Attributes of a Poisson Experiment

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.

The Poisson distribution is an appropriate model for count data. Examples of such data are mortality of infants in a city, the number of misprints in a book, the number of bacteria on a plate, 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.

The only thing we have to know to specify the Poisson distribution is the mean number of occurrences. In the graph the mean number of events is equal to 1/2 From the picture we learn the probability of zero events .60, the probability of one event is .30, and so on. Generally, for small values of μ, the distribution is not symmetric but skewed. This is a general property when the mean is small. The distribution becomes more symmetric when the mean is larger. A property of this distribution is that the variance is equal to the mean. The Poisson distribution resembles the binomial distribution if the probability of an event is very small

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 Formula. Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is μ. Then, the Poisson probability is:

P(x; μ) = (e^-μ) (μ^x) / x!

where x is the actual number of successes that result from the experiment, and e is approximately equal to 2.71828.

The Poisson distribution has the following properties:

• The mean of the distribution is equal to μ .
• The variance is also equal to μ

Example 1

The average number of homes sold by the Acme Realty company is 2 homes per day. What is the probability that exactly 3 homes will be sold tomorrow?

Solution: This is a Poisson experiment in which we know the following:

• μ = 2; since 2 homes are sold per day, on average.
• x = 3; since we want to find the likelihood that 3 homes will be sold tomorrow.
• e = 2.71828; since e is a constant equal to approximately 2.71828.

We plug these values into the Poisson formula as follows:

P(x; μ) = (e^-μ) (μ^x) / x!

P(3; 2) = (2.71828^-2) (2³) / 3!

P(3; 2) = (0.13534) (8) / 6

P(3; 2) = 0.180

Cumulative Poisson Probability

A cumulative Poisson probability refers to the probability that the Poisson random variable is greater than some specified lower limit and less than some specified upper limit.