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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]
The distribution is also applicable to a special case of the difference of dependent Poisson random variables, but just the obvious case where the two variables have a common additive random contribution which is cancelled by the differencing: see Karlis & Ntzoufras (2003) for details and an application.
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.
If X 1 and X 2 are Poisson random variables with means μ 1 and μ 2 respectively, then X 1 + X 2 is a Poisson random variable with mean μ 1 + μ 2. The sum of gamma (α i, β) random variables has a gamma (Σα i, β) distribution. If X 1 is a Cauchy (μ 1, σ 1) random variable and X 2 is a Cauchy (μ 2, σ 2), then X 1 + X 2 is a Cauchy (μ ...
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
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 probability theory, the probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density ...
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 ...