A classic example used to motivate the Poisson distribution is the number of radioactive decay events during a fixed observation period.

[5]: 219 [6]: 14-15 [7]: 193 [8]: 157  This makes it an example of Stigler's law and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre.

[9][10] In 1860, Simon Newcomb fitted the Poisson distribution to the number of stars found in a unit of space.

Bortkiewicz showed that the frequency with which soldiers in the Prussian army were accidentally killed by horse kicks could be well modeled by a Poisson distribution.[12]: 23-25 .

if it has a probability mass function given by:[2]: 60 where The positive real number λ is equal to the expected value of X and also to its variance.

It has been reported that the average number of goals in a World Cup soccer match is approximately 2.5 and the Poisson model is appropriate.

The higher non-centered moments mk of the Poisson distribution are Touchard polynomials in λ:

Obtaining the sign of the second derivative of L at the stationary point will determine what kind of extreme value λ is.

Evaluating the second derivative at the stationary point gives: which is the negative of n times the reciprocal of the average of the ki.

is the quantile function (corresponding to a lower tail area p) of the chi-squared distribution with n degrees of freedom and

When quantiles of the gamma distribution are not available, an accurate approximation to this exact interval has been proposed (based on the Wilson–Hilferty transformation):[46] where

[47] Let denote that λ is distributed according to the gamma density g parameterized in terms of a shape parameter α and an inverse scale parameter β: Then, given the same sample of n measured values ki as before, and a prior of Gamma(α, β), the posterior distribution is Note that the posterior mean is linear and is given by It can be shown that gamma distribution is the only prior that induces linearity of the conditional mean.

Moreover, a converse result exists which states that if the conditional mean is close to a linear function in the

as[51] Some applications of the Poisson distribution to count data (number of events):[52] More examples of counting events that may be modelled as Poisson processes include: In probabilistic number theory, Gallagher showed in 1976 that, if a certain version of the unproved prime r-tuple conjecture holds,[61] then the counts of prime numbers in short intervals would obey a Poisson distribution.

In the case of the Poisson distribution, one assumes that there exists a small enough subinterval for which the probability of an event occurring twice is "negligible".

With this assumption one can derive the Poisson distribution from the binomial one, given only the information of expected number of total events in the whole interval.

In several of the above examples — such as, the number of mutations in a given sequence of DNA—the events being counted are actually the outcomes of discrete trials, and would more precisely be modelled using the binomial distribution, that is

For example, the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution with the events appearing frequent to the operator, but they are rare from the point of view of the average member of the population who is very unlikely to make a call to that switchboard in that hour.

The Law of Small Numbers is a book by Ladislaus Bortkiewicz about the Poisson distribution, published in 1898.

More specifically, if D is some region space, for example Euclidean space Rd, for which |D|, the area, volume or, more generally, the Lebesgue measure of the region is finite, and if N(D) denotes the number of points in D, then Poisson regression and negative binomial regression are useful for analyses where the dependent (response) variable is the count (0, 1, 2, ... ) of the number of events or occurrences in an interval.

[65] When ACh is present, ion channels on the membrane would be open randomly at a small fraction of the time.

Subtracting the effect of noise, Katz and Miledi found the mean and variance of membrane potential to be

(p. 64 [66]) In a Poisson process, the number of observed occurrences fluctuates about its mean λ with a standard deviation

The correlation of the mean and standard deviation in counting independent discrete occurrences is useful scientifically.

For example, the charge e on an electron can be estimated by correlating the magnitude of an electric current with its shot noise.

By correlating the graininess with the degree of enlargement, one can estimate the contribution of an individual grain (which is otherwise too small to be seen unaided).

[citation needed] In causal set theory the discrete elements of spacetime follow a Poisson distribution in the volume.

Namely, for a quantum harmonic oscillator system in a coherent state, the probability of measuring a particular energy level has a Poisson distribution.

However, the conventional definition of the Poisson distribution contains two terms that can easily overflow on computers: λk and k!.

Other solutions for large values of λ include rejection sampling and using Gaussian approximation.