Gaussian univariate functions: two useful features

Two interesting/useful features of Gaussian functions of a single variable:

• the product of two Gaussian functions is still a Gaussian function
• the convolution product of two Gaussian functions is still a Gaussian function with mean equal to the sum of the means and variance equal to the sum of the variances.

Proof of both properties .... still to come ... it's not so immediate ... but it's not too difficult !