Applied Statistics and Probability for Engineers

5-5 COVARIANCE AND CORRELATION 173

If the points in the joint probability distribution of Xand Ythat receive positive probabil-

ity tend to fall along a line of positive (or negative) slope, XYis positive (or negative). If the

points tend to fall along a line of positive slope, Xtends to be greater than Xwhen Yis greater

than Y. Therefore, the product of the two terms xXand yYtends to be positive.

However, if the points tend to fall along a line of negative slope, xXtends to be positive

when yYis negative, and vice versa. Therefore, the product of xXand yYtends

to be negative. In this sense, the covariance between Xand Ydescribes 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 relationshipbetween 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,

 



(^) 

3 xyX yxYXY 4 fXY 1 x, y 2 dx dy

E 31 YY 21 XX 24  



(^) 

1 xX 21 yY 2 fXY 1 x, y 2 dx dy
x
y
x
y
x
y
x
y
(a) Positive covariance (b) Zero covariance
(c) Negative covariance (d) Zero covariance
All points are of
equal probability
Figure 5-13 Joint probability distributions and the sign of covariance betweenXandY.
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