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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 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)
The Skellam distribution, the distribution of the difference between two independent Poisson-distributed random variables. The skew elliptical distribution; The Yule–Simon distribution; The zeta distribution has uses in applied statistics and statistical mechanics, and perhaps may be of interest to number theorists.
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 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 functions respectively.
If a random variable admits a probability density function, then the characteristic function is the Fourier transform (with sign reversal) of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions .
A visual depiction of a Poisson point process starting. In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...