Poisson Distribution

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Poisson Distribution Overview


References: Dr. Mikel J. Harry, Juran's Quality Control Handbook

Purpose

The Poisson distribution is used to calculate the probability of occurrence of an event in a population when there are many opportunities, but the probability of each trial is low (less than 0.10). To describe the behavior of discrete variables when the above conditions are met. A discrete random variable only takes whole values.

Suppose you typically get 4 blemishes on a critical surface per day. That becomes your expectation, but there will be a certain spread: sometimes you'll see a little more and other times a little less, once in a while nothing at all.

Given only the average rate, for a certain period of observation and assuming that the process, or mix of processes, that produce the event flow are essentially random, the Poisson distribution will tell you how likely it is that you will get 3, or 5, or 11, or any other number, during one period of observation.

That is, it predicts the degree of spread around a known average rate of occurrence. The average rate of occurrence is the hump on each of the Poisson curves shown below. 


Anatomy

Poisson DistributionPoisson Distribution

Terminology

A. Vertical axis - Scale to measure the probability of occurrence of an event.

B. Horizontal axis - Scale of measure the number of occurrences.

C. Probability - The area under the curve represents the probability of occurrence of an event. The total area under the curve is equal to one (1).

D. Curve of the distribution for various levels of lambda - The Poisson distribution is a probability distribution for the number of occurrences per unit interval which can be a unit of time or space. The distribution is a good approximation of the binomial distribution for the case where n is large and p is small.

E. Lambda (Parameter which represents the average number of occurrences per interval).

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Poisson Distribution Formula

The Poisson distribution is applicable when:

  1. the event is something that can be counted in whole numbers; 
  2. occurrences are independent, so that one occurrence neither diminishes nor increases the chance of another;
  3. the average frequency of occurrence for the time period in question is known; and
  4. it is possible to count how many events have occurred, such as the number of surface blemishes on a panel, but meaningless to ask how many such events have not occurred.

Major Considerations

Applicable when sample size is at least 16, the population size is at least 10 times the sample size and the probability of occurrence p on each trial is less than 0.1.

Events occur at random order and they are roughly proportional to the length of time, volume of space or area under study. Also, there is no overlapping of events (“clumping”).

Application Cookbook

1. Use Excel functions to calculate the probability of an occurrence for a discrete variable that follows the Poisson distribution. For example, the probability that 12 or less occurrences of an event that has an average number of occurrences of 5.2 is equal to 0.997.

2. In Minitab use the following menu to generate a Poisson distribution. CALC>PROBABILITY DISTRIBUTIONS>POISSON
3. Alternatively, the tables printed at the end of most statistics books can be used.

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