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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