Random Vectors
Vectors :: Covariance :: Functions :: Application :: Markov Model
Characteristic functions
As before, this is just an
dimensional Fourier transform.
Properties of Gaussian random vectors:
 .
 independent if and only if is a diagonal matrix.

If
, then
is also Gaussian,
Linear functions of Gaussians are Gaussians.
Said another way: Family of Gaussians closed under affine transformations.
Suppose is positive definite . Then it can be factored as
Suppose , with p.d. Let . Then is normal with and .
This process of diagonalizing the covariance matrix is called whitening. We say that uncorrelated i.i.d. components are white .

If
(i.e., p.d.) then
is a continuous r.v. with

Important:
Suppose
with
. Partition
,
Consider conditioned on :
Discuss implications. Draw pictures.
Note: For a Gaussian vector, the conditional density is Gaussian.