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5-3 TWO CONTINUOUS RANDOM VARIABLES 161

y

2000

0
0 2000 x
Figure 5-10 Region of integration
for the probability that Y 2000 is
darkly shaded and it is partitioned
into two regions with x 2000 and
x
2000.

The first integral is

2000 0.002y
6 2000
e 0.001x 6 10 4 0.001x
6
6 10 ° ` ¢ e dx e e dx
0.002 2000 0.002
0 0
6 10 6 1 e 2
e 4 a b 0.0475
0.002 0.001
The second integral is

0.002y
6
e 0.001x 6 10 0.003x
6
6 10 ° ` ¢ e dx e dx
0.002 x 0.002
2000 2000
6 10 6 e 6
a b 0.0025
0.002 0.003

Therefore,

P 1Y
20002 0.0475 0.0025 0.05.

Alternatively, the probability can be calculated from the marginal probability distribution of Y
as follows. For y
0
y y
6 0.002y
f 1y2 6 10 e dx 6 10 e e 0.001x dx
6 0.001x 0.002y
Y

0 0
e 0.001x y 6 0.002y 1 e 0.001y
6 0.002y
6 10 e a ` b 6 10 e a b
0.001 0 0.001
6 10 3 0.002y 11 e 0.001y 2 for y
0
e