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 -
If the rate parameter λ is itself random, the resulting marginal distribution can take other forms; for
Estimation of λ from data is commonly done by maximum likelihood, with the MLE for λ being the
Historically, the Poisson distribution is named after Siméon-Denis Poisson, who introduced it in the early 19th