Example 1 Suppose X and Y are two jointly-Gaussian 0-mean random variables with variances of 1 and 4 respectively and a covariance of 1. Find the joint PDF

We have

Joint Characteristic Functions of Two Random Variables

   The joint characteristic function of two random variables X and Y is defined by
                                 

   If and are jointly continuous random variables, then
                              
   Note that is same as the two-dimensional Fourier transform with the basis function instead of

   is related to the joint characteristic function by the Fourier inversion formula