563489578934

B–9 Multivariate Statistics 711
Proof. First, show that this result is correct if N = 2 and L = 1:
q
q
f(x 1 , x 2 ) dx 2 = f(x 1 )f(x 2 ƒ x 1 ) dx 2
L-q
L -q
q
= f(x 1 ) f(x 2 ƒ x 1 ) dx 2 = f(x 1 )
L
(B–90)
since the area under f(x 2 x 1 ) is unity. This procedure is readily extended to prove the
Lth-dimensional case of Eq. (B–89).
Bivariate Statistics
Bivariate (or joint) distributions are the N = 2-dimensional case. In this section, the definitions
from the previous section will be used to evaluate two-dimensional moments. As shown in
Chapter 6, bivariate statistics have some very important applications to electrical engineering
problems, and some additional definitions need to be studied.
-q
DEFINITION.
DEFINITION.
The correlation (or joint mean) of x 1 and x 2 is
m 12 = x 1 x 2 =
L
q
q
-q L-q
x 1 x 2 f(x 1 , x 2 ) dx 1 dx 2
Two random variables x 1 and x 2 are said to be uncorrelated if
m 12 = x 1 x 2 = x 1 x 2 = m 1 m 2
(B–91)
(B–92)
If x 1 and x 2 are independent, it follows that they are also uncorrelated, but the converse is not generally
true. However, as we will see, the converse is true for bivariate Gaussian random variables.
DEFINITION. Two random variables are said to be orthogonal if
m 12 = x 1 x 2 K 0
(B–93)
Note the similarity of the definition of orthogonal random variables to that of orthogonal
functions given by Eq. (2–73).
DEFINITION.
The covariance is
u 11 = (x 1 - m 1 )(x 2 - m 2 )
=
L
q
q
-q L-q
(x 1 - m 1 )(x 2 - m 2 )f(x 1 , x 2 ) dx 1 dx 2
(B–94)
It should be clear that if x 1 and x 2 are independent, the covariance is zero (and x 1 and x 2 are
uncorrelated). The converse is not generally true, but it is true for the case of bivariate
Gaussian random variables.
DEFINITION.
The correlation coefficient is
r = u 11
s 1 s 2
=
(x 1 - m 1 )(x 2 - m 2 )
3(x 1 - m 1 ) 2 3(x 2 - m 2 ) 2
(B–95)