Applied Statistics and Probability for Engineers

172 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

That is, E[h(X,Y)] can be thought of as the weighted average of h(x,y) for each point in the

range of (X,Y). The value of E[h(X,Y)] represents the average value of h(X,Y) that is expected

in a long sequence of repeated trials of the random experiment.

EXAMPLE 5-27 For the joint probability distribution of the two random variables in Fig. 5-12, calculate

The result is obtained by multiplying xXtimes yY, times fXY(x,y) for each point

in the range of (X, Y). First, Xand Yare determined from Equation 5-3 as

and

Therefore,

The covariance is defined for both continuous and discrete random variables by the same formula.

 13 2.4 212 2.0 2 0.2 13 2.4 213 2.0 2 0.30.2

 11 2.4 212 2.0 2 0.2 13 2.4 211 2.0 2 0.2

E 31 XX 21 YY 24  11 2.4 211 2.0 2 0.1

Y 1 0.3 2 0.4 3 0.32.0

X 1 0.3 3 0.72.4

E 31 XX 21 YY 24.

E 3 h 1 X, Y 24 μ (5-27)

b

R

h^1 x, y^2 fXY^1 x, y^2 X, Y^ discrete



R

h 1 x, y 2 fXY 1 x, y 2 dx dy X, Y continuous

Definition

The covariancebetween the random variables Xand Y, denoted as cov(X,Y) or is

XYE 31 XX 21 YY 24 E 1 XY 2 XY (5-28)

XY,

Definition

Figure 5-12 Joint

distribution ofXandY

for Example 5-27.^1

1

23

3

y

2

x

0.1

0.2

0.2

0.2

0.3

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