Cauchy distribution. The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The simplest Cauchy distribution is called the standard Cauchy distribution. It is the distribution of a random variable that is the ratio of two independent standard normal variables and has the probability density function Its cumulative distribution function has the shape of an arctangent function arctan(x): Its importance in physics is the result of it being the solution to the differential equation describing forced resonance.[2] In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane.
Characterisation[edit] Probability density function[edit] The Cauchy distribution has the probability density function Properties[edit] where. List of probability distributions. Many probability distributions are so important in theory or applications that they have been given specific names. Discrete distributions[edit] With finite support[edit] With infinite support[edit] Continuous distributions[edit] Supported on a bounded interval[edit] Supported on semi-infinite intervals, usually [0,∞)[edit] Supported on the whole real line[edit] With variable support[edit] The generalized extreme value distribution has a finite upper bound or a finite lower bound depending on what range the value of one of the parameters of the distribution is in (or is supported on the whole real line for one special value of the parameterThe generalized Pareto distribution has a support which is either bounded below only, or bounded both above and belowThe Tukey lambda distribution is either supported on the whole real line, or on a bounded interval, depending on what range the value of one of the parameters of the distribution is in.The Wakeby distribution Joint distributions[edit]
Stable distribution. The importance of stable probability distributions is that they are "attractors" for properly normed sums of independent and identically-distributed (iid) random variables. The normal distribution is one family of stable distributions. By the classical central limit theorem the properly normed sum of a set of random variables, each with finite variance, will tend towards a normal distribution as the number of variables increases. Without the finite variance assumption the limit may be a stable distribution. Stable distributions that are non-normal are often called stable Paretian distributions,[citation needed] after Vilfredo Pareto. q-analogs of all symmetric stable distributions have been defined, and these recover the usual symmetric stable distributions in the limit of q → 1.[1] Definition[edit] A non-degenerate distribution is a stable distribution if it satisfies the following property: Let X1 and X2 be independent copies of a random variable X.
For all α except α = 1 in which case: Normal distribution. In probability theory, the normal (or Gaussian) distribution is a very commonly occurring continuous probability distribution—a function that tells the probability that an observation in some context will fall between any two real numbers. Normal distributions are extremely important in statistics and are often used in the natural and social sciences for real-valued random variables whose distributions are not known.[1][2] The normal distribution is immensely useful because of the central limit theorem, which states that, under mild conditions, the mean of many random variables independently drawn from the same distribution is distributed approximately normally, irrespective of the form of the original distribution: physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have a distribution very close to the normal.
The Gaussian distribution is sometimes informally called the bell curve. A normal distribution is The factor . . Lévy distribution. Definition[edit] The probability density function of the Lévy distribution over the domain is where is the location parameter and is the scale parameter. The cumulative distribution function is is the complementary error function. Has the effect of shifting the curve to the right by an amount , and changing the support to the interval [ ).
Where y is defined as Note that the characteristic function can also be written in the same form used for the stable distribution with and Assuming which diverges for all n > 0 so that the moments of the Lévy distribution do not exist. Which diverges for and is therefore not defined in an interval around zero, so that the moment generating function is not defined per se.
This is illustrated in the diagram below, in which the probability density functions for various values of c and are plotted on a log-log scale. Probability density function for the Lévy distribution on a log-log scale. Related distributions[edit] Applications[edit] [edit] Notes[edit] References[edit] Yule–Simon distribution. In probability and statistics, the Yule–Simon distribution is a discrete probability distribution named after Udny Yule and Herbert A. Simon. Simon originally called it the Yule distribution.[1] The probability mass function of the Yule–Simon (ρ) distribution is for integer and real , where is the beta function. Where is the gamma function. Is an integer, The parameter can be estimated using a fixed point algorithm.[2] The probability mass function f has the property that for sufficiently large k we have This means that the tail of the Yule–Simon distribution is a realization of Zipf's law: can be used to model, for example, the relative frequency of the th most frequent word in a large collection of text, which according to Zipf's law is inversely proportional to a (typically small) power of Occurrence[edit] The distribution also arises as a compound distribution, in which the parameter of a geometric distribution is treated as a function of random variable having an exponential distribution. for .
Generalized extreme value distribution. In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. Note that a limit distribution need not exist: this requires regularity conditions on the tail of the distribution. Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables.
In some fields of application the generalized extreme value distribution is known as the Fisher–Tippett distribution, named after R. A. Specification[edit] The generalized extreme value distribution has cumulative distribution function for , where , while for . . And. Gumbel distribution. In probability theory and statistics, the Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. Such a distribution might be used to represent the distribution of the maximum level of a river in a particular year if there was a list of maximum values for the past ten years.
It is useful in predicting the chance that an extreme earthquake, flood or other natural disaster will occur. The potential applicability of the Gumbel distribution to represent the distribution of maxima relates to extreme value theory which indicates that it is likely to be useful if the distribution of the underlying sample data is of the normal or exponential type. The Gumbel distribution is a particular case of the generalized extreme value distribution (also known as the Fisher-Tippett distribution). The Gumbel distribution is named after Emil Julius Gumbel (1891–1966), based on his original papers describing the distribution.[1][2] Gompertz function. A Gompertz curve or Gompertz function, named after Benjamin Gompertz, is a sigmoid function. It is a type of mathematical model for a time series, where growth is slowest at the start and end of a time period. The right-hand or future value asymptote of the function is approached much more gradually by the curve than the left-hand or lower valued asymptote, in contrast to the simple logistic function in which both asymptotes are approached by the curve symmetrically.
It is a special case of the generalised logistic function. Formula[edit] where Differentiation[edit] The function curve can be derived from a Gompertz law of mortality, which states the rate of mortality (decay) falls exponentially with current size. Is the rate of growth.k is an arbitrary constant. Example uses[edit] Examples of uses for Gompertz curves include: Growth of tumors[edit] In the 1960s A.K. Where: independently on X(0)>0. Α is a constant related to the proliferative ability of the cells.log() refers to the natural log.