If and are jointly Gaussian, then


                           

   which is the characteristic function of a Gaussian random variable with
   mean    and       variance

   Thus the linear transformation of two Gaussian random variables is a Gaussian random variable.

   Example 4 If Z = X + Y and X and Y are independent, then
                           

   Using the property of the Fourier transform, we get
                              

which proves the result in lecture 20.