PoissonFamilien

PoissonFamilien, or the Poisson family, denotes the set of Poisson distributions parameterized by a rate λ > 0. The probability that X equals k is P(X=k) = e^{-λ} λ^k / k!, for k = 0,1,2,... This family is used to model counts of independent events occurring in a fixed interval or region when each event occurs with low probability and independently of others.

The mean and variance of X are both λ. The moment generating function is M_X(t) = exp(λ (e^t - 1)). The Poisson distribution is an example of a member of an exponential family of distributions. In the context of a Poisson process, the number of events in a fixed time interval follows a Poisson distribution with parameter λ t, and counts in disjoint intervals are independent.

If the rate parameter λ is itself random, the resulting marginal distribution can take other forms; for example, mixing Poisson with a Gamma distribution yields a Negative Binomial distribution.

Estimation of λ from data is commonly done by maximum likelihood, with the MLE for λ being the sample mean. Confidence intervals can be constructed using normal approximations or exact methods based on the Poisson distribution.

Historically, the Poisson distribution is named after Siméon-Denis Poisson, who introduced it in the early 19th century. It finds wide application in physics, telecommunications, ecology, quality control, and risk assessment where events are rare and independent.