Category: Discrete distributions Image File history File links Bvn-small. ... In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... 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). ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... A crude and approximate statement of Benfords law, also called the first-digit law, is that in lists of numbers from many real-life sources of data, the leading digit is 1 almost one-third of the time, and further, larger numbers occur as the leading digit with less... 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... In probability theory, a compound Poisson distribution is the probability distribution of a Poisson-distibuted number of independent identically-distributed random variables. ... 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. ... 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 − N2 of two correlated or uncorrelated random variables N1 and N2 having Poisson distributions with different expected values μ1 and μ2. ... 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 α and β 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 γ 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 χ2  distribution) is one of the theoretical probability distributions most widely used in inferential statistics, i. ... The Dirac delta function, 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 δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ... 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. ... 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 (1891–1966)) 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 that represents the sum of exponentially distributed random variables, each of which has mean . ... 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, Fréchet 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 distribution—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. ... Edit ... 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 Lévy distribution, named after Paul Pierre Lévy, is one of the few distributions that are stable and that have probability density functions that are analytically expressible. ... In probability theory, a Lévy skew alpha-stable distribution or just stable distribution, developed by Paul Lévy, 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 introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)) is a probability distribution of great importance in many fields. ... 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. ... The Pearson distribution is a family of probability distributions that are a generalisation of the normal distribution. ... 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 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 [−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. ... In probability and statistics, the Dirichlet distribution (after Johann Peter Gustav Lejeune Dirichlet) is a continuous multivariate probability distribution. ... 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 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). ... 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. ... 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 probability and statistics, an exponential family is any class of probability distributions having a certain form. ... The concept of infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). ... 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 μ and a scale parameter σ 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. ... A phase-type distribution is a probability distribution that results from a system one or more inter-related poisson processes occurring in sequence, or phases. ... 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. ... // Introduction 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 ) 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. ...