Applied Statistics and Probability for Engineers

174 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

Now

Therefore,

EXAMPLE 5-28 In Example 5-1, the random variables Xand Yare the number of acceptable and suspect bits

among four bits received during a digital communication, respectively. Is the covariance

between Xand Ypositive or negative?

Because Xand Yare the number of acceptable and suspect bits out of the four received,

XY4. If Xis near 4, Ymust be near 0. Therefore, Xand Yhave 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.

 





(^) 


xyfXY 1 x, y 2 dx dyXYE 1 XY 2 XY

E 31 XX 21 YY 24  





(^) 


xy fXY 1 x, y 2 dx dyXYXYXY







(^) 



Xy fXY 1 x, y 2 dx dyX £





(^) 


yfXY 1 x, y 2 dx dy§XY
The correlationbetween random variables Xand Y, denoted as is
XY (5-29)
cov 1 X, Y 2
1 V 1 X 2 V 1 Y 2

(^) XY
(^) X Y
XY,
Definition
Because (^) X 0 and (^) Y 0, if the covariance between Xand Yis positive, negative, or zero,
the correlation between Xand Yis positive, negative, or zero, respectively. The following
result can be shown.
For any two random variables Xand 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.
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