Is the following density a bivariate gaussian?:

Code:

f(x,y)=[sqrt(1-p^2)/(2*pi)]*exp[-1/2*(x^2+y^2-2*p*x*y)]

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.