Introduction to Poisson Distribution

7/27/2019 Introduction to Poisson Distribution

1/16

What do the following statistics have incommon?

The number of telephone calls you receivein an hour.

The number of cars per mile broken down the hard shoulder of a motorway.

The number of dandelions per square metre

on the college playing field.

The number of raisins per 100 cm3 portionfruit cake.

Each of these situations is about the occurrence of unpredictable events within a given

of time, distance, area or volume. Sometimes we are surprised by the number ofcoincidences that occur in such random processes. For example, you may get threetelephone calls within a space of a few minutes, and then have no calls for several hourOr, you may travel 30 miles on the motorway without seeing a single breakdown, then ysee two cars broken down within the space of two miles.

The probability that a certain number of random events occur in a given space of time(distance, etc.) can be calculated using the Poisson probability distribution, provided thacertain conditions hold.

To continue on the recommended route click on the History button at the top of the pa

Copyright 1997 Neville Hunt and Sidney TyrrellCoventry UniversityAll rights reserved. Last updated: 19 February 2011
http://nestor.coventry.ac.uk/~nhunt/poisson/links.htmlhttp://-/?-


2/16

This module aims to teach you about the Poisson probability distribution.

It covers:Introduction - what is it all about?

History - who discovered it?

Derivation - where does it come from?

Graphs - what does it look like?

Applications - what can it be used for?

Calculation - using a spreadsheet

Approximation - by a Normal distribution

You may work through the various parts of this unit in any order, but the recommendedis indicated at each stage by the red option button. You are advised to start by clickingthe Introduction button at the top of the page.

On any page you may use the button to visit other relevant web resources

the button to return to this page.

The interactive elements of this module use Microsoft Excel 97 and will not run unless software (or a later version) is installed on your computer (or network). Excel uses sett

taken from the first worksheet opened in the session, so please ensure that you do not hExcel already running when you start to use any of our spreadsheets.

http://-/?-http://nestor.coventry.ac.uk/~nhunt/poisson/links.html


3/16

by Neville Hunt and Sidney Tyrrell, Coventry University

Contents Back to DISCUSS Contents

Introduction & What is it all about?

& History

& Derivation of the Poisson model

Queueing & Queueing as a Poisson process

? Excel queueing simulation

Graphs & Probability distribution

? Excel graphs of Poisson distributions

Patterns & Spatial distributions

? Excel analysis of Coventry data

Doodlebugs & Flying bombs in WW2

? Excel analysis of WW2 data

Comparisons & Links with the Binomial distribution

? Excel Poisson and Binomial graphs

Probabilities & Example calculation

Excel calculator

& Normal approximations

Normal approximation calculator

Links Links to external selected sites

Copyright 1997 Neville HuntCoventry UniversityAll rights reserved. Last updated: 28 February 2011 .
http://www.mis.coventry.ac.uk/~nhunthttp://www.coventry.ac.uk/ec/~styrrell/http://www.mis.coventry.ac.uk/~nhunt/home/http://nestor.coventry.ac.uk/~nhunt/poisson/discus41.xlshttp://nestor.coventry.ac.uk/~nhunt/poisson/discus42.xlshttp://nestor.coventry.ac.uk/~nhunt/poisson/discus43.xlshttp://nestor.coventry.ac.uk/~nhunt/poisson/discus44.xlshttp://nestor.coventry.ac.uk/~nhunt/poisson/comparis.htmlhttp://nestor.coventry.ac.uk/~nhunt/poisson/discus46.xlshttp://nestor.coventry.ac.uk/~nhunt/poisson/discus45.xlshttp://nestor.coventry.ac.uk/~nhunt/poisson/discus48.xlshttp://nestor.coventry.ac.uk/~nhunt/poisson/links.htmlhttp://nestor.coventry.ac.uk/~nhunt/poisson/links.htmlhttp://nestor.coventry.ac.uk/~nhunt/poisson/links.htmlhttp://nestor.coventry.ac.uk/~nhunt/poisson/discus48.xlshttp://nestor.coventry.ac.uk/~nhunt/poisson/discus45.xlshttp://nestor.coventry.ac.uk/~nhunt/poisson/discus46.xlshttp://nestor.coventry.ac.uk/~nhunt/poisson/comparis.htmlhttp://nestor.coventry.ac.uk/~nhunt/poisson/discus44.xlshttp://nestor.coventry.ac.uk/~nhunt/poisson/discus43.xlshttp://nestor.coventry.ac.uk/~nhunt/poisson/discus42.xlshttp://nestor.coventry.ac.uk/~nhunt/poisson/discus41.xlshttp://www.mis.coventry.ac.uk/~nhunt/home/http://www.coventry.ac.uk/ec/~styrrell/http://www.mis.coventry.ac.uk/~nhunt


4/16

Simeon Denis Poisson was born in 1781 in the small Frenchtown of Pithviers. His father was a private soldier who, onretirement, became the town administrator when the FrenchRevolution broke out. His family wanted him to become adoctor but he quickly abandoned this idea when his firstpatient died.

Instead he entered the Ecole Polytechnique in Paris in 1798,where he learned from such famous mathematicians asLagrange and Laplace. As a student he mixed with Cauchy,Galois, Fourier and Ampere, all of whom became well knownin their different fields. On graduation he became a lecturerand later a professor.

The Poisson distribution occupies just a single page of a paper entitled "Researches on tprobability of criminal and civil verdicts" published in 1837. In this paper Poisson lookethe form of the binomial distribution when the number of trials was large. He derived t

cumulative Poisson distribution as the limiting case of the binomial when the probabilitysuccess tends to zero.

Poisson's main interest lay in the field of mathematical physics where he is rememberedPoisson's integral in potential theory, Poisson brackets in differential equations, the Poisratio in elasticity and Poisson's constant in electricity. In all, during his career Poissonpublished more than 300 papers.

Although brought up as a republican, Poisson later became a peer of France in 1837. Hein Paris in 1840. Poisson is quoted as having said "life is good for only two things: doinmathematics and teaching mathematics"!

To continue on the recommended route click on the Derivation button at the top of thepage.

http://nestor.coventry.ac.uk/~nhunt/poisson/links.html


5/16

One of the most useful applications of the Poisson distribution is inthe field of queueing theory. In many situations where queuesoccur it has been shown that the number of people joining thequeue in a given time period follows the Poisson model.

A spreadsheet can be used to simulate the queue at a supermarketcheckout. You can simulate customers arriving and departing by

setting the arrival and departure rates, and watch the queue buildup. Have you ever wondered how it is that queues can grow solong?

Just click on the link below to load the Excel spreadsheet and accompanying exercises.When you have finished, use the browser's Back button to return here.

Load spreadsheet

To continue on the recommended route click on the Graphs button at the top of the pa

http://nestor.coventry.ac.uk/~nhunt/poisson/discus41.xlshttp://nestor.coventry.ac.uk/~nhunt/poisson/links.htmlhttp://nestor.coventry.ac.uk/~nhunt/poisson/discus41.xls


6/16

For the Poisson distribution to apply to a particular situation the following conditionmust apply:

Certain events are occurring at random over a continuous period of time (or intervdistance, region of area, etc).

These events occur singly, i.e. it is not possible for two events to occur exactly

simultaneously.

The events also occur independently of each other, i.e. the fact that an event haoccurred (or not occurred) does not affect the chance of another event occurring.

The events occur at a constant average rate, usually denoted by the Greek letter

(lambda).

The derivation is quite mathematical, so some users may wish to skip this section.

What is the probability of obtaining r events in unit time?

The easiest way to derive the probability distribution is to develop it as a limiting case oBinomial distribution. For simplicity we shall talk in terms of events happening over timImagine an interval of 1 unit of time divided into n tiny sub-intervals, each so small thanot physically possible for more than one event to occur in a sub-interval. If p is theprobability that an event does occur in a sub-interval, then the average number of even

1 unit of time will be np, which must equal l. So, p=l/n.

So, from the Binomial formula we have:

Prob(r events in 1 unit of time)=Prob(r successes in n trials)

= nCr pr (1-p)n-r = nCr (l/n)

r (1-l/n)n-r

If r=0, Prob(0 events in 1 unit of time)=(1-l/n)n and, if we allow n to tend to infinity, th

becomes e-l . If you have difficulty believing this, check it out with an Excel spreadshee

If r=1, Prob(1 event in 1 unit of time)=n (l/n) (1-l/n)n-1 and, again, if we allow n to ten

infinity, this becomes le-l.Continuing in this fashion, the general result is that:

Prob(r events in 1 unit of time)=lre-l/r! r=0,1,2,...

To continue on the recommended route click on the Queueing button at the top of the

Copyright 1997 Neville Hunt and Sidney Tyrrell

Coventry UniversityAll rights reserved. Last updated: 19 February 2011


7/16


8/16

The shape of the Poisson probability distribution varies according to the mean number ofevents per unit time or space.

This is the PoissonDistribution withmean l=2.

This is the PoissonDistribution withmean l=10.

You can investigate this further using a spreadsheet. Click on the link below to load the spreadsheet and accompanying exercises. When you have finished, use the browser's Bacbutton to return here.

Load spreadsheet

To continue on the recommended route click on the Patterns button at the top of the pa

http://nestor.coventry.ac.uk/~nhunt/poisson/discus42.xlshttp://nestor.coventry.ac.uk/~nhunt/poisson/links.htmlhttp://-/?-http://nestor.coventry.ac.uk/~nhunt/poisson/discus42.xls


9/16

Another important application of the Poisson distributionis in explaining the apparently random distribution ofobjects in an area. This might be the distribution ofweeds on a lawn, or the distribution of blemishes on alarge sheet of glass.

Here you can use a spreadsheet to to investigate the

distribution of various facilities across the city ofCoventry. For example, the picture to the right showsthe distribution of fish-and-chip shops in Coventry(2003).

Click on the link below to load the Excel spreadsheet and accompanying exercises. Whehave finished, use the browser's Back button to return here.

Load spreadsheet

To continue on the recommended route click on the Doodlebugs button at the top of t

page.



10/16

In June 1943 during the second World War the Germanslaunched a much feared secret weapon - a small, pilotless, jet-propelled plane carrying a ton of explosives that detonated oncontact. The Germans called it the V1 Vergeltung (meaningreprisal) - the British called it the flying bomb.

Of the first 10 launched, 5 crashed almost immediately and only

one found its way to Britain where it killed 6 people. However,subsequent flying bomb attacks on Britain caused widespreaddeath and destruction. Within a year 1600 people had beenkilled by flying bombs.

The British authorities were anxious to know if these weapons could be accurately aimed

particular target, or whether they were landing at random. A 36 km2 area of South Lonwas divided into 1/4 km squares and the number of hits received in each square wasrecorded. If the hits were random then the resulting distribution of hits per square shohave followed the Poisson model.

You can analyse this set of data yourself using a spreadsheet. Click on the link below tothe Excel spreadsheet and accompanying exercises. When you have finished, use thebrowser's Back button to return here.

Load spreadsheet

To continue on the recommended route click on the Comparisons button at the top of

page.



11/16


12/16



13/16

Normal Approximation to Poisson

The Normal distribution can be used to approximate Poisson probabilities when l is large

answer to the question "How large is large?", a rule of thumb is that the approximationshould only be used when l 10.

Example

Suppose that in the queueing example (1.8 arrivals per minute) we wanted to know theprobability of more than 70 arrivals in half an hour.

For the approximation to work we must consider three things:

we must match the mean of the Normal distribution to the mean of the Poisson- otherwise the Normal curve will be centred in the wrong place,we must match the standard deviation of the Normal to that of the Poisson- otherwise the Normal curve will not be the correct widthwe must make an adjustment to take account of the fact that the Poisson variable

discrete but the Normal variable is continuous - the continuity correction.

The mean rate is 30 x 1.8 = 54 per hour.The standard deviation is therefore sqrt(54) = 7.35.

More than 70 defectives means 71 or more on the discrete scale, but 71 extends down aas 70.5 on the continuous scale. So the approximation is the area under the Normal curabove 70.5.

Hence, Pr( arrivals > 70) = Pr (arrivals > 70.5) ~ Pr( Z > [70.5 - 54]/7.35 ) = Pr( Z > 2.25which, from tables of the Normal distribution, is 0.0122.

The exact Poisson probability is 0.0152.

The spreadsheet provided below allows you to check how well the Normal distribution fivalues ofl up to 100 - for values ofl beyond 100 it only calculates the Normal probabilit


14/16

Click on the link below to load the Excel spreadsheet and accompanying exercises. Whehave finished, use the browser's Back button to return here.

Load spreadsheet

You have now completed the recommended route through this unit on the Poisson

Distribution. Make sure you visit our Links page to find out about other online resourcerelevant to this topic.



15/16

Campus Map Social Media Directory J obs Contact Us Po

re here: Home / Error 404

RROR 404

ORRY, WE CAN'T FIND THE PAGE YOU

ERE LOOKING FOR...

age you are looking for may have been removed, had its name changed or be temporarily unavailable.

E YOU LOOKING FOR ONE OF THESE PLACES?

oventry University Home

eful Links

tudy at Coventry

usiness

ourse Finder

nline Store

rder a Prospectus

itemap

Help

Contact Us

Press Enquiries

Home

Ask Coventry

Legal Information

Terms and Conditions

Accessibility

Copyright Policy

Freedom of Information Policy

Intellectual Property Policy

Cookie Policy

Contact Us

Coventry University

Priory Street

Coventry, United Kingdom

CV1 5FB

Telephone: +44 (0) 24 7688 76

Study Life on Campus Graduation and Beyond Research Business
http://www.coventry.ac.uk/shared-content/utility-links/google-map/http://www.coventry.ac.uk/shared-content/utility-links/social-media-directory/http://www.coventry.ac.uk/shared-content/utility-links/recruitment/http://www.coventry.ac.uk/shared-content/utility-links/contact-us/http://www.coventry.ac.uk/shared-content/utility-links/login/http://www.coventry.ac.uk/http://www.coventry.ac.uk/http://www.coventry.ac.uk/study-at-coventry/http://www.coventry.ac.uk/business/http://www.coventry.ac.uk/study-at-coventry/course-search/http://store.coventry.ac.uk/http://www.coventry.ac.uk/order-a-prospectus/http://www.coventry.ac.uk/sitemap/http://www.coventry.ac.uk/shared-content/utility-links/contact-us/http://www.coventry.ac.uk/life-on-campus/faculties-and-schools/experts-directory/http://www.coventry.ac.uk/http://www.coventry.ac.uk/ask-coventry/http://www.coventry.ac.uk/legal-documents/terms-and-conditions/http://www.coventry.ac.uk/legal-documents/accessibility-statement/http://www.coventry.ac.uk/legal-documents/copyright-statement/http://www.coventry.ac.uk/legal-documents/freedom-of-information-policy/http://www.coventry.ac.uk/legal-documents/intellectual-property/http://www.coventry.ac.uk/legal-documents/cookie-policy/http://www.coventry.ac.uk/study-at-coventry/http://www.coventry.ac.uk/life-on-campus/http://www.coventry.ac.uk/graduation-and-beyond/http://www.coventry.ac.uk/research/http://www.coventry.ac.uk/business/http://www.coventry.ac.uk/business/http://www.coventry.ac.uk/business/http://www.coventry.ac.uk/research/http://www.coventry.ac.uk/research/http://www.coventry.ac.uk/graduation-and-beyond/http://www.coventry.ac.uk/graduation-and-beyond/http://www.coventry.ac.uk/life-on-campus/http://www.coventry.ac.uk/life-on-campus/http://www.coventry.ac.uk/study-at-coventry/http://www.coventry.ac.uk/study-at-coventry/http://www.coventry.ac.uk/http://www.coventry.ac.uk/legal-documents/cookie-policy/http://www.coventry.ac.uk/legal-documents/intellectual-property/http://www.coventry.ac.uk/legal-documents/freedom-of-information-policy/http://www.coventry.ac.uk/legal-documents/copyright-statement/http://www.coventry.ac.uk/legal-documents/accessibility-statement/http://www.coventry.ac.uk/legal-documents/terms-and-conditions/http://www.coventry.ac.uk/ask-coventry/http://www.coventry.ac.uk/http://www.coventry.ac.uk/life-on-campus/faculties-and-schools/experts-directory/http://www.coventry.ac.uk/shared-content/utility-links/contact-us/http://www.coventry.ac.uk/sitemap/http://www.coventry.ac.uk/order-a-prospectus/http://store.coventry.ac.uk/http://www.coventry.ac.uk/study-at-coventry/course-search/http://www.coventry.ac.uk/business/http://www.coventry.ac.uk/study-at-coventry/http://www.coventry.ac.uk/http://www.coventry.ac.uk/http://www.facebook.com/coventryuniversityhttp://www.coventry.ac.uk/shared-content/utility-links/login/http://www.coventry.ac.uk/shared-content/utility-links/login/http://www.coventry.ac.uk/shared-content/utility-links/contact-us/http://www.coventry.ac.uk/shared-content/utility-links/contact-us/http://www.coventry.ac.uk/shared-content/utility-links/recruitment/http://www.coventry.ac.uk/shared-content/utility-links/recruitment/http://www.coventry.ac.uk/shared-content/utility-links/social-media-directory/http://www.coventry.ac.uk/shared-content/utility-links/social-media-directory/http://www.coventry.ac.uk/shared-content/utility-links/google-map/http://www.coventry.ac.uk/shared-content/utility-links/google-map/

Documents

Introduction to Poisson Distribution