Poisson Distribution

A Poisson distribution is the probability distribution that results from a Poisson experiment.

Attributes of a Poisson Experiment
A Poisson experiment is a statistical experiment that has the following properties:

The experiment results in outcomes that can be classified as successes or failures.
The average number of successes (μ) that occurs in a specified region is known.
The probability that a success will occur is proportional to the size of the region.
The probability that a success will occur in an extremely small region is virtually zero.
Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a period of time, etc.

Notation
The following notation is helpful, when we talk about the Poisson distribution.

e: A constant equal to approximately 2.71828. (Actually, e is the base of the natural logarithm system.) μ: The mean number of successes that occur in a specified region. x: The actual number of successes that occur in a specified region.
P(x; μ): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is μ.
Poisson Distribution
A Poisson random variable is the number of successes that result from a Poisson experiment. The probability distribution of a Poisson random variable is called a Poisson distribution.

Given the mean number of successes (μ) that occur in a specified region, we can compute the Poisson