SADC Course in Statistics

The Poisson distribution

(Session 07)

2To put your footer here go to View > Header and Footer

Learning Objectives

At the end of this session, you will be able to:

• describe the Poisson probability distribution including the underlying assumptions

• calculate Poisson probabilities using a calculator, or Excel software

• apply the Poisson model in appropriate practical situations

3To put your footer here go to View > Header and Footer

Examples of data on counts

A common form of data occurring in practiceare data in the form of counts, e.g.• number of road accidents per year at

different locations in a country• number of children in different families• number of persons visiting a given website

across different days• number of cars stolen in the city each month

An appropriate probability distribution for thistype of random variable is the Poissondistribution.

4To put your footer here go to View > Header and Footer

The Poisson distribution

• The Poisson is a discrete probability distribution named after a French mathematician Siméon-Denis Poisson, 1781-1840.

• A Poisson random variable is one that counts the number of events occurring within fixed space or time interval.

• The occurrence of individual outcomes are assumed to be independent of each other.

5To put your footer here go to View > Header and Footer

• While the number of successes in the binomial distribution has n as the maximum, there is no maximum in the case of Poisson.

• This distribution has just one unknown parameter, usually denoted by (lambda).

• The Poisson probabilities are determined by the formula:

,3,2,1,0,!

)(

kfork

ekXP

k

Poisson Distribution Function

6To put your footer here go to View > Header and Footer

• Suppose the number of cars stolen per month follows a Poisson distribution with parameter = 3

What is the probability that in a given month

• Exactly 2 cars will be stolen?

• No cars will be stolen?

• 3 or more cars will be stolen?

Example: Number of cars stolen

7To put your footer here go to View > Header and Footer

Example: Number of cars stolen

For the first two questions, you will need:

=

=

The 3rd is computed as

= 1 – P(X=0) – P(X=1) – P(X=2)

=

2λ eP(X = 2) =

2!

0λ eP(X = 0) =

0!

8To put your footer here go to View > Header and Footer

Graph of Poisson with = 15

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 4 8 12 16 20 24 28

X

Pro

bab

ility

9To put your footer here go to View > Header and Footer

Graph of Poisson with = 10

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 4 8 12 16 20 24 28

X

Pro

bab

ility

10To put your footer here go to View > Header and Footer

Graph of Poisson with = 7

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0 4 8 12 16 20 24 28

X

Pro

bab

ility

11To put your footer here go to View > Header and Footer

Graph of Poisson with = 4

0.00

0.05

0.10

0.15

0.20

0.25

0 4 8 12 16 20 24 28

X

Pro

bab

ility

12To put your footer here go to View > Header and Footer

Graph of Poisson with = 1

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 4 8 12 16 20 24 28

X

Pro

bab

ility

13To put your footer here go to View > Header and Footer

Practical quiz

• What do you observe about the shapes of the Poisson distribution as the value of the Poisson parameter increases?

• Approximately where does the peak of the distribution occur?

14To put your footer here go to View > Header and Footer

Properties of the Poisson distribution

• The mean of the Poisson distribution is the parameter .

• The standard deviation of the Poisson distribution is the square root of . This implies that the variance of a Poisson random variable = .

• The Poisson distribution tends to be more symmetric as its mean (or variance) increases.

15To put your footer here go to View > Header and Footer

• The expected value of the Poisson random variable (r.v.) with parameter is equal to

0

x

x

E( X ) x e .x!

.1!0

exx

x

Note that, since Poisson is a probability distribution,

Expected value of a Poisson r.v.

16To put your footer here go to View > Header and Footer

• The second moment, E(X2) can be shown to be:

2 2

0

2

2 2 2

x

x

E( X ) x ex!

.

Var( X ) E( X )

Variance of a Poisson r.v.

Hence

• The standard deviation of a Poisson random variable is therefore .

17To put your footer here go to View > Header and Footer

Cumulative probability distribution

Poisson cumulative distribution with mean = 5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 3 6 9 12 15 18 21 24 27 30

X

Pro

ba

bil

ity

18To put your footer here go to View > Header and Footer

• Note that for X larger than about 12, the cumulative probability is almost equal to 1.

• In applications this means that, if say, the family size follows a Poisson distribution with mean 5, then it is almost certain that every family will have less than 12 members.

• Of course there is still the possibility of rare exceptions.

Interpreting the cumulative distn

19To put your footer here go to View > Header and Footer

In example above, we assumed X=family size, has a Poisson distribution with =5.

Thus P(X=x) = 5x e-5/x! , x=0, 1, 2, …etc.

(a)What is the chance that X=15?

Answer: P(X=15) = 515 e-5/15!

= 0.000157

This is very close to zero. So it would be reasonable to assume that a family size of 15 was highly unlikely!

Class Exercise

20To put your footer here go to View > Header and Footer

(b) What is the chance that a randomly selected household will have family size < 2 ?

To answer this, note that

P(X < 2) = P(X = 0) + P(X = 1)

=

(c) What is the chance that family size will be 3 or more?

Class Exercise – continued…

21To put your footer here go to View > Header and Footer

Further practical examples follow…