Is the following density a bivariate gaussian?:
where p is the correlation between x and y. It does integrate to 1 and the marginal densities for x and y are indeed normal (both x and y are distributed Normal(0,1/(1-p^2)), as you can see from completing the squares). I have a book that claims that f(x,y) is 2d gaussian, but I can't figure out what the covariance matrix would be... it isn't [[1 p] [p 1]] because f doesn't have the 1/(1-p^2) in the denominator of the exponential... help?
More general question: are all exponentials of quadratic functions gaussian? This seems to be true for the univariate case but can't confirm for multivariate.