Page 158 - Applied statistics and probability for engineers

136 Chapter 4/Continuous Random Variables and Probability Distributions

Because we have already been waiting for three minutes, we feel that a detection is “due.’’ That is, the probability of
a detection in the next 30 seconds should be higher than 0.3. However, for an exponential distribution, this is not true.
(
.
The requested probability can be expressed as the conditional probability that P X < 3 5u X > 3). From the dei nition
of conditional probability,
(
P X > 3)
.
. /
P X < 3 5u X > 3) = ( < X < 3 5) (
P 3
where
F 3)
P 3 ( < X < 3 5. ) = ( ) − (
F 3 5.
/
.
= ⎡ ⎣ 1 − e − 3 5 1 4 ⎤ − ⎡ ⎣ 1 − e − 3 1 4 ⎤ = .
/
.
.
0 035
⎦
⎦
and
(
P X > 3) = − ( e − / 3 1 4 = 0 117
.
F 3) =
.
1
Therefore,
(
.
.
.
.
/
P X < 3 5u X > 3) = 0 035 0 117 = 0 30
Practical Interpretation: After waiting for three 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 starting the counter. The fact that
we have waited three 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 variable,
and a general statement of the property follows. In fact, the exponential distribution is the only
continuous distribution with this property.
Lack of Memory
Property For an exponential random variable X,
P X < t 2)
(
P X < t 1 + t 2 u X > t 1) = ( (4-16)
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 A + B C + D = ) 1 equals P X < t 2).
(
The area of region C divided by the area C + D equals P X < t 1 + t 2 u X > t 1). The lack of
memory property implies that the proportion of the total area that is in A equals the propor-
tion of the area in C and D that is in C. The mathematical veriication of the lack of memory

property is left as a Mind-Expanding exercise.
The lack of memory property is not so surprising when we 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 experiment; knowledge of previous results does not affect the
probabilities of events in future subintervals. An exponential random variable is the continu-
ous analog of a geometric random variable, and it shares a similar lack of memory property.
f(x)

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