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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 poisson process is a process where events occur randomly in an interval of time or space. [2] [8] The probability distribution for Poisson processes with constant rate λ per time interval is given by the following equation. [4] =!
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 Blackwell-Girshick equation is used in actuarial mathematics to calculate the variance of composite distributions, such as the compound Poisson distribution. Wald's equation provides similar statements about the expectation of composite distributions.
In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator . The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own ...
The (a,b,0) class of distributions is also known as the Panjer, [1] [2] the Poisson-type or the Katz family of distributions, [3] [4] and may be retrieved through the Conway–Maxwell–Poisson distribution. Only the Poisson, binomial and negative binomial distributions satisfy the full form of this
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 ...
The function is the intensity of an underlying Poisson process. The first arrival occurs at time t 1 {\textstyle t_{1}} and immediately after that, the intensity becomes μ ( t ) + ϕ ( t − t 1 ) {\textstyle \mu (t)+\phi (t-t_{1})} , and at the time t 2 {\textstyle t_{2}} of the second arrival the intensity jumps to μ ( t ) + ϕ ( t − t 1 ...