SADC Course in Statistics The Poisson distribution (Session 07)

SADC Course in Statistics

The Poisson distribution

(Session 07)

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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

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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.

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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.

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• 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

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• 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

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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!

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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

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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

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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

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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

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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

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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?

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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.

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• 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.

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• 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 .

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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

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• 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

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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

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(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…

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Further practical examples follow…