2.2. CONTINUOUS DENSITY FUNCTIONS
67
1  z
1  z
1  z
1  z
E
Figure 2.19: Calculation of
F
Z
.
20
40
60
80
100
120
0.005
0.01
0.015
0.02
0.025
0.03
f (t) = (1/30) e
 (1/30) t
Figure 2.20: Exponential density with
λ
= 1
/
30.
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CHAPTER 2. CONTINUOUS PROBABILITY DENSITIES
0
20
40
60
80
100
0
0.005
0.01
0.015
0.02
0.025
0.03
Figure 2.21: Residual lifespan of a hard drive.
is distributed according to the exponential density. We will assume that this model
applies here, with
λ
= 1
/
30.
Now suppose that we have been operating our computer for 15 months.
We
assume that the original hard drive is still running. We ask how long we should
expect the hard drive to continue to run.
One could reasonably expect that the
hard drive will run, on the average, another 15 months.
(One might also guess
that it will run more than 15 months, since the fact that it has already run for 15
months implies that we don’t have a lemon.) The time which we have to wait is
a new random variable, which we will call
Y
.
Obviously,
Y
=
X

15.
We can
write a computer program to produce a sequence of simulated
Y
values. To do this,
we ±rst produce a sequence of
X
’s, and discard those values which are less than
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 Spring '09
 scf
 Probability theory, probability density function, hard drive, exponential density

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