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Applied Statistics and Computing Lab

POISSON DISTRIBUTION

Applied Statistics and Computing Lab

Indian School of Business

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Applied Statistics and Computing Lab

Learning goals

To understand the idea behind Poisson

distribution

To study situations where Poisson distribution

can help evaluate probabilities of events

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Example

Every day, about 7 million commuters travel

on the suburban local trains in Mumbai. On anaverage, 10 persons die in local train related

accidents, every day!

Then what is the probability that I will die

travelling in the train tomorrow?

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Example (contd.) We can use Binomial distribution to evaluate this

probabilitySuccess = death of a person while travelling on the

suburban local train, on a given day

= =

= 7

Would our computer be able to solve thisequation?

Prob X = 1 =7,000,000

10(

10

7,000,000)(

7,000,000 10

7,000,000),,

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Framework for Poisson distribution This is where Poisson distribution comes to rescue!

When is very large, is very small (7 million and

respectively, in our example) and is finite,

the PMF of a variable following Poisson distributionwith parameter = is given by

= =

!where = 0,1,2,3, and > 0

Here, is the value of average occurrences

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More examples There are many other situations in which we

observe that the total number of trials is verylarge and the probability of success is very small.Here are some examples:

Probability of finding 1 out of 10 items to be defective,when produced on a machine that produces 0.1%defective items

Probability of one of the policy holders of a life

insurance plan dying during the next one year

Probability of the tubelight in your room going off,when it generally works about 18 months

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Properties of Poisson distribution

For ~ ,

= = =

It is an important property of Poisson distribution

that its mean and variance is equal to each other

In case of a Binomial distribution where is very

large and is very small, Poisson distributionhelps us approximate the probabilities

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Thank you