Applied Statistics and Probability for Engineers

126 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

and

Therefore,

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 propertyof 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.

P 1 X3.5ƒX 32 0.035 0.1170.30

P 1 X 32  1 F 132 e^3 /1.40.117

For an exponential random variable X,

P 1 Xt 1

 t 20 Xt 12 P 1 Xt 22 (4-16)

Lack of

Memory

Property

Figure 4-24 graphically illustrates the lack of memory property. The area of region Adivided

by the total area under the probability density function equals

. The area of region Cdivided by the area equals The
lack of memory property implies that the proportion of the total area that is in Aequals the
proportion of the area in Cand Dthat 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

P 1 Xt 22 C D P 1 Xt 1

 t 20 X t 12.

1 A B C D 12

Figure 4-24 Lack of

memory property of

an exponential

distribution. t^2 x

C D

B

A

t 1 t 1 + t 2

f (x)

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