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In probability theory, the law of rare events or Poisson limit theorem states that the Poisson distribution may be used as an approximation to the binomial distribution, under certain conditions. [1] The theorem was named after Siméon Denis Poisson (1781–1840). A generalization of this theorem is Le Cam's theorem
The free Poisson distribution [40] with jump size and rate arises in free probability theory as the limit of repeated free convolution (() +) as N → ∞. In other words, let X N {\displaystyle X_{N}} be random variables so that X N {\displaystyle X_{N}} has value α {\displaystyle \alpha } with probability λ N {\textstyle {\frac {\lambda }{N ...
The binomial distribution converges towards the Poisson distribution as the number of trials goes to infinity while the product np converges to a finite limit. Therefore, the Poisson distribution with parameter λ = np can be used as an approximation to B( n , p ) of the binomial distribution if n is sufficiently large and p is sufficiently small.
The formula in the definition of characteristic function allows us to compute φ when we know the distribution function F (or density f). If, on the other hand, we know the characteristic function φ and want to find the corresponding distribution function, then one of the following inversion theorems can be used. Theorem.
An alternative version uses the fact that the Poisson distribution converges to a normal distribution by the Central Limit Theorem. [5]Since the Poisson distribution with parameter converges to a normal distribution with mean and variance , their density functions will be approximately the same:
Fisher–Tippett–Gnedenko theorem – limit theorem for extremum values (such as max{X n}) Irwin–Hall distribution; Markov chain central limit theorem; Normal distribution; Tweedie convergence theorem – a theorem that can be considered to bridge between the central limit theorem and the Poisson convergence theorem [56] Donsker's theorem
The negative binomial distribution has a variance /, with the distribution becoming identical to Poisson in the limit for a given mean (i.e. when the failures are increasingly rare). This can make the distribution a useful overdispersed alternative to the Poisson distribution, for example for a robust modification of Poisson regression .
This is justified by considering the central limit theorem in the log domain (sometimes called Gibrat's law). The log-normal distribution is the maximum entropy probability distribution for a random variate X —for which the mean and variance of ln( X ) are specified.