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
