The Poisson probability distribution, first described by Siméon Poisson in 1838, is widely used to model and simulate many real events. It is especially useful to describe low frequency events, and for higher-frequency events it becomes very similar to a Gaussian probability distribution.
What Is The Poisson Distribution?
The Poisson probability distribution is used to describe discreet events, such as number of defects per metre, number of mortgages granted per week, or the number of times a server is accessed per second. When the average number of events for a given unit is known, it is possible to estimate the probability that the event will occur any other number of times.
The Poisson distribution has the property that the standard deviation is a function only of the mean:
σ² = µ
Siméon Denis Poisson Short Biography
Siméon Denis Poisson was born on the 21st June, 1781, in Loiret, France. His brilliance was spotted by the great French mathematicians Adrien-Marie Legendre, Joseph Louis Lagrange, and Pierre-Simon Laplace, at L'Ecole Polytechnique in Paris. Indications of his brilliance surfaced early, when Legendre recommended that his memoir on Bézout's "method of elimination" be published in the prestigious "Recueil des savants étrangers". As well as developing his theory on probability, and the Poisson Distribution in particular, he published hundreds of significant papers, but he is best known for correcting Laplace's equation for potential ².
Poisson Distribution Formula
If an event happens, on average, µ times per unit, then the probability that the event will happen n times in the same unit is given by the formula
e^(-µ).µ^n / n!
Where
e is the base of natural logarithms, or 2.71828
n is a positive integer
n! = factorial n, i.e. n × (n-1) × (n-2) ... 3.2.1
It will be noted that since e^µ = 1 + µ + µ² / 2! + µ³ / 3! + ... the term e^(-µ) acts like a "normalizing factor", that makes the sum of the probabilities equal to 1, since e^µ × e^(-µ) = 1.
Poisson Distribution Uses
The Poisson distribution is particularly useful because there is only one parameter, µ, to consider. When the average number of occurrences is known, then probabilities may be calculated on which decisions may be based. For example, if the average number of emergency surgeries per day is 2.5 then the probability that there will be zero is:
e^(-2.5) × 2.5^0 / 0!
= 0.082 × 1 / 1 (2.5^0 = 1 and 0! = 1)
= 0.082
The probability that there will be four emergency surgeries is
e^(-2.5) × 2.5^4 / 4!
= 0.082 × 39.063 / 24
= 0.134
The same method is used to calculate the probability for other numbers.
Knowing what the probability of each number of operations is allows managers to plan for eventualities: if there is a legal requirement, for example, that a hospital theater must be able to cope with admissions 99% of the time, then their capacity plan must reflect this.
Poisson Distribution Summary
The Poisson probability distribution was first described by Siméon Denis Poisson in 1838. It is useful to examine the probabilities of discreet events that occur a known average number of times per unit time, length, volume etc. It is often used in modeling and simulations.
Many references to the Poisson distribution refer to the mean by the Greek letter lambda (λ). The Microsoft Excel function POISSON is easily used.
Poisson Distribution References
¹ - Poisson, S. D., Recherches sur la Probabilité des Jugements en Matière Criminelle et en Matière Civile (Researches on the Probability of Criminal and Civil Verdicts), 1838
Martin Bell - Martin holds a B.Sc. degree in chemical engineering, and an M.Sc. degree in electronics and computing. He has spent more than 25 years ...
Join the Conversation