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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.
A mixed Poisson distribution is a univariate discrete probability distribution in stochastics. It results from assuming that the conditional distribution of a random variable, given the value of the rate parameter, is a Poisson distribution , and that the rate parameter itself is considered as a random variable.
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]
In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. [1] Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters.
Via the law of total cumulance it can be shown that, if the mean of the Poisson distribution λ = 1, the cumulants of Y are the same as the moments of X 1. [citation needed] Every infinitely divisible probability distribution is a limit of compound Poisson distributions. [1] And compound Poisson distributions is infinitely divisible by the ...
One of the limitations of the Poisson distribution is that it assumes equidispersion – the mean and variance of the variable are equal. [2] The displaced Poisson distribution may be useful to model underdispersed or overdispersed data, such as: the distribution of insect populations in crop fields; [3] the number of flowers on plants; [1]
For example, count data are commonly modeled using the Poisson distribution, whose variance is equal to its mean. The distribution may be generalized by allowing for variability in its rate parameter, implemented via a gamma distribution, which results in a marginal negative binomial distribution. This distribution is similar in its shape to ...