Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 181

ð

2

1
2 x 4 d x ¼ 16

X 4 ¼ = 5 : (3.148)

0

Placing ( 3.148 ) and ( 3.142 ) into ( 3.147 ), we can obtain the variance of the
random variable Y :

2

s Y ¼ 16

= 5 ð 4 = 3 Þ ¼ 1 : 422 : (3.149)

The standard deviations are obtained from ( 3.146 ) and ( 3.149 ):

q

s X

p

1 s X ¼ ¼ = 3 ¼ 0 : 5774 ; (3.150)

q

s Y

p

1 s Y ¼ ¼ : 422 ¼ 1 : 1925 : (3.151)

Using values for the covariance ( 3.145 ) and standard deviations ( 3.150 ) and
( 3.151 ), we calculate the coefficient of correlation using ( 3.130 ):

C XY
s X s Y ¼

=

2 3

r XY ¼

1925 ¼ 0 : 9682 :

(3.152)

0 : 5774 1 :

As opposed to case (a), in case (b), the variables are dependent and correlated.
Based on the previous discussion, we can conclude that the dependence is a
stronger condition than correlation. That is, if the variables are independent, they
are also uncorrelated. However, if the random variables are dependent they can be
either correlated or uncorrelated , as summarized in Fig. 3.12 .

INDEPENDENT R.V.
f XY =f X f Y

DEPENDENT R.V.

UNCORRELATED
C XY =0
XY =0

UNCORRELATED
C XY =0
XY =0

CORRELATED
C XY 0
XY 0

Positive
Correlation

Negative
Correlation

r XY > 0 r XY < 0

Fig. 3.12 Correlation and dependency between r.v.