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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]
A discrete power-law distribution, the most famous example of which is the description of the frequency of words in the English language. The Zipf–Mandelbrot law is a discrete power law distribution which is a generalization of the Zipf distribution. Conway–Maxwell–Poisson distribution Poisson distribution Skellam distribution
This distribution is also known as the conditional Poisson distribution [1] or the positive Poisson distribution. [2] It is the conditional probability distribution of a Poisson-distributed random variable, given that the value of the random variable is not zero. Thus it is impossible for a ZTP random variable to be zero.
The shift geometric distribution is discrete compound Poisson distribution since it is a trivial case of negative binomial distribution. This distribution can model batch arrivals (such as in a bulk queue [5] [9]). The discrete compound Poisson distribution is also widely used in actuarial science for modelling the distribution of the total ...
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.
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.
Returning to our example, if we pick the Gamma distribution as our prior distribution over the rate of the Poisson distributions, then the posterior predictive is the negative binomial distribution, as can be seen from the table below.
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]