Poisson distribution

In statistics and probability theory, the Poisson distribution is a discrete probability distribution (discovered by Siméon-Denis Poisson (1781-1840) and published, together with his probability theory, in 1838 in his work Recherches sur la probabilité des jugements en matières crimineiles et matiere civile) belonging to certain random variables N that count, among other things, a number of discrete occurrences (sometimes called "arrivals") that take place during a time-interval of given length. The probability that there are exactly k occurrences (k being a natural number including 0, k = 0, 1, 2, ...) is:

<math>P(N=k)=\frac{e^{-\lambda}\lambda^k}{k!}.</math>

Where:

e is the base of the natural logarithm (e = 2.71828...),
k! is the factorial of k,
λ is a positive real number, equal to the expected number of occurrences that occur during the given interval. For instance, if the events occur on average every 2 minutes, and you are interested in the number of events occcurring in a 10 minute interval, you would use as model a Poisson distribution with λ = 5.

Sometimes λ is taken to be the rate, i.e., the average number of occurrences per unit time. In that case, if N_t is the number of occurrences before time t then we have

<math>P(N_t=k)=\frac{e^{-\lambda t}(\lambda t)^k}{k!},</math>

and the waiting time T until the first occurrence is a continuous random variable with an exponential distribution; this probability distribution may be deduced from the fact that

Occurrence

The Poisson distribution arises in connection with Poisson processes. It applies to various phenomena of discrete nature (that is, those that may happen 0, 1, 2, 3, ... times during a given period of time or in a given area) whenever the probability of the phenomenon happening is constant in time or space. Examples include:

The number of unstable nuclei that decayed within a given period of time in a piece of radioactive substance.
The number of cars that pass through a certain point on a road during a given period of time.
The number of spelling mistakes a secretary makes while typing a single page.
The number of phone calls you get per day.
The number of times your web server is accessed per minute.
The number of roadkill[?] you find per mile of road.
The number of mutations in a given stretch of DNA after a certain amount of radiation.
The number of pine trees per square mile of mixed forest.
The number of stars in a given volume of space.
(An example made famous by a book of Ladislaus Josephovich Bortkiewicz (1868-1931): The number of soldiers killed by horse-kicks each year in each corps in the Prussian cavalry.
The number of bombs falling on each square mile of London during a German air raid in the early part of the Second World War.

How does this distribution arise? -- The limit theorem

The binomial distribution with parameters n and λ/n, i.e., the probability distribution of the number of successes in n trials, with probability λ/n of success on each trial, approaches the Poisson distribution with expected value λ as n approaches infinity.

Properties

The expected value of a Poisson distributed random variable is equal to λ and so is its variance.

The most likely value ("mode") of a Poisson distributed random variable is equal to the largest integer ≤ λ, which is also written as floor(λ).

If λ is big enough (λ > 10 say), then the Normal distribution with mean λ and standard deviation √ λ is an excellent approximation to the Poisson distribution.

If N and M are two independent random variables, both following a Poisson distribution with parameters λ and μ, respectively, then N + M follows a Poisson distribution with parameter λ + μ.

The word law is sometimes used as a synonym of probability distribution, and convergence in law means convergence in distribution. Accordingly, the Poisson distribution is sometimes called the law of small numbers because it is the probability distribution of the number of occurrences of an event that happens rarely but has very many opportunities to happen. The Law of Small Numbers is the title of a book by Ladislaus Bortkiewicz about the Poisson distribution, published in 1898. Some historians of mathematics have argued that the Poisson distribution should have been called the Bortkiewicz distribution.