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STAT 5112
Spring 2012
Lecture 17
Feb 20, 2012
Jun Xie
4.4 The exponential distribution
The exponential distribution is frequently used as a model for the distribution of times between the
occurrence of successive events, such as customers arriving at a service facility or calls coming in to a
switchboard.
Proposition
Suppose that the number of events occurring in any time interval of length
t
has a Poisson distribution
with parameter
t
(where
, the rate of the event process, is the expected number of events occurring
in 1 unit of time) and that numbers of occurrences in nonoverlapping intervals are independent of one
another. Then the distribution of elapsed time between the occurrence of two successive events is
exponential with parameter
=
.
Although a complete proof is beyond the scope of the text, the result is easily verified for the time X
1
until the first event occurs:
P
(
X
1
t
) = 1 –
P
(
X
1
>
t
) = 1 –
P
[no events in (0,
t
)]
which is exactly the cdf of the exponential distribution.
Another important application of the exponential distribution is to model the distribution of component
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 Fall '08
 BUD
 Normal Distribution, Variance, Probability theory, Exponential distribution, probability density function

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