The Gaussian bivariate distribution is given by

(1) 
where

(2) 
and

(3) 
is the Covariance. Let and be normally and independently distributed variates with Mean 0 and
Variance 1. Then define
These new variates are normally distributed with Mean and , Variance
and Covariance

(8) 
The Covariance matrix is

(9) 
where

(10) 
The joint probability density function for and is

(11) 
However, from (4) and (5) we have

(12) 
Now, if

(13) 
then this can be inverted to give
Therefore,

(15) 
Expanding the Numerator gives
so
But

(18) 
The Denominator is



(19) 
so

(20) 
and

(21) 
Solving for and and defining

(22) 
gives
The Jacobian is
Therefore,

(26) 
and

(27) 
where

(28) 
Now, if

(29) 
then

(30) 
so
where

(35) 
The Characteristic Function is given by
where

(37) 
and

(38) 
Now let
Then

(41) 
where
Complete the Square in the inner integral



(43) 
Rearranging to bring the exponential depending on outside the inner integral, letting

(44) 
and writing

(45) 
gives
Expanding the term in braces gives
But
is Odd, so the integral over the sine term vanishes, and we are left with
Now evaluate the Gaussian Integral
to obtain the explicit form of the Characteristic Function,
Let and be two independent Gaussian variables with Means and
for ,
2. Then the variables and defined below are Gaussian bivariates with unit Variance and
CrossCorrelation Coefficient :

(51) 

(52) 
The conditional distribution is

(53) 
where
The marginal probability density is
See also BoxMuller Transformation, Gaussian Distribution, McMohan's Theorem,
Normal Distribution
References
Abramowitz, M. and Stegun, C. A. (Eds.).
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 936937, 1972.
Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGrawHill, p. 118, 1992.
© 19969 Eric W. Weisstein
19990525