Poisson distribution predicts the degree of spread around a known average rate of occurrence.
Application
The Poisson distribution applies when:
(1) the event 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
(4) it is possible to count how many events have occurred, such as the number of times a firefly lights up in my garden in a given 5 seconds, some evening
The Poisson situation is most often invoked for rare events, and it is only with rare events that it can successfully mimic the binomial distribution.
Poisson Probability
When the average rate of occurrence of some event per module of observation is λ, we can calculate the probability of any given number of actually observed occurrences, x. Then, the Poisson probability is:
P(x; λ) = (e-λ) (λx) / x!
where x is the actual number of successes that result from the experiment, and e is approximately equal to 2.71828.
ex: 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?
Properties of Poisson Distribution
1. Only one parameter: the average frequency of the event (λ)
2. Asymmetrical, skewed toward the infinity end
3. The mean is equal to the variance
