sigma_B^2)If instead you define random variable D to take on the value of the product A * B, then D will not be distributed normally. As an example, if A = B and mu_A = mu_B = 0 and sigma_A = sigma_B = 1, then D is distributed according to a chi-square distribution with 1 degree of freedom. The "trick" (if you want to call it that) comes from the loose wording people use when they say things like "the product of two Gaussians." In the first case, you are actually multiplying probability distributions. In the second case, you are multiplying the values of draws from probability distributions -- it's kind of subtle. Unfortunately, both interpretations are reasonable and used in practice. The first one comes up most for me, because if you have two independent beliefs about the value of a variable, then the right thing to do to combine the evidence is to multiply the distributions. The second comes up in places like multiplicative models.