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5-5 COVARIANCE AND CORRELATION 173

If the points in the joint probability distribution of X and Y that receive positive probabil-
ity tend to fall along a line of positive (or negative) slope, XY is positive (or negative). If the
points tend to fall along a line of positive slope, X tends to be greater than when Y is greater
X
than . Therefore, the product of the two terms x and y tends to be positive.
Y
X
Y
However, if the points tend to fall along a line of negative slope, x X tends to be positive
when y Y is negative, and vice versa. Therefore, the product of x X and y Y tends
to be negative. In this sense, the covariance between X and Y describes the variation between
the two random variables. Figure 5-13 shows examples of pairs of random variables with
positive, negative, and zero covariance.
Covariance is a measure of linear relationship between the random variables. If the re-
lationship between the random variables is nonlinear, the covariance might not be sensitive to
the relationship. This is illustrated in Fig. 5-13(d). The only points with nonzero probability
are the points on the circle. There is an identifiable relationship between the variables. Still,
the covariance is zero.
The equality of the two expressions for covariance in Equation 5-28 is shown for contin-
uous random variables as follows. By writing the expectations as integrals,

E31Y 21X 24 1x 21y 2 f 1x, y2 dx dy

X
Y XY
Y
X

3xy y x 4 f XY 1x, y2 dx dy
Y
X Y
X

y

y

x x
(a) Positive covariance (b) Zero covariance
y y
All points are of
equal probability

x

x
(c) Negative covariance (d) Zero covariance
Figure 5-13 Joint probability distributions and the sign of covariance between X and Y.