Page 105 - Fundamentals of Probability and Statistics for Engineers

88 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:

Z 1 Z 1 Z 1 Z 1
10 ˆ EfXgˆ xf XY x; y†dxdy ˆ x f XY x; y†dydx
1 1 1 1
1
Z
ˆ xf x†dx;
X
1
where f (x) is the marginal density function of X. We thus see that the result is
X
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

2
9
2
20 ˆ 20 2 10 ˆ 20 m =
X
X
or 4:22†
2
02 ˆ 02 2 ˆ 02 m 2 ;
01 Y Y
4.3.1 COVARIANCE AND CORRELATION COEFFICIENT
The first and simplest joint moment of X and Y that gives some measure of
their interdependence is 11 ˆ Ef X m X ) Y m Y )g. 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

11 ˆ 11 10 01 ˆ 11 m X m Y : 4:23†
Proof of Property 4.1: Property 4.1 is obtained by expanding
X m X ) Y m Y ) and then taking the expectation of each term. We have:

11 ˆ Ef X m X † Y m Y †g ˆ EfXY m Y X m X Y ‡ m X m Y g
ˆ EfXYg m Y EfXg m X EfYg‡ m X m Y
ˆ 11 10 01 10 01 ‡ 10 01
ˆ 11 10 01 :

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

11 11
ˆ 1=2 ˆ : 4:24†
20 02 † X Y

Then, j j 1.

TLFeBOOK