Pearson distribution
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The Pearson distribution is a family of
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
s. It was first published by
Karl Pearson Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician and biostatistician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university st ...
in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on
biostatistics Biostatistics (also known as biometry) are the development and application of statistical methods to a wide range of topics in biology. It encompasses the design of biological experiments, the collection and analysis of data from those experimen ...
.


History

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
cumulant In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s or
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s of observed data: Any
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
can be extended straightforwardly to form a location-scale family. Except in
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cases, a location-scale family can be made to fit the observed
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' ar ...
(first cumulant) and
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
(second cumulant) arbitrarily well. However, it was not known how to construct probability distributions in which the skewness (standardized third cumulant) and
kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurt ...
(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. In his original paper, Pearson (1895, p. 360) identified four types of distributions (numbered I through IV) in addition to the
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
(which was originally known as type V). The classification depended on whether the distributions were
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
ed on a bounded interval, on a half-line, or on the whole
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
; 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 In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
, but now the
inverse-gamma distribution In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
) and introduced the type VI distribution. Together the first two papers cover the five main types of the Pearson system (I, III, IV, V, and VI). In a third paper, Pearson (1916) introduced further special cases and subtypes (VII through XII). 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 In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant ...
. The second is the traditional
kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurt ...
, 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 (β1, β2)) belongs to. 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 In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
had been used by Thomas Bayes as a posterior distribution of the parameter of a
Bernoulli distribution In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probabi ...
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. (Pearson's type II distribution is a special case of type I, but is usually no longer singled out.) The
gamma distribution In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma di ...
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. Pearson's 1895 paper introduced the type IV distribution, which contains Student's ''t''-distribution as a special case, predating
William Sealy Gosset William Sealy Gosset (13 June 1876 – 16 October 1937) was an English statistician, chemist and brewer who served as Head Brewer of Guinness and Head Experimental Brewer of Guinness and was a pioneer of modern statistics. He pioneered small sa ...
's subsequent use by several years. His 1901 paper introduced the
inverse-gamma distribution In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
(type V) and the beta prime distribution (type VI).


Definition

A Pearson
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
''p'' is defined to be any valid solution to the
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
(cf. Pearson 1895, p. 381) :\frac + \frac = 0. \qquad (1) with: :\begin b_0 &= \frac \mu_2, \\ pta = b_1 &= \sqrt \sqrt\frac, \\ ptb_2 &= \frac. \end According to Ord, Pearson devised the underlying form of Equation (1) on the basis of, firstly, the formula for the derivative of the logarithm of the density function of the
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
(which gives a linear function) and, secondly, from a recurrence relation for values in the
probability mass function In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
of the hypergeometric distribution (which yields the linear-divided-by-quadratic structure). In Equation (1), the parameter ''a'' determines a
stationary point In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" in ...
, and hence under some conditions a
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of the distribution, since :p'(\mu-a) = 0 follows directly from the differential equation. Since we are confronted with a first-order linear differential equation with variable coefficients, its solution is straightforward: :p(x) \propto \exp\left( -\int\frac \,dx \right). 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 In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
(and hence the number of real
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
s) of the
quadratic function In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomia ...
:f(x) = b_2x^2 + b_1x + b_0. \qquad (2)


Particular types of distribution


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 :\begin y &= x + \frac, \\ pt\alpha &= \frac. \end Observe that is a well-defined real number and , because by assumption 4 b_2 b_0 - b_1^2 > 0 and therefore . 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 \exp\left(- \frac \int\frac \,dy \right). Pearson (1895, p. 362) called this the "trigonometrical case", because the integral :\int\frac \,dy = \frac \ln(y^2 + \alpha^2) - \frac\arctan\left(\frac\right) + C_0 involves the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
trigonometric arctan function. Then :p(y) \propto \exp\left -\frac \ln\left(1+\frac\right) -\frac +\frac \arctan\left(\frac\right) + C_1 \right Finally, let :\begin m &= \frac, \\ pt\nu &= -\frac. \end Applying these substitutions, we obtain the parametric function: :p(y) \propto \left + \frac\right \exp\left \nu \arctan\left(\frac\right) \right This unnormalized density has
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on the entire
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
. It depends on a
scale parameter In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family o ...
α > 0 and
shape parameter In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. t ...
s ''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: :p(x) = \frac \left + \left(\frac\right)^2 \right \exp\left \nu \arctan\left(\frac \alpha \right)\right The normalizing constant involves the complex
Gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...
(Γ) and the Beta function (B). Notice that the location parameter ''λ'' here is not the same as the original location parameter introduced in the general formulation, but is related via :\lambda = \lambda_ + \frac.


The Pearson type VII distribution

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 :p(x) = \frac \left + \left(\frac \alpha \right)^2 \right, where B is the Beta function. An alternative parameterization (and slight specialization) of the type VII distribution is obtained by letting :\alpha = \sigma\sqrt, which requires ''m'' > 3/2. This entails a minor loss of generality but ensures that the
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
of the distribution exists and is equal to σ2. Now the parameter ''m'' only controls the
kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurt ...
of the distribution. If ''m'' approaches infinity as ''λ'' and ''σ'' are held constant, the
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
arises as a special case: :\begin &\lim_\frac \left + \left(\frac\right)^2 \right \\ pt= & \frac \cdot \lim_ \frac \cdot \lim_ \left + \frac \right \\ pt= & \frac \cdot 1 \cdot \exp\left \frac12 \left(\frac\right)^2 \right \end 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. 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 In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
of ''σ'', skewness of zero, and positive excess kurtosis of γ2.


Student's ''t''-distribution

The Pearson type VII distribution is equivalent to the non-standardized Student's ''t''-distribution with parameters ν > 0, μ, σ2 by applying the following substitutions to its original parameterization: :\begin \lambda &= \mu, \\ pt\alpha &= \sqrt, \\ ptm &= \frac, \end Observe that the constraint is satisfied. The resulting density is :p(x\mid\mu,\sigma^2,\nu) = \frac \left(1+\frac\frac\right)^, which is easily recognized as the density of a Student's ''t''-distribution. This implies that the Pearson type VII distribution subsumes the standard Student's ''t''-distribution and also the standard
Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
. In particular, the standard Student's ''t''-distribution arises as a subcase, when ''μ'' = 0 and ''σ''2 = 1, equivalent to the following substitutions: :\begin \lambda &= 0, \\ pt\alpha &= \sqrt, \\ ptm &= \frac, \end The density of this restricted one-parameter family is a standard Student's ''t'': :p(x) = \frac \left(1 + \frac \right)^,


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 ''a''1 and ''a''2 (not necessarily distinct): :\begin a_1 &= \frac, \\ pta_2 &= \frac. \end In the presence of real roots the quadratic function (2) can be written as :f(x) = b_2(x-a_1)(x-a_2), and the solution to the differential equation is therefore :p(x) \propto \exp\left( -\frac \int\frac \,dx \right). Pearson (1895, p. 362) called this the "logarithmic case", because the integral :\int\frac \,dx = \frac + C involves only the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
function and not the arctan function as in the previous case. Using the substitution :\nu = \frac, we obtain the following solution to the differential equation (1): :p(x) \propto (x-a_1)^ (x-a_2)^. 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\right)^ \left(1-\frac\right)^.


The Pearson type I distribution

The Pearson type I distribution (a generalization of the
beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
) arises when the roots of the quadratic equation (2) are of opposite sign, that is, a_1 < 0 < a_2. Then the solution ''p'' is supported on the interval (a_1, a_2). Apply the substitution :x = a_1 + y (a_2 - a_1), where 0, which yields a solution in terms of ''y'' that is supported on the interval (0, 1): :p(y) \propto \left(\fracy\right)^ \left(\frac(1-y)\right)^. One may define: :\begin m_1 &= \frac, \\ ptm_2 &= \frac. \end Regrouping constants and parameters, this simplifies to: :p(y) \propto y^ (1-y)^, Thus \frac follows a \Beta(m_1+1,m_2+1) with \lambda=\mu_1-(a_2-a_1) \frac-a_1. It turns out that ''m''1, ''m''2 > −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, :y = y_0\left(1-\frac\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 In statistics, Spearman's rank correlation coefficient or Spearman's ''ρ'', named after Charles Spearman and often denoted by the Greek letter \rho (rho) or as r_s, is a nonparametric measure of rank correlation (statistical dependence betwee ...
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 In probability and statistics, Student's ''t''-distribution (or simply the ''t''-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in situ ...
. For the table of values, certain values are used as the constants in the previous equation: :\begin m &= \frac, \\ pta^2 &= \frac, \\ pty_0 &= \frac. \end The moments of ''x'' used are :\begin \mu_2 &= (n-1) n^2+n)/122, \\ pt\beta_2 &= \frac. \end


The Pearson type III distribution

Defining :\lambda= \mu_1 + \frac - (m+1) b_1, b_0+b_1 (x-\lambda) is \operatorname(m+1,b_1^2). The Pearson type III distribution is a
gamma distribution In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma di ...
or
chi-squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squar ...
.


The Pearson type V distribution

Defining new parameters: :\begin C_1 &= \frac, \\ \lambda &= \mu_1-\frac , \end x-\lambda follows an \operatorname(\frac-1,\frac). The Pearson type V distribution is an
inverse-gamma distribution In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
.


The Pearson type VI distribution

Defining :\lambda=\mu_1 + (a_2-a_1) \frac - a_2, \frac follows a \beta^(m_2+1,-m_2-m_1-1). The Pearson type VI distribution is a beta prime distribution or ''F''-distribution.


Relation to other distributions

The Pearson family subsumes the following distributions, among others: *
Beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
(type I) * Beta prime distribution (type VI) *
Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
(type IV) *
Chi-squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squar ...
(type III) *
Continuous uniform distribution In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies bet ...
(limit of type I) *
Exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
(type III) *
Gamma distribution In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma di ...
(type III) * ''F''-distribution (type VI) *
Inverse-chi-squared distribution In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distributionBernardo, J.M.; Smith, A.F.M. (1993) ''Bayesian Theory'' ,Wiley (pages 119, 431) ) is a continuous probability distribution of a positive-val ...
(type V) *
Inverse-gamma distribution In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
(type V) *
Normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
(limit of type I, III, IV, V, or VI) * Student's ''t''-distribution (type VII, which is the non-skewed subtype of type IV) Alternatives to the Pearson system of distributions for the purpose of fitting distributions to data are the quantile-parameterized distributions (QPDs) and the metalog distributions. QPDs and metalogs can provide greater shape and bounds flexibility than the Pearson system. Instead of fitting moments, QPDs are typically fit to empirical CDF or other data with linear least squares.


Applications

These models are used in financial markets, given their ability to be parametrized 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. In the United States, the Log-Pearson III is the default distribution for flood frequency analysis. Recently, there have been alternatives developed to the Pearson distributions that are more flexible and easier to fit to data. See the metalog distributions.


Notes


Sources


Primary sources

* * * * *


Secondary sources

* Milton Abramowitz and Irene A. Stegun (1964). '' Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables''.
National Bureau of Standards The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...
. *
Eric W. Weisstein Eric Wolfgang Weisstein (born March 18, 1969) is an American mathematician and encyclopedist who created and maintains the encyclopedias ''MathWorld'' and ''ScienceWorld''. In addition, he is the author of the '' CRC Concise Encyclopedia of M ...
et al
Pearson Type III Distribution
From MathWorld.


References

*Elderton, Sir W.P, Johnson, N.L. (1969) ''Systems of Frequency Curves''. Cambridge University Press. *Ord J.K. (1972) ''Families of Frequency Distributions''. Griffin, London. {{DEFAULTSORT:Pearson Distribution Continuous distributions Systems of probability distributions