Page 148 - Applied Statistics And Probability For Engineers

c04.qxd 5/10/02 5:20 PM Page 126 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files:

126 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

and
P1X 32 1 F132 e 3/1.4 0.117

Therefore,

P1X 3.5 ƒ X 32 0.035 0.117 0.30

After waiting for 3 minutes without a detection, the probability of a detection in the next 30
seconds is the same as the probability of a detection in the 30 seconds immediately after start-
ing the counter. The fact that you have waited 3 minutes without a detection does not change
the probability of a detection in the next 30 seconds.

Example 4-22 illustrates the lack of memory property of an exponential random vari-
able and a general statement of the property follows. In fact, the exponential distribution is the
only continuous distribution with this property.

Lack of
Memory For an exponential random variable X,
Property
P1X t
t 0 X t 2 P1X t 2 (4-16)
1
2
1
2
Figure 4-24 graphically illustrates the lack of memory property. The area of region A divided
by the total area under the probability density function 1A
B
C
D 12 equals
P1X t 2 . The area of region C divided by the area C
D equals P1X t
t 0 X t 2. The
2
1
2
1

lack of memory property implies that the proportion of the total area that is in A equals the
proportion of the area in C and D that is in C. The mathematical verification of the lack of
memory property is left as a mind-expanding exercise.
The lack of memory property is not that surprising when you consider the development
of a Poisson process. In that development, we assumed that an interval could be partitioned
into small intervals that were independent. These subintervals are similar to independent
Bernoulli trials that comprise a binomial process; knowledge of previous results does not af-
fect the probabilities of events in future subintervals. An exponential random variable is the
continuous analog of a geometric random variable, and they share a similar lack of memory
property.
The exponential distribution is often used in reliability studies as the model for the
time until failure of a device. For example, the lifetime of a semiconductor chip might be
modeled as an exponential random variable with a mean of 40,000 hours. The lack of

f(x)

Figure 4-24 Lack of
A
memory property of
B
an exponential C D
distribution. t 2 t 1 t + t 2 x
1