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Chapter 5 Random Variables

 

5.2 Poisson distribution

 

Some events are rather rare, they don't happen that often. For instance, car accidents are the exception rather than the rule. Still, over a period of time, we can say something about the nature of rare events. An example is the improvement of traffic safety, where the government wants to know if wearing seat belts reduce the number of death in car accidents. Here, the Poisson distribution can be a useful tool to answer question about benefits of seat belt use. 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 the number of activations of a geiger counter. The Poisson distribution was derived by the French mathematician Poisson in 1837, and the first application was the description of the number of death by horse kicking in the prussian army (Bortkiewicz, 1898).

The Poisson distribution is a mathematical rule that assigns probabilities to the number occurrences. The only thing we have to know to specify the Poisson distribution is the mean number of occurrences for which the symbol lambda ($ \lambda$) is often used.

The Poisson distribution resembles the binomial distribution in that is models counts of events. For example, a Poisson distribution could be used to model the number of accidents at an intersection in a week. However, if we want to use the binomial distribution we have to know both the number of people who make enter the intersection, and the number of people who have an accident at the intersection, whereas the number of accidents is sufficient for applying the Poisson distribution. Thus, the Poisson distribution is cheaper to use because the number of accidents is usually recorded by the police department, whereas the total number of drivers is not.

Conditions underwhich a Poisson distribution holds

·         counts of rare events

·         all events are independent

·         average rate does not change over the period of interest

Examples of experiments where a Poisson distribution holds:

·         birth defects

·         number of sample defects on a car

·         number of typographical errors on a page

Examples of experiments where a Poisson distribution may not hold

·         number of insects on a tree - contagion?

·         number of males in families of size 4 - not `rare' events

Computing Poisson probabilities

·         Parameters

o        $ \lambda$= rate of occurrence for this sample = standardized rate x sample size

o        $ \lambda$can range from 0 upwards and does NOT have to be an integer.

o        X must be an integer = 0, 1, 2, 3, .....

·         Poisson formula

         -don't worry about formula;

          We won't be computing it by hand.

·         Some values of the distribution have been tabulated here .

Properties of Poisson experiment

·         If X is Poisson (with parameter $ \lambda$) then expected value for the Poisson Distributed random variable and variance

Examples of Poisson application :

·         birth defects

·         insect parts in chocolate bars

·         drunk drivers stopped at ALERT stations

 

Applications

Example

The Poisson distribution can be applied to the hit rate on a web site.  The graph below shows the distribution of hits/hour on a recipe web-site before the evening feed.

Profile

 

Applications

The Poisson process finds application in diverse areas of science such as physics, teletraffic modeling, queueing theory, manufacturing, and biology. From http://www.bath.ac.uk/~ma1lt/examples.html

·         INDUSTRY - Finding the reliability of machinery by looking at the number of breakdowns in a given period.

·         AGRICULTURE AND ECOLOGY - The distribution of plants and animals in space often follows a Poisson Distribution.

·         BIOLOGY - The sampling of bacteria in a given volume, and estimating numbers in dilution series.

·         MEDICINE - The Poisson Distribution can be used to count the number of victims of specific diseases, such as the number of cancer deaths per house, or the number of malaria deaths per year.

·         TELEPHONERY - The number of calls placed in a given time are Poisson distributed, this is used to work out the number of lines available, for example in a switchboard.

·         QUEUEING THEORY - The Poisson Distribution can be applied to many queues including: doctor's waiting rooms and hospitals, road, rail and air traffic queues.

·         OTHER EXAMPLES - Number of words misread in a text; number of misprints in text; number of times certain words appear in text; the amount of rain in a month; the number of reported fires in a fixed time period.

Problems

1.       Events occur according to a Poisson process with rate 2 per hour.
(a) What is the probability that no event occurs between 8 p.m. and 9 p.m.?
(b) What is the probability that two or more events occur between 6 p.m. and 8 p.m.?

Solution

(a)
Poisson Process - Problem 1a
(b)
Poisson Process - Problem 1b

2.       Many internet service providers suggest modeling internet behavior using a Poisson Distribution. Suppose the number of times the University of Massachusetts Dartmouth's home page is accessed over the noon hour (EST) has an average rate of 10 times per hour.
(a) What is the probability no more than 10 hits occur during the noon hour?
(b) What is the probability that the number of hits will exceed 15 during the noon hour?

Solution

Poisson(λt)=Poisson(10*1)

References

1.       Engineering Statistics Handbook

2.       Kvanli. et al (2003) Intoduction to Business Statistics. Thomson.

 

The Poisson distribution is the continuous limit of the discrete binomial distribution. It depends on the following four assumptions:

1.       It is possible to divide the time interval of interest into many small subintervals (like an hour into seconds).

2.       The probability of an occurrence remains constant throughout the large time interval (random).

3.       The probability of two or more occurrences in a subinterval is small enough to be ignored.

4.       Occurrences are independent.

Clearly, bank arrivals might have problems with assumption number four where payday, lunch hour, and car pooling may affect independence. However, the Poisson Distribution finds applicability in a surprisingly large variety of situations.

The equation for the Poisson Distribution is:

P(x) = µx · e-µ ÷ x!

The number e in the equation above is the base of the natural logarithms or approximately 2.71828182845904523... The variance is equal to the mean. In fact, this can be a quick check to see if this distribution can be applied. Traditionally the Greek letter lambda () is often used instead of µ. The differences between the Poisson distribution and the binomial distribution are:

1.       The binomial distribution is affected by the sample size and the probability while the Poisson distribution is ONLY affected by the mean.

2.       The binomial distribution has values from x = 0 to n but the Poisson distribution has values from x = 0 to infinity.

Example: On average there are three babies born a day with hairy backs. Find the probability that in one day two babies are born hairy. Find the probability that in one day no babies are born hairy.
Solution: a. P(2) = 32 · e-3 ÷ 2 = .224     b. P(0) = 30 · e-3 = .0498

Example: Suppose a bank knows that on average 60 customers arrive between 10 A.M. and 11 A.M. daily. Thus 1 customer arrives per minute. Find the probability that exactly two customers arrive in a given one-minute time interval between 10 and 11 A.M.
Solution: Let µ = 1 and x = 2. P(2)=e-1/2!=0.3679÷2=0.1839.

Example: Other examples include, the number of typographical errors on a page, the number of white blood cells in a blood suspension, or the number of imperfections in a surface of wood or metal. I assume one could apply it to finding four-leaf clovers, but a corresponding class activity has not yet been developed.

Homework for Poisson Distribution

  1. Suppose a bank knows that on average 60 customers arrive in a certain service hour. Using a time interval of 1 minute, calculate the probability of exactly one customer arriving in a given one minute interval within that hour. Use the example in the lecture for exactly two as a pattern. 

 

  1. Suppose a bank knows that on average 60 customers arrive in a certain service hour. Using a time interval of 1 minute, calculate the probability of no customers arriving in a given one minute interval within that hour.  

 

  1. Suppose a bank knows that on average 60 customers arrive in a certain service hour. Using a time interval of 1 minute, calculate the probability of exactly three customers arriving in a given one minute interval within that hour.

 

  1. Suppose a bank knows that on average 60 customers arrive in a certain service hour. Using a time interval of 1 minute, calculate the probability of more than three customers arriving in a given one minute interval within that hour.

 

  1. Graph the probability distribution determined in the problems above (and the lecture example).