# Poisson distribution

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(where $\Gamma(x, y)\,\!$ is the Incomplete gamma function and $\lfloor k\rfloor$ is the floor function)

(for large λ) $\frac{1}{2}\log(2 \pi e \lambda) - \frac{1}{12 \lambda} - \frac{1}{24 \lambda^2} -$
$\frac{19}{360 \lambda^3} + O(\frac{1}{\lambda^4})$

In probability theory and statistics, the Poisson distribution (pronounced [pwasɔ̃]) (or Poisson law of small numbers[1]) is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. (The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.)

The distribution was first introduced 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 matière criminelle et en matière civile (“Research on the Probability of Judgments in Criminal and Civil Matters”). The work focused on certain random variables N that count, among other things, the number of discrete occurrences (sometimes called “arrivals”) that take place during a time-interval of given length.

If the expected number of occurrences in this interval is λ, then the probability that there are exactly k occurrences (k being a non-negative integer, k = 0, 1, 2, ...) is equal to

where

• e is the base of the natural logarithm (e = 2.71828...)
• k is the number of occurrences of an event — the probability of which is given by the function
• 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 4 times per minute, and one is interested in the probability of an event occurring k times in a 10 minute interval, one would use a Poisson distribution as the model with λ = 10×4 = 40.

As a function of k, this is the probability mass function. The Poisson distribution can be derived as a limiting case of the binomial distribution.

The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. A classic example is the nuclear decay of atoms.

The Poisson distribution is sometimes called a Poissonian, analogous to the term Gaussian for a Gauss or normal distribution.
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