Fundamentals of Probability and Statistics for Engineers

here, the means of X and Y are, respectively, 10 and 01. Using Equation
(4.19), for example, we obtain:

where fX(x) is the marginal density function of X. We thus see that the result is
identical to that in the single-random-variable case.
This observation is, of course, also true for the individual variances. They are,
respectively, 20 and 02 , and can be found from Equation (4.21) with appropriate
substitutions for n and m. As in the single-random-variable case, we also have

or

4.3.1 Covariance and Correlation Coefficient

The fir st and simplest joint moment of X and Y that gives so me measure of
their interdependence is It is called the covar-
iance of X and Y. Let us first note some of its properties.
Property 4.1: the covariance is related to nm by

Proof of Property 4.1: Property 4.1 is obtained by expanding

and then taking the expectation of each term. We have:

P roperty 4. 2: let the correlation coefficient of X and Y be defined by

88 Fundamentals of Probability and Statistics for Engineers

10 ˆEfXgˆ

Z 1

1

Z 1

1

xfXY…x;y†dxdyˆ

Z 1

1

x

Z 1

1

fXY…x;y†dydx

ˆ

Z 1

1

xfX…x†dx;

 

 20 ˆ 20  210 ^2 Xˆ 20 m^2 X

 02 ˆ 02  201 ^2 Yˆ 02 m^2 Y

9

=

;

… 4 : 22 †

 11 ˆEfXmX)YmY)g.

 11 ˆ 11  1001 ˆ 11 mXmY: … 4 : 23 †

XmX)YmY)

 11 ˆEf…XmX†…YmY†gˆEfXYmYXmXY‡mXmYg

ˆEfXYgmYEfXgmXEfYg‡mXmY

ˆ 11  1001  1001 ‡ (^1001)
ˆ 11  1001 :

ˆ

 11

… 20  02 †^1 =^2

ˆ

 11

XY

: … 4 : 24 †

Then,jj1.