Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 261

This expression can be rewritten in the following form:

j P i P j

e

1
2 C X

ðx i m i Þðx j m j Þ D i;j

1 j

f X ðXÞ¼ p

C X

i; j ¼ 1 ; ... ; N; (4.136)

;

N= 2

ð 2 pÞ j j

where D i,j is the cofactor of the matrix C X .
Example 4.6.2 In this example, we demonstrate how the PDF of two jointly
normal random variables ( 4.124 ) can be derived from the general expression
( 4.136 ).

Solution The covariance matrix C X is

s 1 C 1 ; 2

C X ¼ ; (4.137)

s 2

C 2 ; 1

where the covariances are:

C 1 ; 2 ¼ C 2 ; 1 ¼ EfX 1 m 1 ÞðX 2 m 2 Þg ¼ r 1 ; 2 s 1 s 2 : (4.138)

The determinant | C X | is given as:

¼ s 1 s 2 C 1 ; 2 :

s 1

C 1 ; 2

j C X j ¼ (4.139)

s 2 C 2 ; 1

Using ( 4.138 ), the determinant ( 4.139 ) is expressed as:

j ¼ s 1 s 2 ð 1 r 1 ; 2 Þ: j C X (4.140)

Its cofactors are:

D 1 ; 1 ¼ s 2 ; D 2 ; 2 ¼ s 1 ; D 1 ; 2 ¼ D 2 ; 1 ¼C 1 ; 2 ¼C 2 ; 1 ¼r 1 ; 2 s 1 s 2 : (4.141)

Placing ( 4.138 )-( 4.141 ) into ( 4.136 ), we get:

h i

1

2 2

ðx 1 m 1 Þ s 2

s 1 s 2 þ

ðx 2 m 2 Þ s 1

s 1 s 2 2 r 1 ; 2 s 1 s 2 ðx 1 m 1 Þðx 2 m 2 Þ

1 r 1 ; 2

1
2 ð 1 r 1 ; 2 Þ

f X 1 ;X 2 ðx 1 ;x 2 Þ¼ q exp

s 1 s 2

2 ps 1 s 2

h i

1

2
s 1 þ ð x 2 m 2 Þ 2

ðx 1 m 1 Þ

s 2 2 r 1 ; 2 ðx 1 m 1 Þðx 2 m 2 Þ

q

1 r 1 ; 2

1
2 ð 1 r 1 ; 2 Þ

¼ exp ;

s 1 s 2

2 ps 1 s 2

(4.142)

which is the same expression as in ( 4.124 ).