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The Poisson Distribution is a discrete distribution. It is named after Simeon-Denis Poisson (1781-1840), a French mathematician, who published its essentials in a paper in 1837. The Poisson distribution and the binomial distribution have some similarities, but also several differences.

The binomial distribution describes a distribution of two possible outcomes designated as successes and failures from a given number of trials. The Poisson distribution focuses only on the number of discrete occurrences over some interval. A Poisson experiment does not have a given have a given number of trials (*n*) as binomial experiment does. For example, whereas a binomial experiment might be used to determine how many black cars are in a random sample of 50 cars, a Poisson experiment might focus on the number of cars randomly arriving at a car wash during a 20-minute interval.

The Poisson distribution has the following characteristics:

- It is a discrete distribution.
- Each occurrence is independent of the occurrence.
- It describes discrete occurrences over an interval.
- The occurrences in each interval can range from zero to infinity.
- The mean number of occurrences must be constant throughout the experiment.

The Poisson distribution is characterized by lambda , the mean number of occurrences in the interval. If a Poisson-distributed phenomenon is studied over a long period of time, is the long-run average of the process. The Poisson formula is used to compute the probability of occurrences over an interval for a given lambda value.

The Poisson Formula is

where

*x* = 0, 1, 2, 3, …

= mean number of occurrences in the interval

e = 2.718282

The following two videos show how to derive the Poisson Formula from the Binomial Formula.

Introduction to Poisson Processes and the Poisson Distribution.

More of the derivation of the Poisson Distribution.

The following video will discuss a situation that can be modeled by a Poisson Distribution, give the formula, and do a simple example illustrating the Poisson Distribution.

Suppose a fast food restaurant can expect two customers every 3 minutes, on average. What is the probability
that four or fewer patrons will enter the restaurant in a 9 minute period?

An Introduction to the Poisson Distribution

An introduction to the Poisson distribution. This video discusses the conditions required for a random variable to have a Poisson distribution. work through a simple calculation example, and briefly discuss the relationship between the binomial distribution and the Poisson distributions.

Statistics 101: Introduction to the Poisson Distribution

In this video, we discuss the basic characteristics of the Poisson Distribution using a real-world example involving a checkout line at a supermarket. A basic understanding of the binomial distribution is helpful, but not necessary. It will also show you how to calculate Poisson probabilities on at TI calculator.

Statistics 101: Poisson Practice Problems

This video goes through two practice problems involving the Poisson Distribution. The first problem examines customer arrivals to a bank ATM and the second analyzes deer-strike probabilities along sections of a rural highway. You are assumed to have a basic understanding of the Poisson Distribution.

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

*(Clicking "View Steps" on the answer screen will take you to the Mathway site, where you can register for a free ten-day trial of the software.)*

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