Lecturpoission distributione

8/12/2019 Lecturpoission distributione

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The Poisson Distribution(Not in Reifs book)


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The Poisson Probability Distribution

Simeon Denis Poisson "Researches on the probability

of criminal civil verdicts" 1837

Looked at the form of the

binomial distribution

When the Number of

Tr ials is Large. He derived the cumulative

Poisson distribution as the

L imiting case of theBinomial When the

Chance of Success

Tends to Zero.


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The Poisson Probability Distribution

Simeon Denis Poisson "Researches on the probability

of criminal civil verdicts" 1837

Looked at the form of the

binomial distribution

When the Number of

Tr ials is Large. He derived the cumulative

Poisson distribution as the

L imiting case of theBinomial When the

Chance of Success

Tends to Zero.Simeon Denis Fish!


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Poisson Distribution:An approximation to binomial

distribution for the SPECIAL CASEwhen the average

number (mean ) of successes is very much smal ler thanthe possible numbern. i.e.


5/12

The Poisson Distribution Models Counts.If events happen at a constant rate over time, the Poisson

Distributiongives

The Probabil i ty of X Number

of Events Occur r ing in a time T. This distribution tells us the

Probabil i ty of All Possible Numbers ofCounts, from 0 to I nf ini ty.

If X= #of counts per second, then the Poisson probability

that X = k(a particular count) is:

Here, the average number of counts per second.

!)(

k

ekXp

k


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Mean and Variance for thePoisson Distribution

Its easy to show that for this distribution,

The Meanis:

Also, its easy to show that The Varianceis:.l

L

So, TheStandard Deviationis:

2

For a Poisson Distr ibution, thevariance and mean are equal!


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Terminology: A Poisson Process

The Poisson parameter can be given as the meannumber of events that occur in a defined time period OR,

equivalently, can be given as a rate, such as = 2events per month. must often be multiplied by a timetin a physical process

(called a Poisson Process)

!

)()( k

etkXP

tk

= t = t

More on thePoisson Distribution


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Example1. If calls to your cell phone are a Poisson process with a

constant rate = 2 calls per hour, what is the probabilitythat, i f you forget to turn your phone offin a 1.5 hour class,your phone rings during that time?

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

P(X 1) = 1 P(X = 0)

P(X 1) = 1 .05 = 95% chance

2.How many phone calls do you expect to get during the class? = t = 2(1.5) = 3

Editorial comment:The students & the instructor in the

class wil l not be very happy with you! !

05.!0

)3(

!0

)5.1*2()0(

330)5.1(20

eee

XP


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9

Conditions Requiredfor the

Poisson Distr ibution to hold:

1. The rate is a constant, independent of time.2.Two events never occurat exactly the same time.

3. Each event is independent. That is, the

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


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Example

= (5 defects/hour)*(0.25 hour)= 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 hourwith

an 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(and incorrectly 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

Probability

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

m

bino mialpo isso n 1

N=10,p=0. 1

0

0.1

0.2

0.3

0.4

0.5

Probability

0 1 2 3 4 5m

poisson

binomial

1

N=3, p=1/3

Comparison of Binomial & Poisson Distributions

with Mean = 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 (Average # Counts) gets large,

this also approaches a Gaussian

l

= 5 = 15

= 25 = 35

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