 ## Seemingly unrelated regressions

In econometrics, the seemingly unrelated regressions or seemingly unrelated regression equations model, proposed by Arnold Zellner in, is a generalization of a linear regression model that consists of several regression equations, each having its ...

## Probit

In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution, which is commonly denoted as N. Mathematically, it is the inverse of the cumulative distribution function of the ...

## Atom (programming language)

Originally intended as a high level hardware description language, Atom was created in early 2007 and released in open-source of April of the same year. Inspired by TRS and Bluespec, Atom compiled circuit descriptions, that were based on guarded ...

## Bhatia–Davis inequality

In mathematics, the Bhatia–Davis inequality, named after Rajendra Bhatia and Chandler Davis, is an upper bound on the variance σ 2 of any bounded probability distribution on the real line. Suppose a distribution has minimum m, maximum M, and expe ...

## Cokurtosis

In probability theory and statistics, cokurtosis is a measure of how much two random variables change together. Cokurtosis is the fourth standardized cross central moment. If two random variables exhibit a high level of cokurtosis they will tend ...

## Combinant

In the mathematical theory of probability, the combinants c n of a random variable X are defined via the combinant-generating function G, which is defined from the moment generating function M as G X t = M X log ⁡ 1 + t) {\displaystyle G_{X}t=M_{ ...

## Comonotonicity

In probability theory, comonotonicity mainly refers to the perfect positive dependence between the components of a random vector, essentially saying that they can be represented as increasing functions of a single random variable. In two dimensio ...

## Concomitant (statistics)

In statistics, the concept of a concomitant, also called the induced order statistic, arises when one sorts the members of a random sample according to corresponding values of another random sample. Let X i, Y i, i = 1., n be a random sample from ...

## Conditional probability distribution

In probability theory and statistics, given two jointly distributed random variables X {\displaystyle X} and Y {\displaystyle Y}, the conditional probability distribution of Y given X is the probability distribution of Y {\displaystyle Y} when X ...

## Convolution of probability distributions

The convolution of probability distributions arises in probability theory and statistics as the operation in terms of probability distributions that corresponds to the addition of independent random variables and, by extension, to forming linear ...

## Coskewness

In probability theory and statistics, coskewness is a measure of how much two random variables change together. Coskewness is the third standardized cross central moment, related to skewness as covariance is related to variance. In 1976, Krauss a ...

## Hellinger distance

In probability and statistics, the Hellinger distance is used to quantify the similarity between two probability distributions. It is a type of f -divergence. The Hellinger distance is defined in terms of the Hellinger integral, which was introdu ...

## Infinite divisibility (probability)

In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed random variables. The characteristic fun ...

## Joint probability distribution

Given random variables X, Y, … {\displaystyle X,Y,\ldots }, that are defined on a probability space, the joint probability distribution for X, Y, … {\displaystyle X,Y,\ldots } is a probability distribution that gives the probability that each of ...

In statistics, the Khmaladze transformation is a mathematical tool used in constructing convenient goodness of fit tests for hypothetical distribution functions. More precisely, suppose X 1, …, X n {\displaystyle X_{1},\ldots,X_{n}} are i.i.d., p ...

## Marginal distribution

In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variable ...

## Memorylessness

In probability and statistics, memorylessness is a property of certain probability distributions. It usually refers to the cases when the distribution of a "waiting time" until a certain event does not depend on how much time has elapsed already. ...

## Mills ratio

In probability theory, the Mills ratio of a continuous random variable X {\displaystyle X} is the function m x:= F ¯ x f x, {\displaystyle mx:={\frac.}

## Neutral vector

In statistics, and specifically in the study of the Dirichlet distribution, a neutral vector of random variables is one that exhibits a particular type of statistical independence amongst its elements. In particular, when elements of the random v ...

## Normalizing constant

The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one.

## Normally distributed and uncorrelated does not imply independent

In probability theory, although simple examples illustrate that linear uncorrelatedness of two random variables does not in general imply their independence, it is sometimes mistakenly thought that it does imply that when the two random variables ...

## Pairwise independence

In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent. Any collection of mutually independent random variables is pairwise independent, but some pairwise indepen ...

## Popovicius inequality on variances

In probability theory, Popovicius inequality, named after Tiberiu Popoviciu, is an upper bound on the variance σ² of any bounded probability distribution. Let M and m be upper and lower bounds on the values of any random variable with a particula ...

## Probability integral transform

In statistics, the probability integral transform or transformation relates to the result that data values that are modelled as being random variables from any given continuous distribution can be converted to random variables having a standard u ...

## Relationships among probability distributions

In probability theory and statistics, there are several relationships among probability distributions. These relations can be categorized in the following groups: Transforms function of a random variable; Duality; Approximation limit relationship ...

## Shape of a probability distribution

In statistics, the concept of the shape of a probability distribution arises in questions of finding an appropriate distribution to use to model the statistical properties of a population, given a sample from that population. The shape of a distr ...

## Smoothness (probability theory)

In probability theory and statistics, smoothness of a density function is a measure which determines how many times the density function can be differentiated, or equivalently the limiting behavior of distribution’s characteristic function. Forma ...

## Stability (probability)

In probability theory, the stability of a random variable is the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters. The distributions of random variables ha ...

## Tail dependence

In probability theory, the tail dependence of a pair of random variables is a measure of their comovements in the tails of the distributions. The concept is used in extreme value theory. Random variables that appear to exhibit no correlation can ...

## Truncated distribution

In statistics, a truncated distribution is a conditional distribution that results from restricting the domain of some other probability distribution. Truncated distributions arise in practical statistics in cases where the ability to record, or ...

## Truncation (statistics)

In statistics, truncation results in values that are limited above or below, resulting in a truncated sample. A random variable y {\displaystyle y} is said to be truncated from below if, for some threshold value c {\displaystyle c}, the exact val ...

## Zero bias transform

The zero-bias transform is a transform from one probability distribution to another. The transform arises in applications of Steins method in probability and statistics.

## Complex inverse Wishart distribution

The complex inverse Wishart distribution is a matrix probability distribution defined on complex-valued positive-definite matrices and is the complex analog of the real inverse Wishart distribution. The complex Wishart distribution was extensivel ...

## Compound probability distribution

In probability and statistics, a compound probability distribution is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with the parameters of that distribut ...

## Beta prime distribution

In probability theory and statistics, the beta prime distribution is an absolutely continuous probability distribution defined for x &gt; 0 {\displaystyle x&gt; 0} with two parameters α and β, having the probability density function: f x = x α − ...

## Beta rectangular distribution

In probability theory and statistics, the beta rectangular distribution is a probability distribution that is a finite mixture distribution of the beta distribution and the continuous uniform distribution. The support is of the distribution is in ...

## Compound Poisson distribution

In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable. ...

## Delaporte distribution

The Delaporte distribution is a discrete probability distribution that has received attention in actuarial science. It can be defined using the convolution of a negative binomial distribution with a Poisson distribution. Just as the negative bino ...

## Exponentially modified Gaussian distribution

In probability theory, an exponentially modified Gaussian distribution describes the sum of independent normal and exponential random variables. An exGaussian random variable Z may be expressed as Z = X + Y, where X and Y are independent, X is Ga ...

## Lomax distribution

The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling. It is named after K. S. Lom ...

## Normal variance-mean mixture

In probability theory and statistics, a normal variance-mean mixture with mixing probability density g {\displaystyle g} is the continuous probability distribution of a random variable Y {\displaystyle Y} of the form Y = α + β V + σ V X, {\displa ...

## Rayleigh mixture distribution

In probability theory and statistics a Rayleigh mixture distribution is a weighted mixture of multiple probability distributions where the weightings are equal to the weightings of a Rayleigh distribution. Since the probability density function f ...

## 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. The probability mass function pmf of the Yule–Simon ρ ...

## Bernoulli distribution

In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability p {\displaystyle p} and the ...

## Generalized Dirichlet distribution

In statistics, the generalized Dirichlet distribution is a generalization of the Dirichlet distribution with a more general covariance structure and almost twice the number of parameters. Random variables with a GD distribution are not completely ...

## Grouped Dirichlet distribution

In statistics, the grouped Dirichlet distribution is a multivariate generalization of the Dirichlet distribution It was first described by Ng et al 2008. The Grouped Dirichlet distribution arises in the analysis of categorical data where some obs ...

## Inverse-gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to t ...

## Normal-inverse-Wishart distribution

In probability theory and statistics, the normal-inverse-Wishart distribution is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate normal distribution with unknown mean and c ...

## Normal-Wishart distribution

In probability theory and statistics, the normal-Wishart distribution is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate normal distribution with unknown mean and precision ...

## Arcsine distribution

In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function is F x = 2 π arcsin ⁡ x = arcsin ⁡ 2 x − 1 π + 1 2 {\displaystyle Fx={\frac {2}{\pi }}\arcsin \left{\sqrt {x}}\right={\frac {\a ...