NationMaster - Encyclopedia: Multivariate normal distribution

In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). It is also closely related to matrix normal distribution. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn. ... In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ... In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory and statistics, a median is a type of average that is described as the number dividing the higher half of a sample, a population, or a probability distribution, from the lower half. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... This article is about the field of statistics. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... The matrix normal distribution is a probability distribution that is a generalization of the normal distribution. ...

General case

A random vector $X = [X_1, dots, X_N]^T$ follows a multivariate normal distribution if it satisfies the following equivalent conditions: A multivariate random variable or random vector is a vector X=(X1,...,Xn) whose components are scalar-valued random variables on the same probability space (Ω, P). ...

If $Sigma$ is non-singular, then the distribution may be described by the following PDF: The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... For the square matrix section, see square matrix. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

where $left| Sigma right|$ is the determinant of $Sigma$ . Note how the equation above reduces to that of the univariate normal distribution if $Sigma$ is a scalar (i.e., a multiple of the identity matrix). In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ...

The vector μ in these conditions is the expected value of X and the matrix $Sigma = A A^T$ is the covariance matrix of the components X_i. In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ...

It is important to realize that the covariance matrix must be allowed to be singular (thus not described by above formula for which $Sigma^{-1}$ is defined). That case arises frequently in statistics; for example, in the distribution of the vector of residuals in ordinary linear regression problems. Note also that the X_i are in general not independent; they can be seen as the result of applying the linear transformation A to a collection of independent Gaussian variables Z. This article is about the field of statistics. ... In statistics and optimization, the concepts of error and residual are easily confused with each other. ... In statistics, linear regression is a regression method that models the relationship between a dependent variable Y, independent variables Xi, i = 1, ..., p, and a random term Îµ. The model can be written as Example of linear regression with one dependent and one independent variable. ...

That the distribution of a random vector X is a multivariate normal distribution can be written in the following notation:

Cumulative distribution function

The cumulative distribution function (cdf)

F (x)

is defined as the probability that all values in a random vector

X

are less than or equal to the corresponding values in vector

x

. Though there is no closed form for

F (x)

, there are a number of algorithms that estimate it numerically. For example, see MVNDST under [1] (includes FORTRAN code) or [2] (includes MATLAB code). In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... Fortran (previously FORTRAN[1]) is a general-purpose[2], procedural,[3] imperative programming language that is especially suited to numeric computation and scientific computing. ... Source code (commonly just source or code) is any series of statements written in some human-readable computer programming language. ... Not to be confused with Matlab Upazila in Chandpur District, Bangladesh. ... Source code (commonly just source or code) is any series of statements written in some human-readable computer programming language. ...

A counterexample

The fact that two or more random variables X and Y are normally distributed does not imply that the pair (X, Y) has a joint normal distribution. A simple example is one in which Y = X if |X| > 1 and Y = −X if |X| < 1.

Normally distributed and independent

If X and Y are normally distributed and independent, then they are "jointly normally distributed", i.e., the pair (X, Y) has a bivariate normal distribution. There are of course also many bivariate normal distributions in which the components are correlated.

Bivariate case

In the 2-dimensional nonsingular case, the probability density function (with mean (0,0)) is In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

where

ρ

is the correlation between

X

and

Y

. In this case, Positive linear correlations between 1000 pairs of numbers. ...

Linear transformation

If $Y = B X ,$ is a linear transformation of $X sim mathcal{N}(mu, Sigma),$ where $B,$ is an $M times N$ matrix then $Y,$ has a multivariate normal distribution with expected value $B mu ,$ and variance $B Sigma B^T ,$ (i.e., $Y sim mathcal{N} left(B mu, B Sigma B^Tright)$ ). In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

Corollary: any subset of the $X_i,$ has a marginal distribution that is also multivariate normal. To see this consider the following example: to extract the subset $(X_1, X_2, X_4)^T ,$ , use

Geometric interpretation

The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean^[1]. The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix

Σ

. The squared relative lengths of the principal axes are given by the corresponding eigenvalues. 3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ... 2-sphere wireframe as an orthogonal projection Just as a stereographic projection can project a spheres surface to a plane, it can also project a 3-spheres surface into 3-space. ...

Σ = U Λ U T = U Λ 1 / 2 (U Λ 1 / 2) T

is an eigendecomposition where the columns of U are unit eigenvectors and

Λ

is a diagonal matrix of the eigenvalues, then we have ... In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ...

Moreover, U can be chosen to be a rotation matrix, as inverting an axis does not have any effect on

N (0,Λ)

, but inverting a column changes the sign of U's determinant. The distribution

N (μ,Σ)

is in effect

N (0, I)

scaled by

Λ 1 / 2

, rotated by U and translated by

μ

. A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but not its magnitude. ...

Conversely, any choice of

μ

, full rank matrix U, and positive diagonal entries

Λ i

yields a non-singular multivariate normal distribution. If any

Λ i

is zero and U is square, the resulting covariance matrix

U Λ U T

is singular. Geometrically this means that every contour ellipsoid is infinitely thin and has zero volume in n-dimensional space, as at least one of the principal axes has length of zero. In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

Correlations and independence

In general, random variables may be uncorrelated but highly dependent. But if a random vector has a multivariate normal distribution then any two or more of its components that are uncorrelated are independent. This implies that any two or more of its components that are pairwise independent are independent. In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent. ...

But it is not true that two random variables that are (separately, marginally) normally distributed and uncorrelated are independent. Two random variables that are normally distributed may fail to be jointly normally distributed, i.e., the vector whose components they are may fail to have a multivariate normal distribution. For an example of two normally distributed random variables that are uncorrelated but not independent, see normally distributed and uncorrelated does not imply independent. In probability theory, it is almost a cliche to say that uncorrelatedness of two random variables does not entail independence. ...

Higher moments

where the sum is taken over all allocations of the set $left{ 1,dots,2lambda right}$ into

λ

(unordered) pairs, giving

(2λ - 1)! / (2 λ - 1 (λ - 1)!)

terms in the sum, each being the product of

λ

covariances. The covariances are determined by replacing the terms of the list $left[ 1,dots,2lambda right]$ by the corresponding terms of the list consisting of

r 1

ones, then

r 2

twos, etc, after each of the possible allocations of the former list into pairs.

For fourth order moments (four variables) there are three terms. For sixth-order moments there are 3 × 5 = 15 terms, and for eighth-order moments there are 3 × 5 × 7 = 105 terms. The sixth-order moment case can be expanded as

Conditional distributions

then the distribution of

x 1

conditional on

x 2 = a

is multivariate normal $(X_1|X_2=a) sim N(bar{mu}, overline{Sigma})$ where

This matrix is the Schur complement of ${mathbfSigma_{22}}$ in ${mathbfSigma}$ . In linear algebra and the theory of matrices, the Schur complement (named after Issai Schur) of a block of a matrix within the larger matrix is defined as follows. ...

Note that knowing the value of

x 2

to be

a

alters the variance; perhaps more surprisingly, the mean is shifted by $Sigma_{12} Sigma_{22}^{-1} left(a - mu_2 right)$ ; compare this with the situation of not knowing the value of

a

, in which case

x 1

would have distribution $N_q left(mu_1, Sigma_{11} right)$ .

The matrix $Sigma_{12} Sigma_{22}^{-1}$ is known as the matrix of regression coefficients. In statistics, regression analysis examines the relation of a dependent variable (response variable) to specified independent variables (explanatory variables). ...

Fisher information matrix

The Fisher information matrix (FIM) for a normal distribution takes a special formulation. The

(m, n)

element of the FIM for $X sim N(mu(theta), Sigma(theta))$ is In statistics, the Fisher information I(θ), thought of as the amount of information that an observable random variable carries about an unobservable parameter θ upon which the probability distribution of X depends, is the variance of the score. ...

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ...

Kullback-Leibler divergence

The Kullback-Leibler divergence from

N 0 N (μ 0,Σ 0)

N 1 N (μ 1,Σ 1)

is: In probability theory and information theory, the Kullback-Leibler divergence (or information divergence, or information gain, or relative entropy) is a natural distance measure from a true probability distribution P to an arbitrary probability distribution Q. Typically P represents data, observations, or a precise calculated probability distribution. ...

Estimation of parameters

The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is perhaps surprisingly subtle and elegant. See estimation of covariance matrices. Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution from a given data set. ... In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter; an estimate is the result from the actual application of the function to a particular set of data. ... In multivariate statistics, the importance of the Wishart distribution stems in part from the fact that it is the probability distribution of the maximum likelihood estimator of the covariance matrix of a multivariate normal distribution. ...

In short, the probability density function (pdf) of an N-dimensional multivariate normal is

which is simply the sample covariance matrix for sample size n. This is a biased estimator whose expectation is In statistics, a biased estimator is one that for some reason on average over_ or underestimates what is being estimated. ...

Entropy

The differential entropy of the multivariate normal distribution is ^[2] Differential entropy (also referred to as continuous entropy) is a concept in information theory which tries to extend the idea of (Shannon) entropy, a measure of average surprisal of a random variable, to continuous probability distributions. ...

where $left| Sigma right|$ is the determinant of the covariance matrix

Σ

. In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

Multivariate normality tests

Multivariate normality tests check a given set of data for similarity to the multivariate normal distribution. The null hypothesis is that the data set is similar to the normal distribution, therefore a sufficiently small p-value indicates non-normal data. Multivariate normality tests include the Cox-Small test ^[3] and Smith and Jain's adaptation ^[4] of the Friedman-Rafsky test. In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternative hypothesis. ... A data set (or dataset) is a collection of data, usually presented in tabular form. ... In statistical hypothesis testing, the p-value of a random variable T used as a test statistic is the probability that T will assume a value at least as extreme as the observed value tobserved, given that a null hypothesis being considered is true. ...

Drawing values from the distribution

A widely used method for drawing a random vector

X

from the

N

-dimensional multivariate normal distribution with mean vector

μ

and covariance matrix

Σ

(required to be symmetric and positive-definite) works as follows: In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ...

In mathematics, the Cholesky decomposition, named after AndrÃ©-Louis Cholesky, is a matrix decomposition of a symmetric positive-definite matrix into a lower triangular matrix and the transpose of the lower triangular matrix. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... Diagram of the Box Muller transform. ...

References

Category: Continuous distributions A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... Image File history File links Bvn-small. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... Univariate describes a concept in statistics or econometrics. ... A multivariate random variable or random vector is a vector X = (X1, ..., Xn) whose components are scalar-valued random variables on the same probability space (Î©, P). ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... A logarithmic scale bar. ... In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ... In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... In physics, the Boltzmann distribution predicts the distribution function for the fractional number of particles Ni / N occupying a set of states i which each has energy Ei: where is the Boltzmann constant, T is temperature (assumed to be a sharply well-defined quantity), is the degeneracy, or number of... Often confused with the multinomial distribution. ... In probability theory, a compound Poisson distribution is the probability distribution of a Poisson-distibuted number of independent identically-distributed random variables. ... The discrete phase-type distribution is a probability distribution that results from a system of one or more inter-related geometric distributions occurring in sequence, or phases. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... In mathematics, the Gauss-Kuzmin distribution gives the probability distribution of the occurrence of a given integer in the continued fraction expansion of an arbitrary real number. ... In probability theory and statistics, the geometric distribution is either of two discrete probability distributions: the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or the probability distribution of the number Y = X âˆ’ 1 of failures before... // In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement. ... In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution) is a discrete probability distribution. ... In probability and statistics the negative binomial distribution is a discrete probability distribution. ... In the parabolic fractal distribution, the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ... In probability theory and statistics, the Rademacher distribution is a discrete probability distribution. ... The Skellam distribution is the discrete probability distribution of the difference N1 âˆ’ N2 of two correlated or uncorrelated random variables N1 and N2 having Poisson distributions with different expected values Î¼1 and Î¼2. ... In probability theory and statistics, the discrete uniform distribution is a discrete probability distribution that can be characterized by saying that all values of a finite set of possible values are equally probable. ... In probability and statistics, the Yule-Simon distribution is a discrete probability distribution. ... In probability theory and statistics, the zeta distribution is a discrete probability distribution. ... Originally, Zipfs law stated that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table. ... In probability theory and statistics, the Zipf-Mandelbrot law is a discrete probability distribution. ... In population genetics, Ewenss sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of n gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are a1 alleles represented once... In probability theory, the multinomial distribution is a generalization of the binomial distribution. ... The multivariate Polya distribution, also called the Dirichlet compound multinomial distribution, is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution and a set of discrete samples x is drawn from the multinomial distribution with probability vector p. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]: where Î± and Î² are parameters that must be greater than zero and B is the beta function. ... A Beta Prime Distribution is a distribution with probability function: where is a Beta function. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and Î³ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... In probability theory and statistics, the chi-square distribution (also chi-squared or Ï‡2 distribution) is one of the theoretical probability distributions most widely used in inferential statistics, i. ... The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0 and the value zero elsewhere. ... A phase-type distribution is a probability distribution that results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. ... The Erlang distribution is a continuous probability distribution with wide applicability primarily due to its relation to the exponential and Gamma distributions. ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... The exponential power distribution, also known as the generalized error distribution, takes a scale parameter a and exponent b. ... In statistics and probability, the F-distribution is a continuous probability distribution. ... In telecommunication, a fading distribution is the probability distribution that signal fading will exceed a given value relative to a specified reference level. ... In statistical mechanics, Fermi-Dirac statistics determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. ... Fishers z-distribution is the distribution of half the logarithm of a F distribution variate: It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto, entitled On a distribution yielding the error functions of several well-known statistics. Nowadays... In probability theory and statistics the Gumbel distribution (named after Emil Julius Gumbel (1891â€“1966)) is used to find the minimum (or the maximum) of a number of samples of various distributions. ... In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. ... In probability theory and statistics, the generalized extreme value distribution (GEV) is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, FrÃ©chet and Weibull families also known as type I, II and III extreme value distributions. ... The generalised hyperbolic distribution is a continuous probability distribution defined by the probability density function where is the modified Bessel function of the second kind. ... In probability theory, the Generalized inverse Gaussian distribution (GIG) is a probability distribution with probability density function It is used extensively in geostatistics, statistical linguistics, finance, etc. ... In probability theory and statistics, the half-logistic distribution is a continuous probability distributionâ€”the distribution of the absolute value of a random variable following the logistic distribution. ... In statistics, Hotellings T-square statistic, named for Harold Hotelling, is a generalization of Students t statistic that is used in multivariate hypothesis testing. ... In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. ... In probability theory, a hyper-exponential distribution is a continuous distribution such that the probability density function of the random variable X is given by: Where is an exponentially distributed random variable with rate parameter , and is the probability that X will take on the form of the exponential distribution... The hypoexponential distribution is a generalization of Erlang distribution in the sense that the n exponential distributions may have different rates. ... In probability and statistics, the inverse-chi-square distribution is the probability distribution of a random variable whose inverse has a chi-square distribution. ... The scaled inverse chi-square distribution arises in Bayesian statistics. ... The probability density function of the inverse Gaussian distribution is given by The Wald distribution is simply another name for the inverse Gaussian distribution. ... The inverse gamma distribution has the probability density function over the support with shape parameter and scale parameter . ... ... In probability theory and statistics, Kumaraswamys double bounded distribution is as versatile as the Beta distribution, but much simpler to use especially in simulation studies as it has a simple closed form solution for both its pdf and cdf. ... The probability distribution for Landau random variates is defined analytically by the complex integral, For numerical purposes it is more convenient to use the following equivalent form of the integral, From GSL manual, used under GFDL. ... In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. ... In probability theory and statistics, the LÃ©vy distribution, named after Paul Pierre LÃ©vy, is one of the few distributions that are stable and that have probability density functions that are analytically expressible. ... In probability theory, a LÃ©vy skew alpha-stable distribution or just stable distribution, developed by Paul LÃ©vy, is a probability distribution where sums of independent identically distributed random variables have the same distribution as the original. ... In probability theory and statistics, the logistic distribution is a continuous probability distribution. ... In probability and statistics, the log-normal distribution is the probability distribution of any random variable whose logarithm is normally distributed. ... The Maxwellâ€“Boltzmann distribution is a probability distribution with applications in physics and chemistry. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... There are very few or no other articles that link to this one. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In probability theory and statistics, the normal-gamma distribution is a four-parameter family of continuous probability distributions. ... The normal-inverse Gaussian distribution is continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. ... The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ... This article or section is incomplete and may require expansion and/or cleanup. ... A phase-type distribution is a probability distribution that results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. ... In probability theory, the polar distribution is the probability distribution of angles occurring in a set of two-dimensional vectors, denoted by It is usually graphically represented as a closed curve , where the radius equals the probability . ... In probability theory and statistics, the raised cosine distribution is a probability distribution supported on the interval []. The probability density function is for and zero otherwise. ... In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. ... NB: The information in this article should be reviewed. ... In probability theory and statistics, the Rice distribution distribution is a continuous probability distribution. ... The shifted Gompertz distribution is the distribution of the largest order statistic of two independent random variables which are distributed exponential and Gompertz with parameters b and b and respectively. ... In probability and statistics, the t-distribution or Students t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ... In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, mode c and upper limit b. ... In probability and statistics, the truncated normal distribution is the probability distribution of a normally distributed random variable whose value is either bounded below or above (or both). ... In probability theory, the Type-1 Gumbel distribution function is for . Reference Taken from the gsl-ref_19. ... In probability theory, the Type-2 Gumbel distribution function is for . Based on gsl-ref_19. ... In probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distributions support are equally probable. ... The variance-gamma distribution is continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the gamma distribution. ... In spectroscopy, the Voigt profile is a spectral line profile named after Woldemar Voigt and found in all branches of spectroscopy in which a spectral line is broadened by two types of mechanisms, one of which alone would produce a Doppler profile, and the other of which would produce a... In probability theory and statistics, the von Mises distribution is a continuous probability distribution. ... In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function where and is the shape parameter and is the scale parameter of the distribution. ... The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval [âˆ’R, R] the graph of whose probability density function f is a semicircle of radius R centered at (0, 0) and then suitably normalized (so that it is really a semi-ellipse... This article or section is in need of attention from an expert on the subject. ... Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors Î±. Clockwise from top left: Î±=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4). ... In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and twice the number of parameters. ... In statistics, the Inverse Wishart distribution, also the inverse Wishart distribution and inverted Wishart distribution is a probability density function defined on matrices. ... The 5-parameter Fisher-Bingham distribution or Kent distribution is a probability distribution on the three-dimensional sphere. ... The matrix normal distribution is a probability distribution that is a generalization of the normal distribution. ... In statistics, a multivariate Student distribution is a multivariate generalization of the Students t-distribution. ... Points sampled from three von Mises-Fisher distributions on the sphere (blue: , green: , red: ). The mean directions are shown with arrows. ... The Wigner quasi-probability distribution was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. ... In statistics, the Wishart distribution, named in honor of John Wishart, is any of a family of probability distributions for nonnegative-definite matrix-valued random variables (random matrices). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In statistics, a bimodal distribution is a distribution with two different peaks — that is, there are two distinct values that measurements tend to center around. ... The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. ... This article defines some terms which characterize probability distributions of two or more variables. ... In mathematics, a Markov chain, named after Andrey Markov, is a discrete-time stochastic process with the Markov property. ... In probability and statistics, an exponential family is any class of probability distributions having a certain form. ... In probability theory, to say that a probability distribution F on the real line is infinitely divisible means that if X is any random variable whose distribution is F, then for every positive integer n there exist n independent identically distributed random variables X1, ..., Xn whose sum is equal in... In probability theory, especially as that field is used in statistics, a location-scale family is a set of probability distributions on the real line parametrized by a location parameter Î¼ and a scale parameter Ïƒ â‰¥ 0; if X is any random variable whose probability distribution belongs to such a family, then... In probability theory, given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y, typically calculated by summing or integrating the joint probability distribution over Y. For discrete random variables, the marginal probability mass function can... In statistics and information theory, a maximum entropy probability distribution is a probability distribution whose entropy is larger than (or equal to) that of all other members of a specified class of distributions. ... The posterior probability can be calculated by Bayes theorem from the prior probability and the likelihood function. ... A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence. ... In the most general form, the dynamics of a quantum mechanical system are determined by a master equation - an equation of motion for the density operator (usually written Ï) of the system. ... In statistics, a sampling distribution is the probability distribution, under repeated sampling of the population, of a given statistic (a numerical quantity calculated from the data values in a sample). ... In probability, a singular distribution is a probability distribution concentrated on a measure zero set where the probability of each point in that set is zero. ...

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