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In probability theory and statistics, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed. The concept is named after Siméon Denis Poisson.
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. According to rules of thumb, this approximation is good if n ≥ 20 and p ≤ 0.05 [ 36 ] such that np ≤ 1 , or if n > 50 and p < 0.1 such that np < 5 , [ 37 ...
The parabolic fractal distribution; 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.
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: If X is a Poisson random variable with ...
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
In probability theory and statistics, the Poisson distribution (/ ˈ p w ɑː s ɒ n /; French pronunciation:) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. [1]
The limiting case n −1 = 0 is a Poisson distribution. The negative binomial distributions, (number of failures before r successes with probability p of success on each trial). The special case r = 1 is a geometric distribution. Every cumulant is just r times the corresponding cumulant of the corresponding geometric distribution.
Poisson-type random measures are a family of three random counting measures which are closed under restriction to a subspace, i.e. closed under thinning. They are the only distributions in the canonical non-negative power series family of distributions to possess this property and include the Poisson distribution, negative binomial distribution, and binomial distribution. [1]