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174 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

Now

y f XY 1x, y2 dx dy £ yf XY 1x, y2 dx dy§ Y
X

X
X

Therefore,

E31X 21Y 24 XY 1x, y2 dx dy Y
xyf
X
Y
X
X
X
Y
Y

xyf XY 1x, y2 dx dy E1XY2
Y
X
X Y

EXAMPLE 5-28 In Example 5-1, the random variables X and Y are the number of acceptable and suspect bits
among four bits received during a digital communication, respectively. Is the covariance
between X and Y positive or negative?
Because X and Y are the number of acceptable and suspect bits out of the four received,
X Y 4. If X is near 4, Y must be near 0. Therefore, X and Y have a negative covariance.
This can be verified from the joint probability distribution in Fig. 5-1.
There is another measure of the relationship between two random variables that is often
easier to interpret than the covariance.

Definition
The correlation between random variables X and Y, denoted as , is
XY
cov 1X, Y2 XY
(5-29)
XY
1V1X2 V1Y2 X Y

0 and
0, if the covariance between X and Y is positive, negative, or zero,
Because X Y
the correlation between X and Y is positive, negative, or zero, respectively. The following
result can be shown.

For any two random variables X and Y

1 XY 1 (5-30)

The correlation just scales the covariance by the standard deviation of each variable.
Consequently, the correlation is a dimensionless quantity that can be used to compare the
linear relationships between pairs of variables in different units.