is a constant vector and 
are positive definite symmetric matrices ( 
Positivity), then
where 
. 
is the joint probability density of a normal distribution of the variables 
. The expectation values of the variables are 
. Their covariance matrix is C.
Lines of constant probability density 
in the 
-plane correspond to constant values of the exponent.
If one chooses this value to be 1 one obtains the condition:
This is the equation of an ellipse, called the covariance ellipse or error ellipse of the bivariate normal distribution. The error ellipse is centred at the point and has as major and minor axes the (uncorrelated)
largest and smallest standard deviation that can be found under any angle. For the size and orientation of the error ellipse see below. The probability of observing of a point (X1,X2) inside the error ellipse is 
.
Note that distances from the point 
to the covariance ellipse do not describe the standard deviation along directions other than along the principal axes. This standard deviation is obtained by error propagation,
and is greater than or equal to the distance to the error ellipse,
the difference being explained by the non-uniform distribution of the second (angular) variable (see figure).
)
the principal axes of the error ellipse are parallel to the coordinate x1, x2 axes, and the principal semi-diameters of the ellipse p1,p2 are equal to 
.
For 
one has the relations 
where a is the angle between the x1 axis and the semi-diameter of length p1. Note that a is determined up to multiples of 
, i.e. for both semi-diameters of both principal axes.
For a rotation angle 
one obtains two variances 
at orthogonal axes, and a corresponding correlation coefficient 
; for any , the relation 
holds.
marginal distributions of the bivariate normal are normal distributions of one variable:
Only for uncorrelated variables, i.e. for , is the bivariate normal the product of two univariate Gaussians
Unbiased estimators for the parameters a1,a2, and the elements Cij are constructed from a sample (X1k X2k), 
as follows:
Estimator of ai:
Estimator of Cij:
