381
Discrete Probability Distributions
(The Poisson and Exponential Distributions)
QSCI 381 – Lecture 15
(Larson and Farber, Sect 4.3)
The Poisson Distribution-I
381

1.
2.
3.
The Poisson distribution is the sampling distribution.
It is used to determine the probability that a specific
number of occurrences takes place within a given
unit of time or space. A random variable X is Poisson
distributed if:
The experiment consists of counting the number of
times, x, an event occurs in a given interval (the
interval can relate to time, area or volume).
The probability of the event occurring is the same
for each interval.
The number of occurrences in one interval is
independent of the number of occurrences in other
intervals.
The Poisson Distribution-II
381

Notation:
P( X  x) 
 x e 
x!
Note that x can be any non-negative
integer.
Mean :  (rate per interval)
Variance : 
An Example of the Poisson
Distribution-I
381

The mean catch of dolphins per month
in the Eastern Tropical Pacific is 3.


What is the probability that the catch will
be 4 in a given month?
What is the probability that the catch will
be 2 or less in a given month.
An Example of the Poisson
Distribution-II
381

The mean catch of dolphins per month
in the Eastern Tropical Pacific is 3.

What is the probability that the catch will
be 4 in a given month?
P[ X  4] 
 e
x

x!
4
3
3e

 0.168
4!
An Example of the Poisson
Distribution-III
381

The mean catch of dolphins per month
in the Eastern Tropical Pacific is 3.
What is the probability that the catch will
be 4 in a given month?
 What is the probability that the catch will
be 2 or fewer in a given month?
P[X2]=P[X=0]+P[X=1]+P[X=2]=

0
1
2
3 3 3
P[ X  2]  e [   ]  0.423
0! 1! 2!
3
Modeling Typhoons-I
381
160
140
Frequency
120
100
80
60
40
20
0
0
1
2
3
4
5
Number of Typhoons
Assuming that typhoons are Poisson distributed, find the mean
number of typhoons per year and the probability that there will
be three or more in a single year.
Modeling Typhoons-II
381

Mean number of typhoons:



i
i
i
 0.5
N
Probability of x typhoons a year:
0
149
0.607

n x
1
24
0.303
2
15
0.076
3
5
0.013
4
4
0.002
5
3
0.000
6
0.000
The probability of the number of typhoons
being at least 3 is 0.015.
Later we will examine methods to determine
whether it is reasonable to assume that these
data are Poisson distributed.
The Poisson Distribution
(Caveats)
381


The Poisson distribution would seem to
be a good distribution to assume for
catch data in fisheries. However, fish
school. What does this imply regarding
the assumptions of the Poisson
distribution?
Be clear when using the Poisson
distribution what the interval is.
The Poisson Distribution
Graphically
381
0.3
0.2
=2
0.25
0.16
0.14
Probability
0.2
0.15
0.12
0.1
0.08
0.1
0.06
0.04
0.05
0.02
0
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.16
=8
0.14
0.12
Probability
Probability
=5
0.18
0.1
0.08
0.06
0.04
0.02
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Summary of Discrete Distributions
381
Distribution
Number of trials
Binomial
Pre-specified
Negative
Binomial
Repeated until
a given number
of successes
Repeated until
one success
occurs
N/A
Geometric
Poisson
Random
Variable
Number of
successes
Parameters
Number of
failures
r, p
Number of
failures
p
Number of
occurrences

n, p
381
Discrete Probability Distributions
and EXCEL-I

BINOMDIST(x,n,p,cum)


Evaluates the probability of obtaining x successes
from n trials when the probability of a success is
p. Set “cum” to True to calculate the cumulative
probability that the number of successes is x or
fewer.
NEGBINOMDIST(x,s,p)

Evaluates the probability of obtaining the s th
success after x failures when the probability of a
success is p. Set s=1 for the Geometric
distribution.
381
Discrete Probability Distributions
and EXCEL-II

POISSON(x,,cumu)

Evaluates the probability of obtaining x
occurrences in an interval when the mean
number is . Set “cum” to True to calculate
the cumulative probability that the number
of occurrences is x or less.