Statistics 510: Notes 15
Reading: Section 5.5-5.6.1, 9.1
I. The Poisson Process (Section 9.1)
The Poisson process is a model for events occurring
randomly in time. Examples include: the arrival of clients
at a bank, the occurrence of serious earthquakes, telephone
calls to a call center, the occurrence of power outages,
urgent calls to an emergency center, etc.
Events over a time period t occur according to a Poisson
process if
(a) the probability of an event occurring in a given small
time period t ' is approximately proportion to t '
(b) the probability of two or more events occurring in a
given small time period t ' is much smaller than t '
(c) the number of events occurring in two non-overlapping
time periods are independent.
Another way of describing the Poisson process is
(1) The events occur one by one.
(2) The number of events occurring during any two
nonoverlapping time intervals are independent of one
another.
(3) The number of arrivals during any given time interval
has a Poisson distribution of which the expected value is
proportional to the duration of the interval.
1
The parameter  is called the rate or arrival intensity of the
Poisson process.
II. Exponential Random Variables
Example (Example 4.7B): Suppose that earthquakes in the
western portion of the United States occur according to a
Poisson process with   2 and with 1 week as the unit of
time, i.e.,
Find the probability distribution of the time, starting from
now, until the next earthquake. What is the probability that
the time is greater than two weeks?
2
Exponential Random Variables: A random variable with
 x
CDF F ( x)  1  e , x  0 is called an exponential
random variable with parameter  .
The probability density function of an exponential random
variable is for x  0,
d
d
f ( x) 
F ( x)  1  e   x   e   x
dx
dx
and 0 for x  0.
The time until the first event in a Poisson process with rate
 is an exponential random variable with parameter  .
Example 1: Suppose customers arrive at a bank according
to a Poisson process with arrival intensity 2 per minute.
What is the probability that starting at 1 p.m., the first
customer arrives within two minutes?
3
Mean and variance of exponential random variable:
1
1
E
(
X
)

,
Var
(
X
)

We will show that

2 .
For n>0, we have

E ( X )   xn e x dx
n
0
Integrating by parts ( dv  e


0
0
 x
, u  x n ) yields
E ( X n )   x n e   x   e   x nx n 1dx
 0

n

n

 0
 e   x nx n 1dx
E[ X n 1 ]
Thus,
4
E( X ) 
1

E( X 2 ) 
E( X 0 ) 
2

E( X ) 
1

2
2
2
1
1
Var ( X )  2     2
  
2
Memorylessness of exponential random variable:
Let X be an exponential random variable with parameter  .
We have for all s, t  0,
P({ X  s  t}  X  t )
P( X  s  t | X  t ) 
P( X  t )
P( X  s  t )

P( X  t )
1  (1  e  ( s t ) )

1  (1  e  t )
 e s
In other words,
P( X  s  t | X  t )  1  e   s
(1.1)
If we think of X as the lifetime of some electrical device,
equation (1.1) states the probability that the device survives
for at least s  t hours given that it has survived t hours is
the same as the initial probability that it survives for at least
s hours. In other words, if the device is alive at age t, the
distribution of the remaining amount of time that it survives
is the same as the original lifetime distribution (that is, it is
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as if the instrument does not remember that it has already
been in use for a time t). This is called the memorylessness
property of the exponential distribution.
Example 2: Is the exponential distribution a good model for
the distribution of human lifetimes?
Example 3: Consider a post office that is staffed by two
clerks. Suppose that when Mr. Smith enters the post office,
he discovers that Ms. Jones is being served by one of the
clerks and Mr. Brown by the other. Suppose also that Mr.
Smith is told that his service will begin as soon as either
Jones or Brown leaves. If the amount of time that a clerk
spends with a customer is exponentially distributed with
parameter  , what is the probability that, of the three
customers, Mr. Smith is the last to leave the post office?
6
III. Gamma distribution:
Suppose events occur according to a Poisson process with
arrival intensity  .
Suppose we start observing the process at some time
(which we denote by time 0). The time until the first event
occurs has an exponential (  ) distribution.
Let X denote the time until the first  events occur. Let W
denote the number of occurrences of the event in the
interval [0, x] . Then W is a Poisson random variable with
parameter  x . The cdf of X can be obtained using W as
follows:
FX ( x)  P( X  x)
 1  P( X  x)
 1  P(fewer than  events occur in the interval [0,x])
 1  FW (  1)
 1
 1  e
k 0
 x
( x ) k
k!
Therefore,
7
d  1   x ( x) k
f X ( x)    e
dx k 0
k!
k
 -1

( x) k 1 
 x
  x ( x )
 x
    e   (  )e
 e ( )

k
!
(k  1)! 
k=1

k
k 1
 1
 1
  x ( x )
  x ( x)
  e
  e
k
!
(k  1)!
k 0
k 1
 1
  e
 x
k 0
 e
 x
( x ) k   2   x (  x ) k
  e
k ! k 0
k!
( x) 1
(  1)!


x 1e   x
(  1)!
What we have just derived is a special case of the gamma
family of probability distributions.
The gamma family can be generalized to cases in which
 is positive but not necessarily an integer. To do this, we
replace (  1)! with a continuous function of (nonnegative)
 , ( ) , the latter reducing to (  1)! when  is a
positive integer.
For any real number   0 , the gamma function (of  ) is
given by

( )   x 1e x dx .
0
Let X be a random variable such that
8
  e  x ( x) 1
x0

f ( x)  
( )
0
x0

Then X is said to have a gamma distribution with
parameters  and  .
The mean and variance of the gamma distribution are
E( X ) 


, Var ( X )  2 .


The gamma family of distributions is a flexible family of
probability distributions for modeling nonnegative valued
random variables. The following plot shows the pdfs for
some gamma distributions.
9
Example 1 continued: Suppose customers arrive at a bank
according to a Poisson process with arrival intensity 2 per
minute. What is the probability that starting at 1 p.m., the
first two customers have arrived within three minutes?
What is the expected value of the amount of time it takes
for two customers to arrive?
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