The document discusses the Poisson distribution, which is a discrete probability distribution that estimates the likelihood of a specified outcome occurring a certain number of times in a fixed interval, given a constant average rate of occurrence. It outlines the formulation, assumptions, and properties of the Poisson distribution, as well as examples illustrating its application in real-world scenarios such as safety assessments and defect patterns in materials. Key parameters are defined, including the shape parameter (λ), the expected value (μ), and standard deviation (σ).

1. BASICS OF THE POISSON
DISTRIBUTION

2. Formulation of the Poisson
Distribution
• If you look closely at a photograph in a
newspaper, you see that the image is
composed of a large number of dots arranged
in rows and columns.
• Each dot is printed larger for a dark area and
smaller for light areas.
• Any one of the dots is printed incorrectly if it
is too light, too dark, missing, or misplaced.

3. Formulation of the Poisson
Distribution
• Define:
– h = area represented by one dot; h > 0 but very small
– µ = constant representing average number of times dot is
printed incorrectly in h; µ > 0
– H = total area, divided into n equal areas, each containing
a dot
• The area h = H/n gets smaller and smaller as n
becomes countably infinite (n = 0, 1, 2, …, no upper
limit).

4. Formulation of the Poisson
Distribution
• We postulate that the probability of an
incorrectly printed dot is proportional to the
size of h; that is, as the area h increases, the
probability of an incorrect dot increases.
Therefore
– µh = probability of one incorrectly printed dot in h
– 1 - µh = probability of a correctly printed dot in h
• Since µh + (1 - µh) = 1, there is no chance of
more than one mistake in any h.

5. Formulation of the Poisson
Distribution
• The small value µh is called a rare event
probability, and it is used to derive the
Poisson pdf from the binomial by substituting
µh for p:
• Take the limit as n  , as p  0, and with np
constant (call its value ), and the Poisson pdf
results:
n
H
h
p 
 

 
!
,
;
limit
0
x
e
p
n
x
b
x
np
p
n










6. Formulation of the Poisson
Distribution
• The right side of the previous equation is the
Poisson pdf, and the descriptor used is P(x; )
to show that X is Poisson with a particular
value of .
• The complete pdf
• Since x is a positive integer, the Poisson
distribution is discrete.
 


 


elsewhere
0
...
2,
1,
0,
!
,
x
x
e
x
P
x




7. Assumptions of the Poisson Distribution
• There are n independent trials where n is very large.
Terdapat percobaan independen sebanyak n di mana n
sangat besar
• Only one outcome is of interest on each trial.
Hanya satu hasil yang kita perhatikan pada setiap
percobaan
• There is a constant probability of occurrence on each
trial.
Probabilitas kejadian konstan pada setiap percobaan
• The probability of more than one occurrence per trial is
negligible.
Probabilitas terjadi lebih dari satu kejadian per
percobaan dapat diabaikan

8. Definition of the Poisson Distribution
The Poisson is a discrete distribution which
estimates the probability that a specified
outcome will occur exactly x times in a
standardized unit when the average rate of
occurrence per unit is a constant .

9. Parameter and Properties
• The Poisson has one parameter which changes
the shape of pdf.
– Shape parameter : 
– Parameter range :  > 0
– Expected value : µ = 
– Standard deviation :  =
• The parameter estimation

n
X
f
X i
i



̂

10. Example 9.3
Dolly and Darrel are engineers with a government-
sponsored safety agency. They have determined the
number of serious accident each month at the
construction sites of new government office buildings to
be Poisson-distributed.
a. Based on the data below, they predict a good change
of one serious accident each month. Are they correct?
b. What are the 1 limits above and below the mean of
this data?
5
4
3
2
1
0
Accidents
per month
0
1
2
8
12
27
Frequency

11. Example 9.4
• Use of the Poisson Distribution Table
– P’(12; 6.60)
– P(5; 12.0)
– P(4; 1.33)
–  


2
80
.
1
;
x
x
P

12. Solving Problems Using the Poisson
• Check assumptions
• Determine the variable and value of the
parameter.
• Obtain a single Poisson probability.
• Obtain the sum of several Poisson probability
values.
• Determine µ and  for the Poisson.
• Plot the pdf or cdf.

13. Example 9.5
An ME is studying the defect pattern in cold-rolled
steel. The engineer inspects 10-m sections to
determine the number of defects D. The results for 50
sections are:
Number of
sections
Defects per
section D
35
8
3
2
1
1
50
0
1
2
3
4
6
a. Determine the probability of
being on or above the
specification limit of three
defects per section.
b. Compute the 3 limits.

14. Example 9.6

15. Problems
• 9.6
• 9.10 (c)
• 9.13
• 9.18