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

Trials is Large.  • He derived the cumulative

Poisson distribution as the Limiting case of the Binomial When the Chance of SuccessTends to Zero.

• Poisson Distribution: An approximation to 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 as radioactive 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 = 1.67σ = 1.29

µ = 10.0σ =

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

The Probability of X Numberof Events Occurring in a time T.

• This distribution tells us theProbability 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:

• Here, λ ≡ the average number of counts per second. !

)(kekXpk

Mean and Variance for thePoisson 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

• So, The Standard Deviation is:

2

For a Poisson Distribution, the variance and mean are equal!

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” )

!)()(ketkXP

tk

μ = t σ = t

More on the Poisson Distribution

Example1. If calls to your cell phone are a Poisson process with 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 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% chance2. How many phone calls do you expect to get during the class?

<X> = t = 2(1.5) = 3Editorial comment: The students & the instructor

in the class will not be very happy with you!!

05.!0

)3(!0

)5.1*2()0( 330)5.1(20

eeeXP

7

Conditions Requiredfor 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 the next event more or less likely to happen.

Example

λ = (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 hour with 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)?

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Prob

abili

ty

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

m

binomialpoisson

N=10,p=0.1

0

0.1

0.2

0.3

0.4

0.5

Prob

abilit

y

0 1 2 3 4 5m

poissonbinomial

N=3, p=1/3

Comparison of Binomial & Poisson Distributions with Mean μ = 1

Clearly, there is not much difference between them!

For N Large & m Fixed:Binomial Poisson

N N

Poisson Distribution: As λ (Average # Counts) gets large, this also approaches a Gaussian

l

λ = 5 λ = 15

λ = 25 λ = 35