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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]
For example: A point is a degenerate circle, whose radius is zero. A parabola can degenerate into two distinct or coinciding parallel lines. An ellipse can degenerate into a single point or a line segment. A hyperbola can degenerate into two intersecting lines.
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 ...
Related to this distribution are a number of other distributions: the displaced Poisson, the hyper-Poisson, the general Poisson binomial and the Poisson type distributions. The Conway–Maxwell–Poisson distribution, a two-parameter extension of the Poisson distribution with an adjustable rate of decay.
The Poisson point process can be defined, studied and used in one dimension, for example, on the real line, where it can be interpreted as a counting process or part of a queueing model; [31] [32] in higher dimensions such as the plane where it plays a role in stochastic geometry [1] and spatial statistics; [33] or on more general mathematical ...
Similar to the examples described above, we consider x, y, φ to be independent uniform random variables over the ranges 0 ≤ x ≤ a, 0 ≤ y ≤ b, − π / 2 ≤ φ ≤ π / 2 . To solve such a problem, we first compute the probability that the needle crosses no lines, and then we take its complement.
Examples: [3] [4] 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.
It is synonymous with a homogeneous spatial Poisson process. [1] Such a process is modeled using only one parameter , i.e. the density of points within the defined area. The term complete spatial randomness is commonly used in Applied Statistics in the context of examining certain point patterns, whereas in most other statistical contexts it is ...