A.23 Bivariate normal distribution

The ordered pair (X, Y)^T of observations has a bivariate normal distribution, denoted

if it has joint density

where

The means and variances are

and X and Y have correlation coefficient and covariance

Most properties follow from those of the ordinary, univariate and normal distribution. One point worth noting (which is clear from the form of the joint density function) is that if X and Y have a bivariate normal distribution, then they are independent if they are uncorrelated (a result which is not true in general).