Download Statistics Chapter 7

EXPONENTIAL DISTRIBUTION
THE PROBABILITY DENSITY FUNCTION
• If a random variable X is exponentially distributed with
parameter  then its probability density function is given by
f x   e  x
x0
• Mean,  = standard
2.0
deviation,  = 1/
• The probability P(Xa) is
obtained as follows:
P X  a   e  a
• If mean,  is given, find
the parameter  first
0.0
(see Example 2):  =1/
Exponential
distribution
with =2
0
3
1
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EXPONENTIAL DISTRIBUTION
Example 1: Let X be an exponential random variable with
=2. Find the following:
1. P X  0.50 
2. P X  2 
2.0
3. P0.50  X  2 
Exponential
distribution
with =2
4. P X  2 
0.0
0
3
2
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EXPONENTIAL DISTRIBUTION
Example 2.1: The length of life of a certain type of electronic
tube is exponentially distributed with a mean life of 500
hours. Find the probability that a tube will last more than
800 hours.
0.3
Exponential
distribution
with =1/5
0.0
0
10
3
20
EXPONENTIAL DISTRIBUTION
Example 2.2: The length of life of a certain type of electronic
tube is exponentially distributed with a mean life of 500
hours. Find the probability that a tube will fail within the
first 200 hours.
0.3
Exponential
distribution
with =1/5
0.0
0
10
4
20
EXPONENTIAL DISTRIBUTION
Example 2.3: The length of life of a certain type of electronic
tube is exponentially distributed with a mean life of 500
hours. Find the probability that the length of life of a tube
will be between 400 and 700 hours.
0.3
Exponential
distribution
with =1/5
0.0
0
10
5
20
EXPONENTIAL DISTRIBUTION
USING EXCEL
Excel function EXPONDIST(a,,TRUE) provides the
probability P(Xa). For example, EXPONDIST(200,1/500,
TRUE) = 0.3297
0.3
Area=0.3297
Exponential
distribution
with =1/5
0.0
0
2
10
6
20
NOTE
• Application:
– Uniform distribution
• Used to generate other distributions
– Normal distribution
• Sum of a large number of random numbers
– Normal distribution is the most widely used
distribution perhaps because of this property.
– If a quantity is obtained by summing up some
other randomly occurring quantities, then it is very
likely that the sum will be normally distributed
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NOTE
– Exponential distribution
• Service times, inter-arrival times, etc. are usually
observed to be exponentially distributed
• If the inter-arrival times are exponentially distributed,
then number of arrivals follows Poisson distribution
and vice versa
• The exponential distribution has an interesting
property called the memory less property: Assume
that the inter-arrival time of taxi cabs are
exponentially distributed and that the probability that
a taxi cab will arrive after 1 minute is 0.8. The above
probability does not change even if it is given that a
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person is waiting for an hour!
READING AND EXERCISES
• Reading: pp. 277-280
• Exercises: 7.26, 7.32
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