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712
Probability and Random Variables
Appendix B
This is also called the normalized covariance. The correlation coefficient is always
within the range
(B–96)
For example, suppose that x 1 = x 2 ; then r =+1. If x 1 =-x 2 , then r =-1; and if x 1 and x 2 are
independent, r = 0. Thus the correlation coefficient tells us, on the average, how likely a
value of x 1 is to being proportional to the value for x 2 . This subject is discussed in more detail
in Chapter 6, where these results are extended to include random processes (time functions).
There the dependence of the value of a waveform at one time is compared with the value of
the waveform that occurs at another time. This will bring the concept of frequency response
into the problem.
Gaussian Bivariate Distribution
A good example of a joint (N = 2) distribution that is of great importance is the bivariate
Gaussian distribution. The bivariate Gaussian PDF is
f(x 1 , x 2 ) =
(B–97)
where is the variance of x 1 , is the variance of x 2 , m 1 is the mean of x 1 , and m 2 is the mean
of x 2 . Examining Eq. (B–97), we see that if r = 0, f(x 1 , x 2 ) = f(x 1 ) f(x 2 ), where f(x 1 ) and f(x 2 ) are
the one-dimensional PDFs of x 1 and x 2 . Thus, if bivariate Gaussian random variables are uncorrelated
(which implies that r = 0), they are independent.
A sketch of the bivariate (two-dimensional) Gaussian PDF is shown in Fig. B–12.
s 1
2
-1 … r … +1
1
1
c (x 1 - m 1 ) 2
2(1 - r 2 2
2ps
) s 1 s 2 21 - r e - 2 1
s 2
2
Multivariate Functional Transformation
- 2r (x 1 - m 1 )(x 2 - m 2 )
s 1 s 2
+ (x 2 - m 2 ) 2
Section B–8 will now be generalized for the multivariate case. Referring to Fig. B–13, we will
obtain the PDF for y, denoted by f y (y) in terms of the PDF for x, denoted by f x (x).
s 2
2
d
f (x 1 , x 2 ) m 2
1
x 2
2ps 1 s 2 1 r 2
m 1
x 1
Figure B–12
Bivariate Gaussian PDF.