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Random Variables and Probability Distributions 59

y

60

y – x =10

R

10
x – y =10

x
0 10 60

Figure 3.16 Region R in Example 3.7

Note that, for a more complicated jpdf, one needs to carry out the volume
RR
integral f 9x, y)dxdy for volume calculations.
R XY
As an exercise, let us determine the joint probability distribution function
and the marginal density functions of random variables X and Y defined in
Example 3.7.
The JPDF of X and Y is obtained from Equation (3.25). It is clear that

0; for x; y† < 0; 0†;
F XY x; y†ˆ
1; for x; y† > 60; 60†:
Within the region (0, 0) (x, y) (60, 60), we have

x
y
Z Z 1 xy
F XY x; y†ˆ dxdy ˆ :
0 0 3600 3600
For marginal density functions, Equations (3.28) and (3.29) give us

8
Z 60
1 1
>
< dy ˆ ; for 0 x 60;
f x†ˆ 0 3600 60
X
>
:
0; elsewhere:

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