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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 ...
A Poisson (counting) process on the line can be characterised by two properties : the number of points (or events) in disjoint intervals are independent and have a Poisson distribution. A Poisson point process can also be defined using these two properties. Namely, we say that a point process is a Poisson point process if the following two ...
Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric.It was introduced by Charles Stein, who first published it in 1972, [1] to obtain a bound between the distribution of a sum of -dependent sequence of random variables and a standard normal distribution in the Kolmogorov (uniform ...
In queueing theory, a discipline within the mathematical theory of probability, a Jackson network (sometimes Jacksonian network [1]) is a class of queueing network where the equilibrium distribution is particularly simple to compute as the network has a product-form solution.
The renewal process is a generalization of the Poisson process. In essence, the Poisson process is a continuous-time Markov process on the positive integers (usually starting at zero) which has independent exponentially distributed holding times at each integer i {\displaystyle i} before advancing to the next integer, i + 1 {\displaystyle i+1} .
The thinning operation entails using some predefined rule to remove points from a point process to form a new point process .These thinning rules may be deterministic, that is, not random, which is the case for one of the simplest rules known as -thinning: [1] each point of is independently removed (or kept) with some probability (or ).
For a Poisson point process, the J function is simply J(r) = 1, hence why it is used as a non-parametric test for whether data behaves as though it were from a Poisson process. It is, however, thought possible to construct non-Poisson point processes for which J ( r ) = 1, [ 10 ] but such counterexamples are viewed as somewhat 'artificial' by ...
Mixed Poisson processes are doubly stochastic in the sense that in a first step, the value of the random variable is determined. This value then determines the "second order stochasticity" by increasing or decreasing the original intensity measure μ {\displaystyle \mu } .