ThePearson distribution is a family ofcontinuousprobability distributions. It was first published byKarl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles onbiostatistics.

The Pearson system was originally devised in an effort to model visiblyskewed observations. It was well known at the time how to adjust a theoretical model to fit the first twocumulants ormoments of observed data: Anyprobability distribution can be extended straightforwardly to form alocation-scale family. Except inpathological cases, a location-scale family can be made to fit the observedmean (first cumulant) andvariance (second cumulant) arbitrarily well. However, it was not known how to construct probability distributions in which theskewness (standardized third cumulant) andkurtosis (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 thenormal distribution (which was originally known as type V). The classification depended on whether the distributions weresupported on a bounded interval, on a half-line, or on the wholereal 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 thenormal distribution, but now theinverse-gamma distribution) 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 theskewness: β₁ = γ₁² where γ₁ is the skewness, or thirdstandardized moment. The second is the traditionalkurtosis, 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 shows which Pearson type a given concrete distribution (identified by a point (β₁, β₂)) belongs to.

Many of the skewed or non-mesokurtic distributions familiar to statisticians today were still unknown in the early 1890s. What is now known as thebeta distribution had been used byThomas Bayes as aposterior distribution of the parameter of aBernoulli distribution in his 1763 work oninverse 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.) Thegamma 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 containsStudent'st-distribution as a special case, predatingWilliam Sealy Gosset's subsequent use by several years. His 1901 paper introduced theinverse-gamma distribution (type V) and thebeta prime distribution (type VI).

A Pearsondensityp is defined to be any valid solution to thedifferential equation (cf. Pearson 1895, p. 381)

${\displaystyle \frac{p^{\prime }(x)}{p(x)}+\frac{a+(x-\mu )}{b_0+b_1(x-\mu )+b_2(x-\mu {)}^2}=0.(1)}$

with:

According to Ord,^[3] 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 thenormal distribution (which gives a linear function) and, secondly, from a recurrence relation for values in theprobability mass function of thehypergeometric distribution (which yields the linear-divided-by-quadratic structure).

In Equation (1), the parametera determines astationary point, and hence under some conditions amode of the distribution, since

${\displaystyle p^{\prime }(\mu -a)=0}$

follows directly from the differential equation.

Since we are confronted with afirst-order linear differential equation with variable coefficients, its solution is straightforward:

${\displaystyle p(x)\propto \exp \left(-\int \frac{x+a}{b_2{x}^2+b_{1}x+b_0}dx\right).}$

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 thediscriminant (and hence the number of realroots) of thequadratic function

${\displaystyle f(x)=b_2{x}^2+b_{1}x+b_0.(2)}$

If the discriminant of the quadratic function (2) is negative (), it has no real roots. Then define

Observe thatα is a well-defined real number andα ≠ 0, because by assumption${\displaystyle 4b_2{b}_0-b_1^2>0}$ and thereforeb₂ ≠ 0. Applying these substitutions, the quadratic function (2) is transformed into

${\displaystyle f(x)=b_2(y^2+{\alpha }^2).}$

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 ofy:

${\displaystyle p(y)\propto \exp \left(-\frac{1}{b_2}\int \frac{y-\frac{b_1}{2b_2}+a}{y^2+{\alpha }^2}dy\right).}$

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

${\displaystyle \int \frac{y-\frac{2b_{2}a-b_1}{2b_2}}{y^2+{\alpha }^2}dy=\frac{1}{2}\ln (y^2+{\alpha }^2)-\frac{2b_{2}a-b_1}{2b_2\alpha }\arctan \left(\frac{y}{\alpha }\right)+C_0}$

involves theinversetrigonometric arctan function. Then

${\displaystyle p(y)\propto \exp \left[-\frac{1}{2b_2}\ln \left(1+\frac{y^2}{{\alpha }^2}\right)-\frac{\ln \alpha }{b_2}+\frac{2b_{2}a-b_1}{2b_2^2\alpha }\arctan \left(\frac{y}{\alpha }\right)+C_1\right].}$

Finally, let

Applying these substitutions, we obtain the parametric function:

${\displaystyle p(y)\propto {\left[1+\frac{y^2}{{\alpha }^2}\right]}^{-m}\exp \left[-\nu \arctan \left(\frac{y}{\alpha }\right)\right].}$

This unnormalized density hassupport on the entirereal line. It depends on ascale parameter α > 0 andshape parametersm > 1/2 and ν. One parameter was lost when we chose to find the solution to the differential equation (1) as a function ofy rather thanx. We therefore reintroduce a fourth parameter, namely thelocation parameterλ. We have thus derived the density of thePearson type IV distribution:

${\displaystyle p(x)=\frac{{\left|\frac{\mathrm{\Gamma }\left(m+\frac{\nu }{2}i\right)}{\mathrm{\Gamma }(m)}\right|}^2}{\alpha \mathrm{B}\left(m-\frac{1}{2},\frac{1}{2}\right)}{\left[1+{\left(\frac{x-\lambda }{\alpha }\right)}^2\right]}^{-m}\exp \left[-\nu \arctan \left(\frac{x-\lambda }{\alpha }\right)\right].}$

Thenormalizing constant involves thecomplexGamma function (Γ) and theBeta function (B).Notice that thelocation parameterλ here is not the same as the original location parameter introduced in the general formulation, but is related via

${\displaystyle \lambda ={\lambda }_{original}+\frac{\alpha \nu }{2(m-1)}.}$

The shape parameterν of the Pearson type IV distribution controls itsskewness. If we fix its value at zero, we obtain a symmetric three-parameter family. This special case is known as thePearson type VII distribution (cf. Pearson 1916, p. 450). Its density is

${\displaystyle p(x)=\frac{1}{\alpha \mathrm{B}\left(m-\frac{1}{2},\frac{1}{2}\right)}{\left[1+{\left(\frac{x-\lambda }{\alpha }\right)}^2\right]}^{-m},}$

where B is theBeta function.

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

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

which requiresm > 3/2. This entails a minor loss of generality but ensures that thevariance of the distribution exists and is equal to σ². Now the parameterm only controls thekurtosis of the distribution. Ifm approaches infinity asλ andσ are held constant, thenormal distribution arises as a special case:

This is the density of a normal distribution with meanλ and standard deviationσ.

It is convenient to require thatm > 5/2 and to let

${\displaystyle m=\frac{5}{2}+\frac{3}{{\gamma }_2}.}$

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 ofλ,standard deviation ofσ,skewness of zero, and positiveexcess kurtosis of γ₂.

The Pearson type VII distribution is equivalent to the non-standardizedStudent'st-distribution with parameters ν > 0, μ, σ² by applying the following substitutions to its original parameterization:

Observe that the constraintm > 1/2 is satisfied.

The resulting density is

${\displaystyle p(x\mid \mu ,{\sigma }^2,\nu )=\frac{1}{\sqrt{\nu {\sigma }^2}\mathrm{B}\left(\frac{\nu }{2},\frac{1}{2}\right)}{\left(1+\frac{1}{\nu }\frac{(x-\mu {)}^2}{{\sigma }^2}\right)}^{-\frac{\nu +1}{2}},}$

which is easily recognized as the density of a Student'st-distribution.

This implies that the Pearson type VII distribution subsumes the standardStudent'st-distribution and also the standardCauchy distribution. In particular, the standard Student'st-distribution arises as a subcase, whenμ = 0 andσ² = 1, equivalent to the following substitutions:

The density of this restricted one-parameter family is a standard Student'st:

${\displaystyle p(x)=\frac{1}{\sqrt{\nu }\mathrm{B}\left(\frac{\nu }{2},\frac{1}{2}\right)}{\left(1+\frac{x^2}{\nu }\right)}^{-\frac{\nu +1}{2}},}$

If the quadratic function (2) has a non-negative discriminant (${\displaystyle b_1^2-4b_2{b}_0\ge 0}$), it has real rootsa₁ anda₂ (not necessarily distinct):

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

${\displaystyle f(x)=b_2(x-a_1)(x-a_2),}$

and the solution to the differential equation is therefore

${\displaystyle p(x)\propto \exp \left(-\frac{1}{b_2}\int \frac{x-a}{(x-a_1)(x-a_2)}dx\right).}$

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

${\displaystyle \int \frac{x-a}{(x-a_1)(x-a_2)}dx=\frac{(a_1-a)\ln (x-a_1)-(a_2-a)\ln (x-a_2)}{a_1-a_2}+C}$

involves only thelogarithm function and not the arctan function as in the previous case.

Using the substitution

${\displaystyle \nu =\frac{1}{b_2(a_1-a_2)},}$

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

${\displaystyle p(x)\propto (x-a_1{)}^{-\nu (a_1-a)}(x-a_2{)}^{\nu (a_2-a)}.}$

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

${\displaystyle p(x)\propto {\left(1-\frac{x}{a_1}\right)}^{-\nu (a_1-a)}{\left(1-\frac{x}{a_2}\right)}^{\nu (a_2-a)}.}$

ThePearson type I distribution (a generalization of thebeta distribution to more general finite region of support) arises when the roots of the quadratic equation (2) are of opposite sign, that is,. Then the solutionp is supported on the interval${\displaystyle (a_1,a_2)}$. Apply the substitution

${\displaystyle x=a_1+y(a_2-a_1),}$

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

${\displaystyle p(y)\propto {\left(\frac{a_1-a_2}{a_1}y\right)}^{(-a_1+a)\nu }{\left(\frac{a_2-a_1}{a_2}(1-y)\right)}^{(a_2-a)\nu }.}$

One may define:

Regrouping constants and parameters, this simplifies to

${\displaystyle p(y)\propto y^{m_1}(1-y{)}^{m_2},}$

Thus${\displaystyle \frac{x-\lambda -a_1}{a_2-a_1}}$ follows abeta distribution${\displaystyle \mathrm{B}(m_1+1,m_2+1)}$ with${\displaystyle \lambda ={\mu }_1-(a_2-a_1)\frac{m_1+1}{m_1+m_2+2}-a_1}$. It turns out thatm₁,m₂ > −1 is necessary and sufficient forp to be a proper probability density function.

ThePearson type II distribution is a special case of the Pearson type I family restricted to symmetric distributions. Using formulae from the type I section, with${\displaystyle m_1=m_2=m}$ and${\displaystyle -a_1=a_2=a}$, on the interval (−a, a) it can be written as:

${\displaystyle p(x)\propto {\left(1-\frac{x^2}{a^2}\right)}^m.}$

Or with

${\displaystyle x=-a+2ya,}$

${\displaystyle y}$ is distributed according to thebeta distribution on the interval (0, 1),

${\displaystyle p(y)\propto {\left(1-4{\left(y-\frac{1}{2}\right)}^2\right)}^m\propto y^m(1-y{)}^m.}$

with appropriate constant of proportionality the PDF becomes

${\displaystyle p(y)=y^m(1-y{)}^m\frac{\mathrm{\Gamma }(2m+2)}{\mathrm{\Gamma }(m+1{)}^2}.}$

Defining

${\displaystyle \lambda ={\mu }_1+\frac{b_0}{b_1}-(m+1)b_1,}$

${\displaystyle b_0+b_1(x-\lambda )}$ is${\displaystyle \mathrm{Gamma}(m+1,b_1^2)}$. The Pearson type III distribution is agamma distribution orchi-squared distribution.

Defining new parameters:

${\displaystyle x-\lambda }$ follows an${\displaystyle \mathrm{InverseGamma}(\frac{1}{b_2}-1,\frac{a-C_1}{b_2})}$. The Pearson type V distribution is aninverse-gamma distribution.

Defining

${\displaystyle \lambda ={\mu }_1+(a_2-a_1)\frac{m_2+1}{m_2+m_1+2}-a_2,}$

${\displaystyle \frac{x-\lambda -a_2}{a_2-a_1}}$ follows a${\displaystyle {\beta }^{\mathrm{\prime }}(m_2+1,-m_2-m_1-1)}$. The Pearson type VI distribution is abeta prime distribution orF-distribution.

The Pearson family subsumes the following distributions, among others:

• Bernoulli distribution (type B, limit of type I)
• Beta distribution (type I and its symmetric subtype II)
• Beta prime distribution (type VI)
• Cauchy distribution (subtype of type VII)
• Chi-squared distribution (subtype of type III)
• Continuous uniform distribution (type R, subtype of types II, VIII, IX, and XII)
• Exponential distribution (type X/E, subtype of type III, limit of types IX and XI)
• Gamma distribution (type III, limit of types I and VI)
• Generalized Pareto distribution (types VIII, IX, X, and XI)
• F-distribution (subtype of type VI)
• Inverse-chi-squared distribution (subtype of type V)
• Inverse-gamma distribution (type V, limit of types IV and VI)
• Normal distribution (type G, limit of types I/II, III, IV/VII, V, and VI)
• Pareto distribution (type XI, subtype of type VI)
• Power function distribution (types VIII and IX, subtypes of type I)
• Student'st-distribution (type VII, symmetric subtype of type IV)

As of 2025, the only types without a name are types IV (seeabove) and XII (beta distribution with${\displaystyle \alpha +\beta =2}$).

Alternatives to the Pearson system of distributions for the purpose of fitting distributions to data are thequantile-parameterized distributions (QPDs) and themetalog distributions. QPDs and metalogs can provide greater shape and bounds flexibility than the Pearson system. Instead of fitting moments, QPDs are typically fit toempirical CDF or other data withlinear least squares.

Examples of modern alternatives to the Pearson skewness-vs-kurtosis diagram are: (i)https://github.com/SchildCode/PearsonPlot and (ii) the "Cullen and Frey graph" in the statistical application R.

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.,^{[which?][citation needed]} and this family of distributions may prove to be one of the more important.

In the United States, thelog-gamma distribution (historically named Log-Pearson III) is the default distribution for flood frequency analysis.^[4]

Recently, there have been alternatives developed to the Pearson distributions that are more flexible and easier to fit to data. See themetalog distributions.

• Pearson, Karl (1893)."Contributions to the mathematical theory of evolution [abstract]".Proceedings of the Royal Society.54 (326–330):329–333.doi:10.1098/rspl.1893.0079.JSTOR 115538.

• Pearson, Karl (1895)."Contributions to the mathematical theory of evolution, II: Skew variation in homogeneous material"(PDF).Philosophical Transactions of the Royal Society.186:343–414.Bibcode:1895RSPTA.186..343P.doi:10.1098/rsta.1895.0010.JSTOR 90649.

• Pearson, Karl (1901)."Mathematical contributions to the theory of evolution, X: Supplement to a memoir on skew variation".Philosophical Transactions of the Royal Society A.197 (287–299):443–459.Bibcode:1901RSPTA.197..443P.doi:10.1098/rsta.1901.0023.JSTOR 90841.

• Pearson, Karl (1916)."Mathematical contributions to the theory of evolution, XIX: Second supplement to a memoir on skew variation".Philosophical Transactions of the Royal Society A.216 (538–548):429–457.Bibcode:1916RSPTA.216..429P.doi:10.1098/rsta.1916.0009.JSTOR 91092.

• Rhind, A. (July–October 1909)."Tables to facilitate the computation of the probable errors of the chief constants of skew frequency distributions".Biometrika.7 (1/2):127–147.doi:10.1093/biomet/7.1-2.127.JSTOR 2345367.

• Milton Abramowitz and Irene A. Stegun (1964).Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.National Bureau of Standards.
• Eric W. Weisstein et al.Pearson Type III Distribution. FromMathWorld.

• 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.

• Continuous distributions
• Systems of probability distributions

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