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Encyclopedia > Pearson distribution
Diagram of the Pearson system, showing distributions of types I, III, VI, V, and IV in terms of β1 (squared skewness) and β2 (traditional kurtosis)

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics. Image File history File links Size of this preview: 406 Ã— 600 pixelsFull resolution (1200 Ã— 1773 pixel, file size: 284 KB, MIME type: image/png) Diagram of the Pearson system of distributions File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old... Image File history File links Size of this preview: 406 Ã— 600 pixelsFull resolution (1200 Ã— 1773 pixel, file size: 284 KB, MIME type: image/png) Diagram of the Pearson system of distributions File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old... By one convention, a probability distribution is called continuous if its cumulative distribution function is continuous. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... Karl Pearson FRS (March 27, 1857 â€“ April 27, 1936) established the discipline of mathematical statistics. ... Year 1895 (MDCCCXCV) was a common year starting on Tuesday (link will display full calendar) of the Gregorian calendar (or a common year starting on Sunday of the 12-day-slower Julian calendar). ... Biostatistics or biometry is the application of statistics to a wide range of topics in biology. ...

The Pearson system was originally devised in an effort to model visibly skewed observations. It was well known at the time how to adjust a theoretical model to fit the first two cumulants or moments of observed data: Any probability distribution can be extended straightforwardly to form a location-scale family. Except in pathological cases, a location-scale family can be made to fit the observed mean (first cumulant) and variance (second cumulant) arbitrarily well. However, it was not known how to construct probability distributions in which the skewness (standardized third cumulant) and kurtosis (standardized fourth cumulant) could be adjusted equally freely. This need became apparent when trying to fit known theoretical models to observed data that exhibited skewness. Pearson's examples include survival data, which are usually asymmetric. 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. ... // Cumulants of probability distributions In probability theory and statistics, the cumulants Îºn of the probability distribution of a random variable X are given by In other words, Îºn/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function. ... -1... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... 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 mathematics, a pathological example is one whose properties are (or should be considered) untypically bad. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... 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. ...

In his original paper, Pearson (1895, p. 360) identified four types of distributions (numbered I through IV) in addition to the normal distribution (which was originally known as type V). The classification depended on whether the distributions were supported on a bounded interval, on a half-line, or on the whole real line; and whether they were potentially skewed or necessarily symmetric. A second paper (Pearson 1901) fixed two omissions: it redefined the type V distribution (originally just the normal distribution, but now the inverse-gamma distribution) and introduced the type VI distribution. Together the first two papers cover the five main types of the Pearson system (I, III, VI, V, and IV). In a third paper, Pearson (1916) introduced further special cases and subtypes (VII through XII). The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... 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, the real line is simply the set of real numbers. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... The inverse gamma distribution has the probability density function over the support with shape parameter and scale parameter . ...

Rhind (1909, pp. 430–432) devised a simple way of visualizing the parameter space of the Pearson system, which was subsequently adopted by Pearson (1916, plate 1 and pp. 430ff., 448ff.). The Pearson types are characterized by two quantities, commonly referred to as β1 and β2. The first is the square of the skewness: $beta_1 = gamma_1^2$ where γ1 is the skewness, or third standardized moment. The second is the traditional kurtosis, or fourth standardized moment: β2 = γ2 + 3. (Modern treatments define kurtosis γ2 in terms of cumulants instead of moments, so that for a normal distribution we have γ2 = 0 and β2 = 3. Here we follow the historical precedent and use β2.) The diagram on the right shows which Pearson type a given concrete distribution (identified by a point 12)) belongs to. 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. ... In probability theory and statistics, the kth standardized moment of a probability distribution is &#956;k/&#963;k, where &#956;k is the kth moment about the mean and &#963; is the standard deviation. ... 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. ...

Many of the skewed and/or non-mesokurtic distributions familiar to us today were still unknown in the early 1890s. What is now known as the beta distribution had been used by Thomas Bayes as a posterior distribution of the parameter of a Bernoulli distribution in his 1763 work on inverse probability. The Beta distribution gained prominence due to its membership in Pearson's system and was known until the 1940s as the Pearson type I distribution. [1] (Pearson's type II distribution is a special case of type I, but is usually no longer singled out.) The gamma distribution originated from Pearson's work (Pearson 1893, p. 331; Pearson 1895, pp. 357, 360, 373–376) and was known as the Pearson type III distribution, before acquiring its modern name in the 1930s and 1940s. [2] Pearson's 1895 paper introduced the type IV distribution, which contains Student's t-distribution as a special case, predating William Gosset's subsequent use by several years. His 1901 paper introduced the inverse-gamma distribution (type V) and the beta prime distribution (type VI). 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. ... 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. ... Thomas Bayes (c. ... In Bayesian probability theory, the posterior probability is the conditional probability of some event or proposition, taking empirical data into account. ... 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, inverse probability is an obsolete term for the probability distribution of an unobserved variable. ... In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. ... 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. ... William Sealy Gosset (June 13, 1876 â€“ October 16, 1937) was a chemist and statistician, better known by his pen name Student. ... The inverse gamma distribution has the probability density function over the support with shape parameter and scale parameter . ... A Beta Prime Distribution is a distribution with probability function: where is a Beta function. ...

## Definition

A Pearson density p is defined to be any valid solution to the differential equation (cf. Pearson 1895, p. 381) In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ...

$p'(x) + frac{x-a_0}{b_2 x^2 + b_1 x + b_0};p(x) = 0. qquad (1) !$

The Pearson family has four parameters (a0, b0, b1, b2), which can be used to freely adjust the first four moments of the distribution, subject to very few constraints. The parameter a0 determines a stationary point, and hence under some conditions a mode of the distribution, since Stationary points (red pluses) and inflection points (green circles). ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ...

$p'(a_0) = 0 !$

follows directly from the differential equation.

Since we are confronted with a linear differential equation with variable coefficients, its solution is straightforward: In mathematics, a linear differential equation is a differential equation of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function. ...

The integral in this solution simplifies considerably when certain special cases of the integrand are considered. Pearson (1895, p. 367) distinguished two main cases, determined by the sign of the discriminant (and hence the number of real roots) of the quadratic function In algebra, the discriminant of a polynomial is a certain expression in the coefficients of the polynomial which equals zero if and only if the polynomial has multiple roots in the complex numbers. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... f(x) = x2 - x - 2 A quadratic function, in mathematics, is a polynomial function of the form , where are real numbers and . ...

$f(x) = b_2,x^2 + b_1,x + b_0. qquad (2)!$

### Case 1, negative discriminant: The Pearson type IV distribution

If the discriminant of the quadratic function (2) is negative ($b_1^2 - 4 b_2 b_0 < 0$), it has no real roots. Then define

$y = x + frac{b_1}{2,b_2} !$  and
$alpha = frac{sqrt{4,b_2,b_0 - b_1^2,}}{2,b_2}. !$

Observe that α is a well-defined real number and $alpha neq 0$, because by assumption $4 b_2 b_0 - b_1^2 > 0$ and therefore $b_2 neq 0$. Applying these substitutions, the quadratic function (2) is transformed into

$f(x) = b_2,(y^2 + alpha^2). !$

The absence of real roots is obvious from this formulation, because α2 is necessarily positive.

We now express the solution to the differential equation (1) as a function of y:

$p(y) propto expleft(- frac{1}{b_2}, intfrac{y - frac{b_1}{2,b_2} - a_0}{y^2 + alpha^2} ,mathrm{d}y right). !$

Pearson (1895, p. 362) called this the "trigonometrical case", because the integral

$intfrac{y - frac{2,b_2,a_0 + b_1}{2,b_2}}{y^2 + alpha^2} ,mathrm{d}y = frac{1}{2} ln(y^2 + alpha^2) - frac{2,b_2,a_0 + b_1}{2,b_2,alpha} arctanleft(frac{y}{alpha}right) + C_0 !$

involves the inverse trigonometic arctan function. Then In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ...

$p(y) propto expleft[ -frac{1}{2,b_2} ln!left(1+frac{y^2}{alpha^2}right) -frac{lnalpha}{2,b_2} +frac{2,b_2,a_0 + b_1}{2,b_2^2,alpha} arctanleft(frac{y}{alpha}right) + C_1 right] !$

Finally, let

$m = frac{1}{2,b_2} !$  and
$nu = -frac{2,b_2,a_0 + b_1}{2,b_2^2,alpha} !$

Applying these substitutions, we obtain the parametric function:

$p(y) propto left[1 + frac{y^2}{alpha^2}right]^{-m} expleft[-nu arctanleft(frac{y}{alpha}right)right] !$

This unnormalized density has support on the entire real line. It depends on a scale parameter α > 0 and shape parameters m > 1 / 2 and ν. One parameter was lost when we chose to find the solution to the differential equation (1) as a function of y rather than x. We therefore reintroduce a fourth parameter, namely the location parameter λ. We have thus derived the density of the Pearson type IV distribution: 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, the real line is simply the set of real numbers. ... In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... In probability theory and statistics, a shape parameter is a special kind of numerical parameter of a parametric family of probability distributions. ... In statistics, if a family of probabiblity densities parametrized by a scalar- or vector-valued parameter &#956; is of the form f&#956;(x) = f(x &#8722; &#956;) then &#956; is called a location parameter, since its value determines the location of the probability distribution. ...

$p(x) = frac{left|frac{Gamma!left(m+frac{nu}{2}iright)}{Gamma(m)}right|^2} {alpha,mathrm{Beta}!left(m-frac12, frac12right)} left[1 + left(frac{x-lambda}{alpha}right)^{!2,} right]^{-m} expleft[-nu arctanleft(frac{x-lambda}{alpha}right)right]. !$

The normalizing constant involves the complex Gamma function (Γ) and the Beta function (B). The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ... In theoretical physics, specifically quantum field theory, a beta-function Î²(g) encodes the dependence of a coupling parameter, g, on the energy scale, of a given physical process. ...

#### The Pearson type VII distribution

Plot of Pearson type VII densities with λ = 0, σ = 1, and: $gamma_2=infty$ (red); γ2 = 4 (blue); and γ2 = 0 (black)

The shape parameter ν of the Pearson type IV distribution controls its skewness. If we fix its value at zero, we obtain a symmetric three-parameter family. This special case is known as the Pearson type VII distribution (cf. Pearson 1916, p. 450). Its density is Image File history File links Size of this preview: 800 Ã— 600 pixel Image in higher resolution (1600 Ã— 1200 pixel, file size: 234 KB, MIME type: image/png) // Probability density function of the Pearson type VII distribution The red curve shows the limiting density with infinite kurtosis; the blue curve shows... Image File history File links Size of this preview: 800 Ã— 600 pixel Image in higher resolution (1600 Ã— 1200 pixel, file size: 234 KB, MIME type: image/png) // Probability density function of the Pearson type VII distribution The red curve shows the limiting density with infinite kurtosis; the blue curve shows... 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. ...

$p(x) = frac{1}{alpha,mathrm{Beta}!left(m-frac12, frac12right)} left[1 + left(frac{x-lambda}{alpha}right)^{!2,} right]^{-m}, !$

where B is the Beta function. In theoretical physics, specifically quantum field theory, a beta-function Î²(g) encodes the dependence of a coupling parameter, g, on the energy scale, of a given physical process. ...

An alternative parameterization (and slight specialization) of the type VII distribution is obtained by letting

$alpha = sigma,sqrt{2,m-3}, !$

which requires m > 3 / 2. This entails a minor loss of generality but ensures that the variance of the distribution exists and is equal to σ2. Now the parameter m only controls the kurtosis of the distribution. If m approaches infinity as λ and σ are held constant, the normal distribution arises as a special case: 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. ... 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. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ...

$lim_{mtoinfty} frac{1}{sigma,sqrt{2,m-3},mathrm{Beta}!left(m-frac12, frac12right)} left[1 + left(frac{x-lambda}{sigma,sqrt{2,m-3}}right)^{!2,} right]^{-m} !$
$= frac{1}{sigma,sqrt{2},Gamma!left(frac12right)} times lim_{mtoinfty} frac{Gamma(m)}{Gamma!left(m-frac12right) sqrt{m-frac32}} times lim_{mtoinfty} left[1 + frac{left(frac{x-lambda}{sigma}right)^2}{2,m-3} right]^{-m} !$
$= frac{1}{sigmasqrt{2,pi}} times 1 times exp!left[-frac12 left(frac{x-lambda}{sigma}right)^{!2,} right] !$

This is the density of a normal distribution with mean λ and standard deviation σ.

It is convenient to require that m > 5 / 2 and to let

$m = frac52 + frac{3}{gamma_2}. !$

This is another specialization, and it guarantees that the first four moments of the distribution exist. More specifically, the Pearson type VII distribution parameterized in terms of (λ,σ,γ2) has a mean of λ, standard deviation of σ, skewness of zero, and excess kurtosis of γ2. In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... 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. ...

#### Student's t-distribution

The Pearson type VII distribution subsumes Student's t-distribution, and hence also the Cauchy distribution. Student's t-distribution arises as the result of applying the following substitutions to its original parameterization: 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. ... 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). ...

$lambda = 0, !$
$alpha = sqrt{nu}, !$  and
$m = frac{nu+1}{2}, !$

where ν > 0. Observe that the constraint m > 1 / 2 is satisfied. The density of this restricted one-parameter family is

$p(x) = frac{1}{sqrt{nu},mathrm{Beta}!left(frac{nu}{2}, frac12right)} left[1 + frac{x^2}{nu} right]^{-frac{nu+1}{2}}, !$

which is easily recognized as the density of Student's t-distribution.

### Case 2, non-negative discriminant

If the quadratic function (2) has a non-negative discriminant ($b_1^2 - 4 b_2 b_0 geq 0$), it has real roots r1 and r2 (not necessarily distinct):

$r_{1,2} = frac{-b_1 pm sqrt{b_1^2 - 4 b_2 b_0}}{2 b_2}, !$

In the presence of real roots the quadratic function (2) can be written as

$f(x) = b_2,(x-r_1)(x-r_2), !$

and the solution to the differential equation is therefore

$p(x) propto expleft( -frac{1}{b_2} int!!frac{x-a_0}{(x - r_1) (x - r_2)} ,mathrm{d}x right). !$

Pearson (1895, p. 362) called this the "logarithmic case", because the integral

$int!!frac{x-a_0}{(x - r_1) (x - r_2)} ,mathrm{d}x = frac{(r_1-a_0)ln(x-r_1) - (r_2-a_0)ln(x-r_2)}{r_1-r_2} + C !$

involves only the logarithm function, and not the arctan function as in the previous case. Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

Using the substitution

$nu = frac{1}{b_2,(r_1-r_2)} !$

we obtain the following solution to the differential equation (1):

$p(x) propto (x-r_1)^{-nu (r_1-a_0)} (x-r_2)^{nu (r_2-a_0)}. !$

Since this density is only known up to a hidden constant of proportionality, that constant can be changed and the density written as follows:

$p(x) propto left(1-frac{x}{r_1}right)^{-nu (r_1-a_0)} left(1-frac{x}{r_2}right)^{ nu (r_2-a_0)} !$

#### The Pearson type I distribution

The Pearson type I distribution (a generalization of the beta distribution) arises when the roots of the quadratic equation (2) are of opposite sign, that is, r1 < 0 < r2. Then the solution p is supported on the interval (r1,r2). Apply the substition Image File history File links This is a lossless scalable vector image. ... 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. ...

$x = r_1 + y (r_2 - r_1) qquad mbox{where} 0

which yields a solution in terms of y that is supported on the interval (0,1):

$p(y) propto left(frac{r_1-r_2}{r_1};yright)^{(-r_1+a_0)nu} left(frac{r_2-r_1}{r_2};(1-y)right)^{(r_2-a_0)nu}. !$

Regrouping constants and parameters, this simplifies to:

$p(y) propto y^m (1-y)^n, !$

where m and n are two arbitrary real parameters. It turns out that $m>-1 land n>-1$ is necessary and sufficient for p to be a proper probability density function.

#### The Pearson type II distribution

The Pearson type II distribution is a special case of the Pearson type I family restricted to symmetric distributions.

For the Pearson Type II Curve [3] ,

$y = y_{0}left(1-frac{x^2}{a^2}right)^m$

where

$x = sum d^2/2 -(n^3-n)/12$

the ordinate, y, is the frequency of $sum d^2$. The Pearson Type II Curve is used in computing the table of significant correlation coefficients for Spearman's rank correlation coefficient when the number of items in a series is less than 100 (or 30, depending on some sources). After that, the distribution mimics a standard Student's t-distribution. For the table of values, certain values are used as the constants in the previous equation: In statistics, Spearmans rank correlation coefficient, named after Charles Spearman and often denoted by the Greek letter Ï (rho), is a non-parametric measure of correlation â€“ that is, it assesses how well an arbitrary monotonic function could describe the relationship between two variables, without making any assumptions about the frequency... 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. ...

$m = frac{5beta_{2}-9}{2(3-beta_{2})}$
$a^2 = frac{2mu_{2}beta_{2}}{3-beta_{2}}$
$y_{0} = frac{N[Gamma(2m+2)]}{a[2^{2m+1}][Gamma(m+1)]}$

The moments of x used are

μ2 = (n − 1)[(n2 + n) / 12]2
$beta_{2}=frac{3(25n^4-13n^3-73n^2+37n+72)}{25n(n+1)^2(n-1)}$

#### The Pearson type III distribution

Pearson type III distribution gamma distribution, chi-square distribution In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. ... 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 Pearson type V distribution

Pearson type V distribution inverse-gamma distribution The inverse gamma distribution has the probability density function over the support with shape parameter and scale parameter . ...

#### The Pearson type VI distribution

Pearson type VI distribution beta prime distribution, F-distribution A Beta Prime Distribution is a distribution with probability function: where is a Beta function. ... In statistics and probability, the F-distribution is a continuous probability distribution. ...

## Relation to other distributions

The Pearson family subsumes the following distributions, among others:

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. ... 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 exponential distributions are a class of continuous probability distribution. ... 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. ... 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 inverse gamma distribution has the probability density function over the support with shape parameter and scale parameter . ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... 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. ...

## Applications

These models are used in financial markets, given their ability to be parametrised in a way that has intuitive meaning for market traders. A number of models are in current use that capture the stochastic nature of the volatility of rates, stocks etc. and this family of distributions may prove to be one of the more important.

## Notes

1. ^ Miller, Jeff; et al. (2006-07-09). Beta distribution. Earliest Known Uses of Some of the Words of Mathematics. Retrieved on December 9, 2006.
2. ^ Miller, Jeff; et al. (2006-12-07). Gamma distribution. Earliest Known Uses of Some of the Words of Mathematics. Retrieved on December 9, 2006.
3. ^ Ramsey, Philip H. (1989-09-01). Critical Values for Spearman's Rank Order Correlation. Retrieved on August 22, 2007.

## Sources

### Primary sources

Karl Pearson FRS (March 27, 1857 â€“ April 27, 1936) established the discipline of mathematical statistics. ... Karl Pearson FRS (March 27, 1857 â€“ April 27, 1936) established the discipline of mathematical statistics. ... Karl Pearson FRS (March 27, 1857 â€“ April 27, 1936) established the discipline of mathematical statistics. ... Karl Pearson FRS (March 27, 1857 â€“ April 27, 1936) established the discipline of mathematical statistics. ...

### Secondary sources

Probability distributionsview  talk  edit ]
Univariate Multivariate
Discrete: BenfordBernoullibinomialBoltzmanncategoricalcompound Poisson • discrete phase-type • degenerate • Gauss-Kuzmin • geometrichypergeometriclogarithmicnegative binomialparabolic fractalPoissonRademacherSkellamuniform • Yule-Simon • zetaZipf • Zipf-Mandelbrot Ewensmultinomialmultivariate Polya
Continuous: BetaBeta primeCauchychi-squareDirac delta function • Coxian • Erlangexponentialexponential powerFfading • Fermi-Dirac • Fisher's z • Fisher-Tippett • Gammageneralized extreme valuegeneralized hyperbolicgeneralized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square) • inverse Gaussianinverse gamma (scaled inverse gamma) • KumaraswamyLandauLaplaceLévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speedNakagaminormal (Gaussian) • normal-gamma • normal inverse Gaussian • ParetoPearson • phase-type • polarraised cosineRayleigh • relativistic Breit-Wigner • Riceshifted Gompertz • Student's t • triangulartruncated normal • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigtvon MisesWeibullWigner semicircleWilks' lambda Dirichlet • Generalized Dirichlet distribution . inverse-Wishart • Kentmatrix normalmultivariate normalmultivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous: Cantorconditionalequilibriumexponential familyinfinitely divisible • location-scale family • marginalmaximum entropyposteriorprior • quasi • samplingsingular

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