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3.2 Approximation by Poisson Distribution. ... the Poisson binomial distribution is the discrete probability distribution of a sum of ... For example, if n = 3 ...
Therefore, it can be used as an approximation of the binomial distribution if n is sufficiently large and p is sufficiently small. The Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if n ≥ 100 and n p ≤ 10. [31]
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 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
If X is a binomial (n, p) random variable and if n is large and np is small then X approximately has a Poisson(np) distribution. If X is a negative binomial random variable with r large, P near 1, and r(1 − P) = λ, then X approximately has a Poisson distribution with mean λ. Consequences of the CLT:
The benefit of this approximation is that is converted from an exponent to a multiplicative factor. This can greatly simplify mathematical expressions (as in the example below) and is a common tool in physics. [1] The approximation can be proven several ways, and is closely related to the binomial theorem.
The Poisson distribution, which describes a very large number of individually unlikely events that happen in a certain time interval. Related to this distribution are a number of other distributions: the displaced Poisson, the hyper-Poisson, the general Poisson binomial and the Poisson type distributions.
A particular example of this is the binomial test, involving the binomial distribution, as in checking whether a coin is fair. Where extreme accuracy is not necessary, computer calculations for some ranges of parameters may still rely on using continuity corrections to improve accuracy while retaining simplicity.