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In probability theory and statistics, the Poisson distribution (/ ˈ p w ɑː s ɒ n /) 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]
In probability theory and statistics, the Conway–Maxwell–Poisson (CMP or COM–Poisson) distribution is a discrete probability distribution named after Richard W. Conway, William L. Maxwell, and Siméon Denis Poisson that generalizes the Poisson distribution by adding a parameter to model overdispersion and underdispersion.
The graph of a probability mass function. All the values of this function must be non-negative and sum up to 1. In probability and statistics, a probability mass function (sometimes called probability function or frequency function [1]) is a function that gives the probability that a discrete random variable is exactly equal to some value. [2]
In probability theory, the zero-truncated Poisson distribution (ZTP distribution) is a certain discrete probability distribution whose support is the set of positive integers. This distribution is also known as the conditional Poisson distribution [ 1 ] or the positive Poisson distribution . [ 2 ]
Displaced Poisson distributions for several values of and . At r = 0 {\displaystyle r=0} , the Poisson distribution is recovered. The probability mass function is only defined at integer values.
The probability mass function of a Poisson-distributed random variable with mean μ is given by (;) =!.for (and zero otherwise). The Skellam probability mass function for the difference of two independent counts = is the convolution of two Poisson distributions: (Skellam, 1946)
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.
The reference [13] discusses techniques of evaluating the probability mass function of the Poisson binomial distribution. The following software implementations are based on it: The following software implementations are based on it: