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Encyclopedia > Beta distribution
Not to be confused with Beta function.
Beta
Probability density function
Probability density function for the Beta distribution
Cumulative distribution function
Cumulative distribution function for the Beta distribution
Parameters α > 0 shape (real)
β > 0 shape (real)
Support x in [0; 1]!
Probability density function (pdf) frac{x^{alpha-1}(1-x)^{beta-1}} {mathrm{B}(alpha,beta)}!
Cumulative distribution function (cdf) I_x(alpha,beta)!
Mean frac{alpha}{alpha+beta}!
Median
Mode for α > 1,β > 1
Variance frac{alphabeta}{(alpha+beta)^2(alpha+beta+1)}!
Skewness frac{2,(beta-alpha)sqrt{alpha+beta+1}}{(alpha+beta+2)sqrt{alphabeta}}
Excess kurtosis see text
Entropy see text
Moment-generating function (mgf) 1 +sum_{k=1}^{infty} left( prod_{r=0}^{k-1} frac{alpha+r}{alpha+beta+r} right) frac{t^k}{k!}
Characteristic function {}_1F_1(alpha; alpha+beta; i,t)!

In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, typically denoted by α and β. It is the special case of the Dirichlet distribution with only two parameters. Since the Dirichlet distribution is the conjugate prior of the multinomial distribution, the beta distribution is the conjugate prior of the binomial distribution. In Bayesian statistics, it can be seen as the posterior distribution of parameter p of a binomial distribution after observing α -1 independent events with probability p and β -1 with probability 1-p. In theoretical physics, specifically quantum field theory, a beta-function β(g) encodes the dependence of a coupling parameter, g, on the energy scale, of a given physical process. ... Download high resolution version (1300x975, 214 KB) See the image on the commons for gnuplot source. ... Image File history File links Download high resolution version (1300x975, 189 KB) See the image on the commons for gnuplot source. ... In probability theory and statistics, a shape parameter is a special kind of numerical parameter of a parametric family of probability distributions. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... 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... This article is about the statistical concept. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... This article is about mathematics. ... Example of 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. ... A probability distribution describes the values and probabilities that a random event can take place. ... In probability theory and statistics, a shape parameter is a special kind of numerical parameter of a parametric family of probability distributions. ... Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors α. Clockwise from top left: α=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4). ... Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors α. Clockwise from top left: α=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4). ... In Bayesian probability theory, a class of prior probability distributions p(θ) is said to be conjugate to a class of likelihood functions p(x|θ) if the resulting posterior distributions p(θ|x) are in the same family as p(θ). For example, the Gaussian family is conjugate to itself (or self-conjugate... In probability theory, the multinomial distribution is a generalization of the binomial distribution. ... 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. ... Bayesian refers to methods in probability and statistics named after the Reverend Thomas Bayes (ca. ...

Contents

Characterization

Probability density function

The probability density function of the beta distribution is: In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

f(x;alpha,beta) = frac{x^{alpha-1}(1-x)^{beta-1}}{int_0^1 u^{alpha-1} (1-u)^{beta-1}, du} !
= frac{Gamma(alpha+beta)}{Gamma(alpha)Gamma(beta)}, x^{alpha-1}(1-x)^{beta-1}!
= frac{1}{mathrm{B}(alpha,beta)}, x ^{alpha-1}(1-x)^{beta-1}!

where Γ is the gamma function. The beta function, B, appears as a normalization constant to ensure that the total probability integrates to unity. The Gamma function along part of the real axis In mathematics, the Gamma function (represented by the capitalized Greek letter Γ) is an extension of the factorial function to real and complex numbers. ... In theoretical physics, specifically quantum field theory, a beta-function β(g) encodes the dependence of a coupling parameter, g, on the energy scale, of a given physical process. ...


Cumulative distribution function

The cumulative distribution function is 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...

F(x;alpha,beta) = frac{mathrm{B}_x(alpha,beta)}{mathrm{B}(alpha,beta)} = I_x(alpha,beta) !

where Bx(α,β) is the incomplete beta function and Ix(α,β) is the regularized incomplete beta function. In mathematics, the incomplete beta function is a generalization of the beta function that replaces the definite integral of the beta function with an indefinite integral. ... In mathematics, the incomplete beta function is a generalization of the beta function that replaces the definite integral of the beta function with an indefinite integral. ...


Properties

Moments

The expected value and variance of a beta random variable X with parameters α and β are given by the formulae: 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... This article is about mathematics. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ...

begin{align} operatorname{E}(X) = & frac{alpha}{alpha+beta} operatorname{Var}(X) = & frac{alpha beta}{(alpha+beta)^2(alpha+beta+1)} end{align}

The skewness is Example of 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. ...

frac{2 (beta - alpha) sqrt{alpha + beta + 1} } {(alpha + beta + 2) sqrt{alpha beta}}. ,!

The kurtosis excess is: In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ...

6,frac{alpha^3-alpha^2(2beta-1)+beta^2(beta+1)-2alphabeta(beta+2)} {alpha beta (alpha+beta+2) (alpha+beta+3)}.,!

Quantities of information

Given two beta distributed random variables, X ~ Beta(α, β) and Y ~ Beta(α', β'), the information entropy of X is Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ...

begin{align} H(X) &= lnmathrm{B}(alpha,beta)-(alpha-1)psi(alpha)-(beta-1)psi(beta)+(alpha+beta-2)psi(alpha+beta) end{align} ,

where ψ is the digamma function. In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions. ...


The cross entropy is In information theory, the cross entropy between two probability distributions measures the overall difference between the two distributions. ...

H(X,Y) = lnmathrm{B}(alpha',beta')-(alpha'-1)psi(alpha)-(beta'-1)psi(beta)+(alpha'+beta'-2)psi(alpha+beta).,

It follows that the Kullback-Leibler divergence between these two beta distributions 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. ...

D_{mathrm{KL}}(X,Y) = lnfrac{mathrm{B}(alpha',beta')} {mathrm{B}(alpha,beta)} - (alpha'-alpha)psi(alpha) - (beta'-beta)psi(beta) + (alpha'-alpha+beta'-beta)psi(alpha+beta)

Shapes

The beta density function can take on different shapes depending on the values of the two parameters:

  • alpha < 1, beta < 1 is U-shaped (red plot)
  • alpha < 1, beta geq 1 or alpha = 1, beta > 1 is strictly decreasing (blue plot)
  • alpha = 1, beta = 1 is the uniform [0,1] distribution
  • alpha = 1, beta < 1 or alpha > 1, beta leq 1 is strictly increasing (green plot)
    • alpha > 2, beta = 1 is strictly convex
    • alpha = 2, beta = 1 is a straight line
    • 1 < alpha < 2, beta = 1 is strictly concave
  • alpha > 1, beta > 1 is unimodal (purple & black plots)

Moreover, if α = β then the density function is symmetric about 1/2 (red & purple plots). In mathematics, convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have Convex function on an interval. ... In calculus, a differentiable function f is convex on an interval if its derivative function f ′ is increasing on that interval: a convex function has an increasing slope. ... 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. ... UniModal or SkyTran is a proposal by Douglas Malewicki for a 160km/h (100mph) personal rapid transit system. ...


Parameter estimation

Let

bar{x} = frac{1}{N}sum_{i=1}^N x_i

be the sample mean and In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set. ...

v = frac{1}{N}sum_{i=1}^N (x_i - bar{x})^2

be the sample variance. The method-of-moments estimates of the parameters are This article is about mathematics. ... In statistics, the method of moments is a method of estimation of population parameters such as mean, variance, median, etc. ...

alpha = bar{x} left(frac{bar{x} (1 - bar{x})}{v} - 1 right),
beta = (1-bar{x}) left(frac{bar{x} (1 - bar{x})}{v} - 1 right).

If the distribution is required over an interval other than [0, 1], say scriptstyle [ell,h] , then replace bar{x} with frac{(bar{x}-ell)}{(h-ell)} , and v with frac{v}{(h-ell)^2} in the above equations.[1][2]


Related distributions

  • If X has a beta distribution, then T=X/(1-X) has a "beta distribution of the second kind", also called the beta prime distribution.
  • The connection with the binomial distribution is mentioned below.
  • The Beta(1,1) distribution is identical to the standard uniform distribution.
  • If X and Y are independently distributed Gamma(α, θ) and Gamma(β, θ) respectively, then X / (X + Y) is distributed Beta(α,β).
  • If X and Y are independently distributed Beta(α,β) and F(2β,2α) (Snedecor's F distribution with 2β and 2α degrees of freedom), then Pr(X ≤ α/(α+xβ)) = Pr(Y > x) for all x > 0.
  • The beta distribution is a special case of the Dirichlet distribution for only two parameters.
  • The Kumaraswamy distribution resembles the beta distribution.
  • If X sim {rm U}(0, 1], has a uniform distribution, then X^2 sim {rm Beta}(1/2,1) or for the 4 parameter case, X^2 sim {rm Beta}(0,1,1/2,1) which is a special case of the Beta distribution called the power-function distribution.
  • Binomial opinions in subjective logic are equivalent to Beta distributions.

A Beta Prime Distribution is a distribution with probability function: where is a Beta function. ... 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 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. ... In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. ... In statistics and probability, the F-distribution is a continuous probability distribution. ... Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors α. Clockwise from top left: α=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4). ... 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. ... Subjective logic is a type of probabilistic logic that explicitly takes uncertainty into account. ...

Applications

B(i, j) with integer values of i and j is the distribution of the i-th order statistic (the i-th smallest value) of a sample of i + j − 1 independent random variables uniformly distributed between 0 and 1. The cumulative probability from 0 to x is thus the probability that the i-th smallest value is less than x, in other words, it is the probability that at least i of the random variables are less than x, a probability given by summing over the binomial distribution with its p parameter set to x. This shows the intimate connection between the beta distribution and the binomial distribution. 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. ... 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. ...


Beta distributions are used extensively in Bayesian statistics, since beta distributions provide a family of conjugate prior distributions for binomial (including Bernoulli) and geometric distributions. The beta(0,0) distribution is an improper prior and sometimes used to represent ignorance of parameter values. Bayesian inference is statistical inference in which probabilities are interpreted not as frequencies or proportions or the like, but rather as degrees of belief. ... In Bayesian probability theory, a conjugate prior is a prior distribution which has the property that the posterior distribution is the same type of distribution. ... 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 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 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... A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence. ...


The Beta distribution can be used to model events which are constrained to take place within an interval defined by a minimum and maximum value. For this reason, the Beta distribution—along with the triangular distribution—is used extensively in PERT, critical path method (CPM) and other project management / control systems to describe the time to completion of a task. In project management, shorthand computations are widely used to estimate the mean and standard deviation of the Beta distribution: In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, mode c and upper limit b. ... PERT network chart for a seven-month project with five milestones (10 through 50) and six activities (A through F). ... The Critical Path Method, abbreviated CPM, or critical path analysis, is a mathematically based algorithm for scheduling a set of project activities. ... Project Management is the discipline of organizing and managing resources (e. ...

begin{align} mu(X) & {} = frac{a + 4b + c}{6} sigma(X) & {} = frac{c-a}{6} end{align}

where a is the minimum, c is the maximum, and b is the most likely value.


References

  1. ^ Engineering Statistics Handbook
  2. ^ Brighton Webs Ltd. Data & Analysis Services for Industry & Education

External links

v • d • e
Probability distributions
 
Continuous univariate supported on a bounded interval
Beta · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle
MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... A probability distribution describes the values and probabilities that a random event can take place. ... A probability distribution describes the values and probabilities that a random event can take place. ... 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. ... Often confused with the multinomial distribution. ... // 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 theory and statistics, the Rademacher distribution is a discrete probability distribution. ... 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. ... Zipfs law, an empirical law formulated using mathematical statistics, refers to the fact that many types of data studied in the physical and social sciences can be approximated with a Zipfian distribution, one of a family of related discrete power law probability distributions. ... In probability theory and statistics, the Zipf-Mandelbrot law is a discrete probability distribution. ... A probability distribution describes the values and probabilities that a random event can take place. ... 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-distributed 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, 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 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. ... 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 and statistics, the Yule-Simon distribution is a discrete probability distribution. ... In probability theory and statistics, the zeta distribution is a discrete probability distribution. ... 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, 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. ... 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 triangular distribution is a continuous probability distribution with lower limit a, mode c and upper limit b. ... In probability theory and statistics, the U-quadratic distribution is a continuous probability distribution defined by a unique quadratic function with lower limit a and upper limit b. ... 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 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... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... A Beta Prime Distribution is a distribution with probability function: where is a Beta function. ... This article is about the mathematics of the chi-square distribution. ... 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. ... In statistics and probability, the F-distribution is a continuous probability distribution. ... This article does not cite its references or sources. ... The folded normal distribution is a probability distribution related to the normal distribution. ... 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, Fréchet and Weibull families also known as type I, II and III extreme value distributions. ... 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, 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 scale-inverse-chi-square distribution arises in Bayesian statistics (spam filtering in particular). ... 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, 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 and statistics, the log-normal distribution is the single-tailed probability distribution of any random variable whose logarithm is normally distributed. ... The Maxwell–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. ... In probability theory and statistics, the noncentral chi-square or noncentral distribution is a generalization of the chi-square distribution. ... 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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 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-2 Gumbel distribution function is for . Based on gsl-ref_19. ... 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. ... This article or section is in need of attention from an expert on the subject. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... 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). ... This article needs cleanup. ... The exponential power distribution, also known as the generalized error distribution, takes a scale parameter a and exponent b. ... 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. ... 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 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. ... 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. ... 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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. ... Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors α. Clockwise from top left: α=(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 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. ... In probability theory and statistics, the normal-gamma distribution is a four-parameter family of continuous probability distributions. ... A probability distribution describes the values and probabilities that a random event can take place. ... In statistics, the Inverse Wishart distribution, also the inverse Wishart distribution and inverted Wishart distribution is a probability density function defined on matrices. ... The matrix normal distribution is a probability distribution that is a generalization of the normal distribution. ... 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. ... Circular or directional statistics is the subdiscipline of statistics that deals with circular or directional data. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... 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. ... Circular or directional statistics is the subdiscipline of statistics that deals with circular or directional data. ... The 5-parameter Fisher-Bingham distribution or Kent distribution is a probability distribution on the three-dimensional sphere. ... In probability theory and statistics, the von Mises distribution is a continuous probability distribution. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... 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 δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. ... 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. ... The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. ... In probability and statistics, an exponential family is any class of probability distributions having a certain form. ... 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 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. ... This article or section is incomplete and may require expansion and/or cleanup. ...

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Beta Tools USA: Beta Procare (179 words)
Beta products are distributed in the United States and Canada by Beta Distribution USA.
The company has now begun opening a limited number of Beta Procare Franchise Stores in selected geographical markets and sales channels here in the U.S. Beta has long been established as one of the world\'s leading hand tool manufacturers and has distributors in over 93 counties.
Individualized responses or communications to you that involve either effecting or attempting to effect the sale of a franchise will be made only if we are first in compliance with state registration requirements, or are covered by an applicable state exclusion or exemption.
1.3.6.6.17. Beta Distribution (417 words)
The beta is different in that we define the general distribution in terms of the lower and upper bounds.
The following is the plot of the beta cumulative distribution function with the same values of the shape parameters as the pdf plots above.
Since the beta distribution is not typically used for reliability applications, we omit the formulas and plots for the hazard, cumulative hazard, survival, and inverse survival probability functions.
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