Information Theory and Principal Component Analysis - Biomedical Signal Analysis: Contemporary Methods and Applications - page 79

0.25

0.2

0.15

0.1

0.05

0

2

1 2

1

0

0

1

1

2

2

Figure 3.1
Smoothed density of a two-dimensional random vector, uniform in [ − 1 , 1] 2 uniform
distribution.

Figure 3.1 shows a plot of the density of a uniform two-dimensional
random vector.

Gaussian Density
Definition 3.12:

n is said to be Gaussian
if its density function p X exists and is of the form

A random vector X :Ω → R

(2 π ) n det C exp

μ )

1 1

2 ( x

μ ) C −1 ( x p X ( x )= − − −

n and C is symmetric and positive-definite.

where μ ∈ R

If X is Gaussian with μ and C ,asabove,then E ( X )= μ and
Cov( X )= C . A white Gaussian random vector is called normal. In
the one-dimensional case a Gaussian random variable with mean μ

∈ R

and variance σ 2 > 0hasthedensity

(2 π ) σ exp μ ) 2 .

1 1

2 σ 2 ( x

p X ( x )= − −