Applied Statistics and Probability for Engineers

5-5 COVARIANCE AND CORRELATION 175

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, is near1 (or1). If

equals1 or1, it can be shown that the points in the joint probability distribution that

receive positive probability fall exactly along a straight line. Two random variables with

nonzero correlation are said to be correlated.Similar to covariance, the correlation is a meas-

ure of the linear relationshipbetween random variables.

EXAMPLE 5-29 For the discrete random variables Xand Ywith the joint distribution shown in Fig. 5-14,

determine XYand XY.

The calculations for E(XY), E(X), and V(X) are as follows.

Because the marginal probability distribution of Yis the same as for X, E(Y)1.8 and

V(Y)1.36. Consequently,

Furthermore,

EXAMPLE 5-30 Suppose that the random variable Xhas the following distribution: P(X1)0.2,

P(X2)0.6, P(X3)0.2. Let Y 2 X5. That is, P(Y7)0.2, P(Y9)0.6,

P(Y11)0.2. Determine the correlation between Xand Y. Refer to Fig. 5-15.

Because Xand Yare linearly related, 1. This can be verified by direct calculations:

Try it.

For independent random variables, we do not expect any relationship in their joint prob-

ability distribution. The following result is left as an exercise.

XY

XY

XY

1.26

11 1.36 211 1.36 2

0.926

XYE 1 XY 2 E 1 X 2 E 1 Y 2 4.5 1 1.8 21 1.8 2 1.26

 13 1.8 22 0.41.36

V 1 X 2  10 1.8 22 0.2 11 1.8 22 0.2 12 1.8 22 0.2

E 1 X 2  0 0.2 1 0.2 2 0.2 3 0.41.8

 22 0.1 33 0.44.5

E 1 XY 2  00 0.2 11 0.1 12 0.1 21 0.1

XY XY

Figure 5-14 Joint distribution for

Example 5-29.

Figure 5-15 Joint distribution for

Example 5-30.

1

1

23

3

y

2

x

0.1

0.1

0.1

0.1

0.4

0.2

0

0 1

7

23

11

y

9

x

0.2

0.6

0.2

ρ= 1

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