Upcoming SlideShare
×

# The Poisson Distribution

7,272 views

Published on

Introduces the Poisson Probability Distribution

4 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• good

Are you sure you want to  Yes  No

Are you sure you want to  Yes  No
Views
Total views
7,272
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
233
2
Likes
4
Embeds 0
No embeds

No notes for slide

### The Poisson Distribution

1. 1. 11 Poisson Distribution 2009. Max Chipulu, University of Southampton 1 The Poisson DistributionThe Poisson Distribution 1. To introduce the Poisson distribution as the situation where a discrete number of successes can be observed in a finite interval 2. To calculate Poisson distribution probabilities using the recurrence formula 3. To recognize and use the Poisson distribution as an approximation of the Binomial distribution in the situation where the number of trials is very large but probability of a success is small, i.e. when the expected number of successes in the Binomial is small. Poisson Distribution 2009. Max Chipulu, University of Southampton 2 This wasThis was SimSimééonon--Denis PoissonDenis Poisson Born 21Born 21 JuneJune 17811781 Died 25Died 25 AprilApril 18401840
2. 2. 22 Poisson Distribution 2009. Max Chipulu, University of Southampton 3 He is the Frenchman that said, ‘Life is good for only two things: to study Mathematics and to teach it.’ Can you imagine that?Can you imagine that? A life good only for studying MathematicsA life good only for studying Mathematics…… Poisson Distribution 2009. Max Chipulu, University of Southampton 4 Poisson was a great Mathematician. HePoisson was a great Mathematician. He achieved many great Mathematical featsachieved many great Mathematical feats For example,For example, ‘‘What are the odds that a horseWhat are the odds that a horse kicks a soldier to death?kicks a soldier to death?’’ He oftenHe often agonisedagonised over Mathematical issuesover Mathematical issues
3. 3. 33 Poisson Distribution 2009. Max Chipulu, University of Southampton 5 He found that the probability of a soldier being killed by horse kick on any given day was very small. Poisson observed ten army corps over twenty years. However, because he had observed each army corps over many days, the number of opportunities of this event happening was very large… so that the event actually happened a few times. Poisson Distribution 2009. Max Chipulu, University of Southampton 6 We don’t know this! But since Poisson was great, lets assume he had considerable powers of Statistical Reasoning Lets also assume he summoned them to ponder this: But how to calculate the probability. . . To die or not to die by horse kickTo die or not to die by horse kick
4. 4. 44 Poisson Distribution 2009. Max Chipulu, University of Southampton 7 My experiment is this: Does a soldier get killed by horse kick? He proceeded, thus:He proceeded, thus: Let each day that I observe be an experiment If a soldier gets killed by horse kick, then the experiment is a success. If not, the experiment has failed. Poisson Distribution 2009. Max Chipulu, University of Southampton 8 Clearly, a horse deciding to kick a soldier toClearly, a horse deciding to kick a soldier to death is a random event.death is a random event. And so, observing each day is aAnd so, observing each day is a BernouliBernouli TrialTrial But I know about Bernoulli trials: I can work outBut I know about Bernoulli trials: I can work out probabilities using the Binomial distribution.probabilities using the Binomial distribution.
5. 5. 55 Poisson Distribution 2009. Max Chipulu, University of Southampton 9 But if I observe a corps over a year, this is 365But if I observe a corps over a year, this is 365 trials!!trials!! I mustn't forget my resources are limited:I mustn't forget my resources are limited: I have never heard of computers or electronicI have never heard of computers or electronic calculators.calculators. How can I work this out?How can I work this out? THIS IS IMPOSSIBLETHIS IS IMPOSSIBLE Poisson Distribution 2009. Max Chipulu, University of Southampton 10 And so, Poisson found a way toAnd so, Poisson found a way to approximate the binomial probabilities:approximate the binomial probabilities: ! )( )( )( i eixP i µµ− == We knowWe know ee is Euleris Euler’’s number. Its value iss number. Its value is 2.71828. It is more commonly known as the2.71828. It is more commonly known as the exponential.exponential. We also know what the productWe also know what the product is: This is theis: This is the expected number of successes in a Binomialexpected number of successes in a Binomial DistributionDistribution
6. 6. 66 Poisson Distribution 2009. Max Chipulu, University of Southampton 11 For Zero Successes, Poisson Formula SimplifiesFor Zero Successes, Poisson Formula Simplifies toto )( 0 )( !0 )( )0( µµ µ −− === eexP Poisson Distribution 2009. Max Chipulu, University of Southampton 12 A little more Mathematics simplifies things as follows:A little more Mathematics simplifies things as follows: )!1()!1( )( )1( 1 )( )1( )( − = − =−= − − − − i e i eixP ii µµµ µµ ! )( )(since,But )( i eixP i µµ− == )1()(Therefore )( !)!1( )1(Then )( 1 )( −=== === − =−= − − − ixP i ixP ixP i i e i i eixP ii µ µ µ µ µµ µµ i i iiiii ! )!1(:grearrangin;)!1(*!:!Expand =−−=
7. 7. 77 Poisson Distribution 2009. Max Chipulu, University of Southampton 13 Therefore, once we know the probability of zeroTherefore, once we know the probability of zero successes, we can calculate other probabilitiessuccesses, we can calculate other probabilities successively as:successively as: )1()( −=== ixP i ixP µ Number ofNumber of successessuccesses minus 1minus 1 ExpectedExpected number ofnumber of successsuccess Number ofNumber of successsuccess Poisson Distribution 2009. Max Chipulu, University of Southampton 14 Poisson collected data on the number ofPoisson collected data on the number of men that died in each the 10 corps over themen that died in each the 10 corps over the twenty years.twenty years. This is equivalent to observing 200 samplesThis is equivalent to observing 200 samples of the same distribution.of the same distribution. Poisson found that the average number ofPoisson found that the average number of men that died in each corps was 0.61 permen that died in each corps was 0.61 per yearyear
8. 8. 88 Poisson Distribution 2009. Max Chipulu, University of Southampton 15 :iesprobabilitthe outworktoablewasPoisson,61.0withThus, =µ 5433.0)0( )61.0( === − exP 3314.05433.0*61.0)0( 1 )1( ===== xPxP µ 1011.03314.0* 2 61.0 )1( 2 )2( ===== xPxP µ etc;0021.01011.0* 3 61.0 )2( 3 )3( ===== xPxP µ Calculating Poisson ProbabilitiesCalculating Poisson Probabilities Poisson Distribution 2009. Max Chipulu, University of Southampton 16
9. 9. 99 Poisson Distribution 2009. Max Chipulu, University of Southampton 17 These are the probabilities of observing 0, 1,These are the probabilities of observing 0, 1, 2 deaths etc in one corps over a year.2 deaths etc in one corps over a year. To find the number of corps out of the 200To find the number of corps out of the 200 sampled in which 0, 1, 2 deaths etc weresampled in which 0, 1, 2 deaths etc were expected, Poisson multiplied theseexpected, Poisson multiplied these probabilities by 200.probabilities by 200. He then compared the numbers resultingHe then compared the numbers resulting from his theory with those he had collected.from his theory with those he had collected. Poisson Distribution 2009. Max Chipulu, University of Southampton 18 Deaths in Army over 20yrs: Actual VsDeaths in Army over 20yrs: Actual Vs Poisson RatesPoisson Rates 0 20 40 60 80 100 120 Frequency 0 1 2 3 4 5 Deaths per year Actual Poisson
10. 10. 1010 Poisson Distribution 2009. Max Chipulu, University of Southampton 19 Besides being suitable for approximating the binomial,Besides being suitable for approximating the binomial, the Poisson distribution is generally suitable forthe Poisson distribution is generally suitable for modelling discrete events that can occur within amodelling discrete events that can occur within a constrained interval of time or spaceconstrained interval of time or space • SPACE • e.g. a square piece of metal can develop a micro crack at any point within its surface. The Poisson distribution can be used to model number of cracks observed • TIME • e.g. the number of arrivals at a queue within a specific period of time such as 1 minute are usually modelled by a Poisson distribution. Poisson Distribution 2009. Max Chipulu, University of Southampton 20 Example: Call CentreExample: Call Centre A call centre receives, on average, three calls per minute. What is the probability that there will be, at most, 1 calls in a half minute? Suggested Steps for Calculation: 1. This is a Poisson Distribution: Discuss with the person nearest you. 2. Take out your calculators and work out -the expected number of calls -the probability of 0 calls -probability of 1 call -the required probability
11. 11. 1111 Poisson Distribution 2009. Max Chipulu, University of Southampton 21 Example: Call Centre SolutionExample: Call Centre Solution Poisson Distribution 2009. Max Chipulu, University of Southampton 22 Example: Call Centre SolutionExample: Call Centre Solution
12. 12. 1212 Poisson Distribution 2009. Max Chipulu, University of Southampton 23 Example:Example: GosportGosport && FarehamFareham Leukemia RatesLeukemia Rates Altogether, there are 170, 000 inhabitants in the townsAltogether, there are 170, 000 inhabitants in the towns GosportGosport andand FarehamFareham. A few years ago, local residents were alarmed to discover that. A few years ago, local residents were alarmed to discover that 13 children in families that lived within a 513 children in families that lived within a 5--mile radius of each other hadmile radius of each other had suffered with leukemia.suffered with leukemia. However, according to a medical doctor from Southampton GeneralHowever, according to a medical doctor from Southampton General hospital,hospital, the alarm was unwarranted as there are fluctuations in any procethe alarm was unwarranted as there are fluctuations in any process thatss that is subject to the rates of chance. He stated that he did not finis subject to the rates of chance. He stated that he did not find 13 casesd 13 cases particularly high given that the incidence of the disease is 8.3particularly high given that the incidence of the disease is 8.3 per million.per million. Not surprisingly, the residents were unconvinced. But lacking exNot surprisingly, the residents were unconvinced. But lacking expertise, theypertise, they were unable to present a coherent rebuttal to the doctorwere unable to present a coherent rebuttal to the doctor’’s statement.s statement. Are the rates of Leukemia inAre the rates of Leukemia in GosportGosport andand FarehamFareham consistent with a randomconsistent with a random causal process?causal process? Poisson Distribution 2009. Max Chipulu, University of Southampton 24 GosportGosport andand FarehamFareham Leukemia Rates SolutionLeukemia Rates Solution
13. 13. 1313 Poisson Distribution 2009. Max Chipulu, University of Southampton 25 GosportGosport andand FarehamFareham Leukemia Rates SolutionLeukemia Rates Solution Poisson Distribution 2009. Max Chipulu, University of Southampton 26 GosportGosport andand FarehamFareham Leukemia Rates SolutionLeukemia Rates Solution