Gaussian Bivariate Distribution

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
 (4) (5)

These new variates are normally distributed with Mean and , Variance
 (6) (7)

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
 (14)

Therefore,

 (15)

Expanding the Numerator gives
 (16)
so

 (17)
But

 (18)

The Denominator is
 (19)
so
 (20)

and

 (21)

Solving for and and defining
 (22)

gives
 (23) (24)

The Jacobian is
 (25)

Therefore,
 (26)

and

 (27)

where

 (28)

Now, if
 (29)

then
 (30)

 (31) (32)

so
 (33) (34)

where
 (35)

The Characteristic Function is given by

 (36)

where
 (37)

and
 (38)

Now let
 (39) (40)

Then

 (41)

where
 (42)

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

 (46)
Expanding the term in braces gives

 (47)
But is Odd, so the integral over the sine term vanishes, and we are left with

 (48)
Now evaluate the Gaussian Integral

 (49)

to obtain the explicit form of the Characteristic Function,

 (50)

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 Cross-Correlation Coefficient :

 (51)

 (52)

The conditional distribution is
 (53)

where
 (54) (55)

The marginal probability density is
 (56)

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 936-937, 1972.

Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 118, 1992.