3. Poisson

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The Poisson Distribution

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The Poisson Probability Distribution

Simeon Denis “Fish”!

• "Researches on the probability ofcriminal civil verdicts" 1837

• Looked at the form of the binomial

distribution

When the Number of

Trials is Large.

• He derived the cumulative Poissondistribution as the

Limiting case of the

Binomial When theChance of Success

Tends to Zero.

Simeon DenisPoisson

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Other phenomena that often follow a Poisson

distribution are

•death of infants,

•the number of misprints in a book,

•the number of customers arriving, and t

•he number of activations of a Geiger counter.

The distribution was derived by the Frenchmathematician Siméon Poisson in 1837, and the

first application was the description of the number of

deaths by horse kicking in the Prussian army.

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• Poisson Distribution: An approximation to the binomial

distribution for the SPECIAL CASE when the average

number (mean µ ) of successes is very much smaller than

the possible number n. i.e. µ << n because p << 1.

• This distribution is important for the study of such phenomena asradioactive decay. This distribution is NOT necessarily symmetric!

Data are usually bounded on one side & not the other.

An advantage of this distribution is that σ2

= μ

The Poisson Distribution

µ = 1.67

σ = 1.29

µ = 10.0

σ =

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The Poisson Distribution Models Counts

• If events happen at a constant rate over time, the PoissonDistribution gives

The Probability of X Number of

Events Occurring in a time T.• This distribution tells us the

Probability of All Possible Numbers of

Counts, from 0 to Infinity.• If X= # of counts per second, then the Poisson probability that

X = k (a particular count) is:

l

λ ≡ the average number of counts per second.

!)(

k

ek X p

k

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Mean and Variance for the

Poisson Distribution• It’s easy to show that for this distribution,

The Mean is:

• Also, it’s easy to show that The Variance is:l

L

The Standard Deviation is:

2

For a Poisson Distribution, the

variance and mean are equal!

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Terminology: A “Poisson Process” • The Poisson parameter can be given as the mean

number of events that occur in a defined time period OR,

equivalently, can be given as a rate, such as

= 2

events per month. must often be multiplied by a time t

in a physical process

(called a “Poisson Process” )

!

)(

)( k

et

k X P

t k

μ =

t σ =

t

More on the Poisson Distribution

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Example

1. If calls to your cell phone are a Poisson processwith a constant rate = 2 calls per hour, what

is the probability that, if you forget to turn your

phone off in a 1.5 hour class, your phone ringsduring that time?

Answer: If X = # calls in 1.5 hours, we want

P(X ≥ 1) = 1 – P(X = 0)

p

P(X ≥ 1) = 1 – .05 = 95% chance

05.!0

)3(

!0

)5.1*2()0( 3

30)5.1(20

eee

X P

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Example Continued

2. How many phone calls do you expectto get during the class?

<X> =

t = 2(1.5) = 3

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10

Conditions Required for the

Poisson Distribution to hold:l

1. The rate is a constant, independent of time.

2. Two events never occur at exactly the same

time.3. Each event is independent. That is, the

occurrence of one event does not make thenext event more or less likely to happen.

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Example

λ = (5 defects/hour)*(0.25 hour) =

λ = 1.25

p(x) = (

xe-)/(x!)x = given number of defects

P(x = 0) = (1.25)0e-1.25)/(0!)

= e-1.25 = 0.287

= 28.7%

• A production line produces 600 parts per hour withan average of 5 defective parts an hour. If you test

every part that comes off the line in 15 minutes,what

is the probability of finding no defective parts (andincorrectly concluding that your process is perfect)?

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0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

P r o b a b i l i t y

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

m

binomial

poisson

N=10,p=0.1

0

0.1

0.2

0.3

0.4

0.5

P r o b a b i l i t y

0 1 2 3 4 5m

poisson

binomial

N=3, p=1/3

Comparison of the Binomial & Poisson

Distributions with Mean μ = 1

Clearly, there is not much difference between them!

For N Large & m Fixed:

Binomial

Poisson

N N

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Poisson Distribution: As λ (Average # Counts)

gets large, this also approaches a Gaussian

l

λ = 5 λ = 15

λ = 25 λ = 35

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14

1. The probability an event occurs inthe interval is proportional to thelength of the interval.

2. An infinite number of occurrencesare possible.

3. Events occur independently at a

rate .

Poisson DistributionX=number of occurrences of event in a given time

period

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15

Poisson Distribution

Source: http://en.wikipedia.org/wiki/Poisson_distribution

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16

lλ= np

= 10,000 x 0.00024 = 2.4

e.g. Probability of an accident in a year is

0.00024. So in a town of 10,000, the rate

Poisson Distribution - Example

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17

Poisson with =2.4

Poisson Distribution

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Examples of possible Poisson distributions

1) Number of messages arriving at a telecommunications system in a day

2) Number of flaws in a metre of fibre optic cable

3) Number of radio-active particles detected in a given time

4) Number of photons arriving at a CCD pixel in some exposure time

(e.g. astronomy observations)

Sum of Poisson variables

The probability of events per unit time does not have to be constant for the total number

of events to be Poisson – can split up the total into a sum of the number of events in

smaller intervals.

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Assuming these are independent random events, the number

of people killed in a given year therefore has a Poisson

distribution:

Answer :

Poisson distribution

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1 2 3 4

0% 0%0%0%


Suppose that trucks arrive at a

receiving dock with an average

arrival rate of 3 per hour. What

is the probability exactly 5 trucks

will arrive in a two-hour period?

Question from Derek Bruff

Countdown

20

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Suppose that trucks arrive at a

receiving dock with an average

arrival rate of 3 per hour. What is

the probability exactly 5 trucks

will arrive in a two-hour period?

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Mean and variance

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Example: Telecommunications

Messages arrive at a switching centre at random and at an average rate of 1.2 persecond.

(a) Find the probability of 5 messages arriving in a 2-sec interval.

(b) For how long can the operation of the centre be interrupted, if the probability oflosing one or more messages is to be no more than 0.05?

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Question: (b) For how long can the operation of the centre be interrupted, if the

probability of losing one or more messages is to be no more than 0.05?

(b) Let the required time = t seconds. Average rate of arrival is 1.2/second.

Answer:

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1 2 3 4 5

0% 0%0%0%0%

Poisson or not?

Which of the following are

likely to be well modelled by a

Poisson distribution?

(can click more than one)

Can you tell what is fish, or will you flounder?

1. Number of duds found when I test

four components2. The number of heart attacks in

Brighton each year

3. The number of planes landing at

Heathrow between 8 and 9am4. The number of cars getting

punctures on the M1 each year

5. Number of people in the UK flooded

out of their home in July Countdown

60

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Number of duds found when I test four components

Number of people in the UK flooded out of their home in July

- NO: this is Binomial(it is not the number of independent random events in a continuous interval)

The number of heart attacks in Brighton each year

- YES: large population, no obvious correlations between heart attacks in

different people

The number of planes landing at Heathrow between 8 and 9am

- NO: 8-9am is rush hour, planes land regularly to land as many as possible

(1-2 a minute) – they do not land at random times or they would hit each other!

The number of cars getting punctures on the M1 each year

- YES (roughly): If punctures are due to tires randomly wearing thin, then expect

punctures to happen independently at random But: may not all be independent, e.g. if there is broken glass in one lane

Are they Poisson? Answers:

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Example

The probability of a certain part failing within ten years is 10-6

. Five million of the parts have been sold so far.

What is the probability that three or more will fail within ten years?

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Poisson Distribution Summary

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An example is the improvement of trafficsafety, where the government wants to know

whether seat belts reduce the number of

death in car accidents. Here, the Poissondistribution can be a useful tool to answer

questions about benefits of seat belt use.

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Example

Arrivals at a bus-stop follow a

Poisson distribution with an averageof 4.5 every quarter of an hour.

Obtain a barplot of the distribution(assume a maximum of 20 arrivals in

a quarter of an hour) and calculate

the probability of fewer than 3 arrivalsin a quarter of an hour.

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The probabilities of 0 up to 2 arrivals canbe calculated directly from the formula

( )

!

xe p x

x

4.5 0

4.5(0)0!

e p

with =4.5

So p(0) = 0.01111

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Similarly p(1)=0.04999 and p(2)=0.11248

So the probability of fewer than 3 arrivalsis 0.01111+ 0.04999 + 0.11248 =0.17358

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Consider a collection of graphs for

different values of

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=3

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=4

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=5

6

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=6

10

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=10

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In the last case, the probability of 20

arrivals is no longer negligible, sovalues up to, say, 30 would have to be

considered.

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3. Poisson