Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 259

The expression ( 4.124 ) shows some interesting properties of normal random
variables:

P.1 The joint density function ( 4.124 ) is completely determined by first two moments
of marginal variables: the mean value and the mean squared value (note that the
variance and the correlation coefficient are given by ( 2.338 ) and ( 3.130 ), respectively).
P.2 If the random variables are jointly normal, then the marginal variables are also
normal.

P.3 If normal random variables X 1 and X 2 are not correlated (i.e., the coefficient of
correlation is equal to zero), then they are also independent,

r 1 ; 2 ¼ 0 : (4.125)

From ( 4.124 ) and ( 4.125 ), we arrive at:

h i

1
2

2
s 1 þ ðx 2 m 2 Þ 2

ð x 1 m 1 Þ

1

2 ps 1 s 2 e

s 2

f X 1 ;X 2 ðx 1 ; x 2 Þ¼

2 2

e ð x 1 m 1 Þ e ð x 2 m 2 Þ

1
2 p

1
2 p

2 s 1 2 s 2

p p

¼ ¼ f X 1 ðx 1 Þf X 2 ðx 2 Þ:

s 1 s 2

(4.126)

From ( 4.126 ), we see that the joint density function is equal to the product of the
corresponding densities indicating that the random variables X 1 and X 2 are

independent.
Therefore, if the normal random variables are not correlated, they are also
independent. This is exceptional and holds only for normal random variables.
Previously, in Chap. 3 , it was stated that the condition of noncorrelation is weaker
than that of independence and that, in general, if random variables are uncorrelated
they can be either dependent or independent.

Example 4.6.1 The joint density of the random variables X 1 and X 2 is given as:

" #

x 1

2

1
2 4 þ ð x 2 2 Þ

1
12 p

9

f X 1 ;X 2 ðx 1 ; x 2 Þ¼ e : (4.127)

Find the PDF of the random variable

X ¼ X 1 X 2 : (4.128)

Solution From ( 4.127 ), we note that X 1 and X 2 are independent normal random
variables with the corresponding mean values and variances,

s 1 ¼ 4 s 2 ¼ 9

m 1 ¼ 0 ; m 2 ¼ 2 ; ; : (4.129)