X x theorem 0 1 x x proof 0 0 0 1 x x x x x x e e e e





x
x .
Theorem:
0
( ) 1 




x
x .
Proof:
   
 

  
  
  
    
  
0 0 0
( ) 1
! !
x x
x x x
x e e e e
x x
Theorem: The mean of the Poisson distribution is

and the variance of a Poisson distribution is

.
Proof:
   
   
  

   
   
   
    
 
   
1
0 1 1 0
[ ]
! ( 1)! ( 1)! ( )!
x x x x
x x x x
E x xe e e e
x x x x
A similar procedure shows
    2 [ ] E x
. Thus,
       
2 2
[ ] ( ) Var x
.
For certain values of

, CDF charts are provided. Using these CDF charts are very straight forward after
identifying

. Using the Poisson CDF chart: The Poisson CDF chart is actually several different CDF
charts with a selection of possible rates (

). First identify

for the problem and use that column to
answer your questions. The value of

could be a non-integer. You can have an arrival rate of 7.6 per
hour. Since
7.6  
is not on our charts, we would have to use the Poisson distribution formula to
determine our probabilities. In the real world you would use a computer program instead of these
charts or the formula.

24
If 9   and we wanted to determine ( 8) P X  , we would go to the appropriate column and look down
to 8 x  and finalize our answer: ( 8) (8) .4557 P X   
Example: Determine ( 6)  P X for a Poisson random variable with
9  
( 6) (6) (5) .2068 .1157 .0909 P X       

.
Exercise: Determine (7 )  P X for a Poisson random variable with   4
Theorem: If the number of events in a time interval t follows a Poisson distribution with mean

, then
the number of events in time period
k t 
follows a Poisson distribution with mean
k  
This is quite a common sense theorem. If arrivals occur at a rate of 10 per hour, we could also say 20 per
2-hour period, 5 per half hour, 2.5 per 15-minute period etc.
Suppose that X counts the number of arrivals in a 20 minute period and
Poi( 8) X  
. (Read X
distributed). Thus, the rate is 8 per 20-minute time period. Suppose that we want to determine
probability questions over a 25-minute period. What would our new

be?
.
.
is

,
Exercise: Determine
(9 ) P Y 
.

25
Exercise: The number of patients requiring a certain type of treatment in a given day well modeled by a Poisson
distribution with a mean rate of
6  
. The facility currently has 4 machines that are required for the treatment.
Each machine can be used on 2 patients per day.
a) Determine the probability that they have enough machines to treat all patients on a given day.
b) How many machines would they need if they wanted the probability that they would have enough machines to
treat all patients on a given day to exceed .99?
Exploratory Example: A person goes out into the hallway and tosses a coin every 10 seconds. Each time
a heads is tossed, the person dings a bell. Let X count the number of dings per minute. Clearly X has a
binomial distribution with parameters
 6 n
and
.5 p 
. Is this a Poisson distribution too? All three
criteria seem to be met. We have a constant arrival rate of three dings per minute. We certainly have
independent increments. Th ere is no way to get two dings in a very short period of time. Let’s compare