Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 271

Exercise 4.11 The joint density function of the normal random variables X 1 and X 2
is given as:

h i

2

x 2
225

ðx 1 5 Þ

ð x 1 5 Þ x 2
150

e 1 : 5

1
300 p

þ

100

f X 1 X 2 ðx 1 ; x 2 Þ¼ p : (4.189)

0 : 75

Find the correlation coefficient and the mean value of the variable

X ¼ aX 1 þ bX 2 : (4.190)

Answer Comparing the general expression ( 4.124 ) and ( 4.189 ), we have:

5 ¼ 2 ð 1 r 2 1 : Þ and r ¼ 0 : 5 : (4.191)

The mean values of the variables X 1 and X 2 are 5 and zero, respectively. From
( 4.190 ), we have:

EfXg¼afX 1 gþbEfX 2 g¼ 5 a þ 0 ¼ 5 a: (4.192)

Exercise 4.12 Three independent normal random variables have the following
parameters:

s i ¼ 4 m i ¼ 0 ; ; i ¼ 1 ; 2 ; 3 : (4.193)

Find the coefficient of correlation for variables Y 1 and Y 2 if

Y 1 ¼ X 1 þ X 2 ; Y 2 ¼ X 2 X 3 : (4.194)

Answer The random variables Y 1 and Y 2 are the sums of normal random variables
and consequently, they are also normal with the parameters,

EfY 1 g¼EfX 1 gþEfX 2 g¼ 0 ;

EfY 2 g¼EfX 2 gEfX 3 g¼ 0 ;

s Y 1 ¼ s 1 þ s 2 ¼ 8

;

s 2

Y 2 ¼ s 2 þ s 3 ¼ 8

: (4.195)

Knowing that the variables X i are independent, it follows that

X i X j ¼ X i X j ¼ 0 ; i 6¼ j: (4.196)

The correlation coefficient is:

X 2
8 ¼

s 2
8 ¼

Y 1 Y 2 Y 1 Y 2
s Y 1 s Y 2

¼ ð X 1 þ X 2 Þð X 2 X 3 Þ
8

4
8 ¼ 0

r ¼ ¼ : 5 : (4.197)