Often one has to compute the probability of getting exactly n occurrences over a period of time or other continuum, for example:
a biologist wants to know if a patient's blood is normal by counting the red corpuscles in the squares of a ruled graticule applied on a slide containing diluted blood;
a quality control professional is asked to accept a batch of electronic components that the supplier guarantees as having at most 233 defectives per million;
an insurance claims officer assesses the likelihood of a home insurance holder having 1 or 2 fires over 5 years;
a highway engineer needs to know the probability of having more than 10 vehicles arriving at a tollgate in 1 minute;
a facilities manager wants to know if 2 switchboard operators can handle 95% of incoming calls;
a design engineer wants to establish the failure rate of a given product with a 95% confidence level;
a hospital manager needs to set up an emergency room knowing that the average rate of arrivals is 1.5 per 1 hour.
The Poisson distribution, named after the 19th century French mathematician Siméon Poisson, applies to such cases as these, where the events take place independently, at a known constant rate derived from a large sample (practically > 50), and with a low probability of occurrence (practically < 10%). The Poisson distribution is discrete, meaning that n can take only certain values, is defined by only 1 parameter (λ or mean), and mean and variance are equal (λ = σ²). For small values of p, Poisson is a good approximation for the binomial distribution.
The solution of the Poisson distribution function is given by the formula :
For: λ = np = mean; n = random variable; e = constant equal to approximately 2.718282.