The
exponential random variable can be used to describe the life time of a machine,
industrial
product
and Human being. Also, it can be used to describe the waiting time of a
customer for
some
service.
Exponential Probability Density Function:
A
random variable X taking values in 
has the
exponential probability density function
if
,
where
The
graph of is
Properties of Exponential Density Function:
Let
X be the random variable with the exponential density function and
the
parameter 
. Then
,
for
any 
.
and
.
Note: 
is called the survival function.
Example 9:
Let X represent the life time of a washing machine. Suppose the average
lifetime for this type
of washing machine is 15
years. What is the probability that this washing machine can be used
for less than 6 years?
Also, what is the probability that this washing machine can be used for
more
than 18 years?
[solution:]
X has the exponential
density function with 
Then,
Thus,
for this washing machine, it is about 30% chance that it can be used for quite
a long time
or a
short time.
Relationship Between Poisson and Exponential Random
Variable:
Let Y be a Poisson random variable representing the number of
occurrences in an time interval
of length t with the
probability distribution
where 
is the mean
number of occurrences in this time interval.
Then, if X represent
the time of one
occurrence, X has the exponential
density function with mean
(t).
The
intuition of the above result is as follows. Suppose the time interval is [0,1]
(in hour)
and 
. Then, on the average, there are 4 occurrences during 1 hour
period. Thus,
the
mean time for one occurrence is 
(hour). The
number of occurrences can
be
described by a Poisson random variable (discrete) with mean 4 while the time of
one
occurrence
can be described by an exponential random variable (continuous) with mean 
.
Example 10:
Suppose the average number
of car accidents on the highway in two days is 8. What is the
probability of no accident
for more than 3 days?
[solutions:]
The
average number of car accidents on the highway in one day is 
. Thus, the mean time
of
one occurrence is 
.
Let
Y be the Poisson random variable with mean 4 representing the number of car
accidents in
one
day while X be the exponential random variable with mean representing the
time
of
one accident occurrence. Thus,
JavaStatSoft:
Probability
density:
Statistics
-> Probability -> Probability Functions -> Probability Density
Function (PDF)
Probability
density:
Statistics
-> Probability -> Probability Functions -> Cumulative Distribution Function
(CDF)
Percentile
(or Quantile):
Statistics
-> Probability -> Probability Functions -> Percentile Function