Pearson distribution

Short description: Family of continuous probability distributions

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.

## Contents

* 1 History
* 2 Definition
* 3 Particular types of distribution
* 3.1 Case 1, negative discriminant
* 3.1.1 The Pearson type IV distribution
* 3.1.2 The Pearson type VII distribution
* 3.1.3 Student's t-distribution
* 3.2 Case 2, non-negative discriminant
* 3.2.1 The Pearson type I distribution
* 3.2.2 The Pearson type II distribution
* 3.2.3 The Pearson type III distribution
* 3.2.4 The Pearson type V distribution
* 3.2.5 The Pearson type VI distribution
* 4 Relation to other distributions
* 5 Applications
* 6 Notes
* 7 Sources
* 7.1 Primary sources
* 7.2 Secondary sources
* 7.3 References

## 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 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.

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, 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: β1 = γ12 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 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 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 Sealy Gosset's subsequent use by several years. His 1901 paper introduced the inverse-gamma distribution (type V) and the beta prime distribution (type VI).

## Definition

A Pearson density p is defined to be any valid solution to the differential equation (cf. Pearson 1902, p. 277)

p′(x)p(x)+A(x)B(x)=0(1)

where:

* A(x) = a1 \+ a2x is a polynomial of degree at most 1,
* B(x) = b0 \+ b1x \+ b2x2 with B ≠ 0 is a non-zero polynomial of degree at most 2,
* (a2, b2) ≠ (0, 0):
* if (a2, b2) = (0, 0) but (a1, b1) ≠ (0, 0), multiply A and B by x+u with arbitrary finite u ≠ ​b0⁄b1,
* if (a2, b2) = (0, 0) and (a1, b1) = (0, 0), multiply A and B by (x+u)(x+v) with arbitrary finite u and v satisfying u ≠ v,
* the domain I of p is an open interval (xmin, xmax) bounded by infinities and/or real zeros of B but not containing any of them, i.e.:
* if B has no real zeros, then I=ℝ,
* if B has one real zero x0, then I = (−∞, x0) or I = (x0, +∞),
* if B has two real zeros x− < x+, then I = (−∞, x−) or I = (x−, x+) or I = (x+, +∞).

Dimensional analysis shows that coefficients ak and bk have dimension [AX−k], with [A] an arbitrary dimension and [X] the dimension of x, hence the shift in the indices of ak.

If the distribution has moments up to order at least 4, with mean μ and standard deviation σ, we have (cf. Pearson 1916, p. 437, and Carver 1924, pp. 103–104)

p′(x)p(x)+1σc1+c2x−μσd0+d1x−μσ+d2(x−μσ)2=0

with

{c2=2(5β2−6β1−9)d0=4β2−3β1c1=d1=γ1(β2+3)d2=2β2−3β1−6

defined up to a non-zero multiplicative constant of proportionality. Moreover, the natural dimension of coefficients ak and bk is then [X8−k].

Proof

The proof below is an expansion of the proof sketched in Carver 1924, pp. 103–104. In particular, the fact that [q(x) D(x) xr]J = 0 was just assumed without any justification:

> [...] if the frequency function, when multiplied by tn, vanishes at the limits of the distribution [...]

* * *

Let X be the distribution with density p satisfying equation (1), with mean μ, standard deviation σ, skewness γ1, finite kurtosis β2, and domain I = (xmin, xmax) following the aforementioned conditions regarding the zeros of B: if xmin (resp. xmax) is finite, then B(xmin) = 0 (resp. B(xmax) = 0). Let Z = ⁠X−μ/σ⁠ be its standardization, which thus has density q(x) = σ p(σx+μ), domain J = Template:Pars, and its raw moments have the same numerical values (but not the same dimensions) as the standardized moments of X: ∫J xr q(x) dx = αr, with α0 = 1, α1 = 0, α2 = 1, α3 = γ1, and α4 = β2.

The density of Z satisfies

q′(x)q(x)=σ2p′(σx+μ)σp(σx+μ)=−σA(σx+μ)B(σx+μ)

which has the same linear-divided-by-quadratic structure as equation (1). We can thus set

q′(x)q(x)+C(x)D(x)=0(1′)

with C(x) = c1 \+ c2x and D(x) = d0 \+ d1x \+ d2x2. As D(x) is proportional to B(σx+μ), the finite boundaries of the domain J are zeros of D.

Conversely, as p(x) = σ−1 qTemplate:Pars, we get

p′(x)p(x)=σ−2q′(x−μσ)σ−1q(x−μσ)=−1σC(x−μσ)D(x−μσ),

hence

p′(x)p(x)+1σc1+c2x−μσd0+d1x−μσ+d2(x−μσ)2=0

with coefficients ck and dk to be determined.

Given a natural number r ∈ ⟦0, 3⟧, we then multiply each term of equation (1') by q(x) D(x) xr, yielding

q′(x)D(x)xr+q(x)(c1xr+c2xr+1)=0.

Integrating the two terms of the equation over J, and integrating by parts the first addend, we obtain

0=∫J0dx=[q(x)D(x)xr]J−∫Jq(x)[D(x)xr]′dx+∫Jq(x)(c1xr+c2xr+1)dx=[q(x)D(x)xr]J+∫Jq(x)[c1xr+c2xr+1−rd0xr−1−(r+1)d1xr−(r+2)d2xr+1]dx=[q(x)D(x)xr]J+c1αr+c2αr+1−rd0αr−1−(r+1)d1αr−(r+2)d2αr+1.

Let t be a boundary, finite or infinite, of the domain J:

* if t is finite:
* the convergence of ∫J q(x) dx = 1 means that we have q(x) = o((x−t)−1) as x → t, hence limx → t q(x) (x−t) = 0,
* t is a zero of the quadratic polynomial D so ⁠D(x)/x−t⁠ is a linear polynomial taking a finite value at t,
* tr is finite,
* thus limx → tq(x) D(x) xr = 0;
* if t is infinite:
* the existence of the kurtosis of X, and thus of Z, means that we have xη q(x) = o(x−1) for at least η ≤ 4 as x → t, hence limx → t q(x) xn = 0 for at least n ≤ 5,
* D(x) xr is a polynomial of degree at most r+2 ≤ 5,
* thus limx → tq(x) D(x) xr = 0.

Consequently, we always have [q(x) D(x) xr]J = 0.

Letting r take all values in ⟦0, 3⟧, we thus obtain a system of 4 linear equations:

{0=c1α0+c2α1−0d0α−1−1d1α0−2d2α1=c1−d10=c1α1+c2α2−1d0α0−2d1α1−3d2α2=c2−d0−3d20=c1α2+c2α3−2d0α1−3d1α2−4d2α3=c1+γ1c2−3d1−4γ1d20=c1α3+c2α4−3d0α2−4d1α3−5d2α4=γ1c1+β2c2−3d0−4γ1d1−5β2d2.

We thus have c1 = d1 and c2 = d0 \+ 3 d2, yielding

{0=d1+γ1(d0+3d2)−3d1−4γ1d2=γ1d0−2d1−γ1d20=γ1d1+β2(d0+3d2)−3d0−4γ1d1−5β2d2=(β2−3)d0−3γ1d1−2β2d2.

Then d1 = ​γ1 (d0 − d2)⁄2, thus

0=2β2d0−6d0−3γ12(d0−d2)−4β2d2=(2β2−6−3γ12)d0+(3γ12−4β2)d2=(2β2−3β1−6)d0−(4β2−3β1)d2.

As the coefficients ck and dk can be all multiplied by a non-zero constant without changing equation (1'), we can then freely set

{d0=4β2−3β1d2=2β2−3β1−6

resulting in

{c2=d0+3d2=2(5β2−6β1−9)c1=d1=γ1(d0−d2)/2=γ1(β2+3).

By setting ξ4 = β2 \+ 3 and ξ6 = 3β2 − 3β1 − 3, we obtain expressions without additive constants as we have:

{c2=2ξ6−ξ4d0=ξ6+ξ4c1=d1=γ1ξ4d2=ξ6−ξ4.

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 the normal distribution (which gives a linear function) and, secondly, from a recurrence relation for values in the probability mass function of the hypergeometric distribution (which yields the linear-divided-by-quadratic structure).

In equation (1), the parameters a1 and a2 determine a stationary point if a2 ≠ 0, and hence under some conditions a mode of the distribution, since

p′(−a1a2)=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)∝exp⁡(−∫a2x+a1b2x2+b1x+b0dx).

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

B(x)=b2x2+b1x+b0.(2)

## Particular types of distribution

### Case 1, negative discriminant

#### The Pearson type IV distribution

If the discriminant of the quadratic function (2) is negative (b12−4b2b0<0), it has no real roots. Then define

y=x+b12b2,α=4b2b0−b122b2.

Observe that α is a well-defined real number and α ≠ 0, because by assumption 4b2b0−b12>0 and therefore b2 ≠ 0. Applying these substitutions, the quadratic function (2) is transformed into

B(x)=b2(y2+α2).

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)∝exp⁡(−1b2∫y−b12b2+ay2+α2dy).

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

∫y−2b2a−b12b2y2+α2dy=12ln⁡(y2+α2)−2b2a−b12b2αarctan⁡(yα)+C0

involves the inverse trigonometric arctan function. Then

p(y)∝exp⁡[−12b2ln⁡(1+y2α2)−ln⁡αb2+2b2a−b12b22αarctan⁡(yα)+C1].

Finally, let

m=12b2,ν=−2b2a−b12b22α.

Applying these substitutions, we obtain the parametric function:

p(y)∝[1+y2α2]−mexp⁡[−νarctan⁡(yα)].

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:

p(x)=|Γ⁡(m+ν2i)Γ(m)|2αB⁡(m−12,12)[1+(x−λα)2]−mexp⁡[−νarctan⁡(x−λα)].

The normalizing constant involves the complex Gamma function (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

λ=λoriginal+αν2(m−1).

#### The Pearson type VII distribution

Plot of Pearson type VII densities with λ = 0, σ = 1, and: γ2 = ∞ (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

p(x)=1αB⁡(m−12,12)[1+(x−λα)2]−m,

where B is the Beta function.

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

α=σ2m−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:

limm->∞1σ2m−3B⁡(m−12,12)[1+(x−λσ2m−3)2]−m=1σ2Γ⁡(12)⋅limm->∞Γ(m)Γ⁡(m−12)m−32⋅limm->∞[1+(x−λσ)22m−3]−m=1σ2π⋅1⋅exp⁡[−12(x−λσ)2].

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=52+3γ2.

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 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:

λ=μ,α=νσ2,m=ν+12,

Observe that the constraint m > 1/2 is satisfied.

The resulting density is

p(x∣μ,σ2,ν)=1νσ2B⁡(ν2,12)(1+1ν(x−μ)2σ2)−ν+12,

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. In particular, the standard Student's t-distribution arises as a subcase, when μ = 0 and σ2 = 1, equivalent to the following substitutions:

λ=0,α=ν,m=ν+12,

The density of this restricted one-parameter family is a standard Student's t:

p(x)=1νB⁡(ν2,12)(1+x2ν)−ν+12,

### Case 2, non-negative discriminant

If the quadratic function (2) has a non-negative discriminant (b12−4b2b0≥0), it has real roots a1 and a2 (not necessarily distinct):

a1=−b1−b12−4b2b02b2,a2=−b1+b12−4b2b02b2.

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

B(x)=b2(x−a1)(x−a2),

and the solution to the differential equation is therefore

p(x)∝exp⁡(−1b2∫x−a(x−a1)(x−a2)dx).

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

∫x−a(x−a1)(x−a2)dx=(a1−a)ln⁡(x−a1)−(a2−a)ln⁡(x−a2)a1−a2+C

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

Using the substitution

ν=1b2(a1−a2),

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

p(x)∝(x−a1)−ν(a1−a)(x−a2)ν(a2−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:

p(x)∝(1−xa1)−ν(a1−a)(1−xa2)ν(a2−a).

#### The Pearson type I distribution

The Pearson type I distribution (a generalization of the beta distribution to more general finite region of support) arises when the roots of the quadratic equation (2) are of opposite sign, that is, a1<0<a2. Then the solution p is supported on the interval (a1,a2). Apply the substitution

x=a1+y(a2−a1),

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

p(y)∝(a1−a2a1y)(−a1+a)ν(a2−a1a2(1−y))(a2−a)ν.

One may define:

m1=a−a1b2(a1−a2),m2=a−a2b2(a2−a1).

Regrouping constants and parameters, this simplifies to

p(y)∝ym1(1−y)m2,

Thus x−λ−a1a2−a1 follows a beta distribution B(m1+1,m2+1) with λ=μ1−(a2−a1)m1+1m1+m2+2−a1. It turns out that m1, m2 > −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. Using formulae from the type I section, with m1=m2=m and −a1=a2=a, on the interval (−a, a) it can be written as:

p(x)∝(1−x2a2)m.

Or with

x=−a+2ya,

y is distributed according to the beta distribution on the interval (0, 1),

p(y)∝(1−4(y−12)2)m∝ym(1−y)m.

with appropriate constant of proportionality the PDF becomes

p(y)=ym(1−y)mΓ(2m+2)Γ(m+1)2.

#### The Pearson type III distribution

Defining

λ=μ1+b0b1−(m+1)b1,

b0+b1(x−λ) is Gamma⁡(m+1,b12). The Pearson type III distribution is a gamma distribution or chi-squared distribution.

#### The Pearson type V distribution

Defining new parameters:

C1=b12b2,λ=μ1−a−C11−2b2,

x−λ follows an InverseGamma⁡(1b2−1,a−C1b2). The Pearson type V distribution is an inverse-gamma distribution.

#### The Pearson type VI distribution

Defining

λ=μ1+(a2−a1)m2+1m2+m1+2−a2,

x−λ−a2a2−a1 follows a β′(m2+1,−m2−m1−1). 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:

* Bernoulli distribution (B; limit of I)
* Beta distribution (I and its symmetric subtype II)
* Beta prime distribution (VI)
* Cauchy distribution (subtype of VII)
* Chi-squared distribution (subtype of III)
* Continuous uniform distribution (R; subtype of II, VIII, IX, and XII)
* Exponential distribution (X/E; subtype of III, limit of IX and XI)
* Gamma distribution (III; limit of I and VI)
* Generalized Pareto distribution (VIII, IX, X/E, and XI)
* F-distribution (subtype of VI)
* Inverse-chi-squared distribution (subtype of V)
* Inverse-gamma distribution (V; limit of IV and VI)
* Normal distribution (G; limit of II (and thus I), III, VII (and thus IV), V, and VI)
* Pareto distribution (XI; subtype of VI)
* Power function distribution (VIII and IX; subtype of I)
* Student's t-distribution (VII; symmetric subtype of IV)
* Wigner semicircle distribution (subtype of II)

As of 2025, the only types without a name are IV (see above) and XII (beta distribution with α+β=2).

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.

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.

## 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.,[which?] and this family of distributions may prove to be one of the more important.

In the United States, the log-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 the metalog distributions.

## Notes

1. ↑ Miller, Jeff (2006-07-09). "Beta distribution". Earliest Known Uses of Some of the Words of Mathematics. http://jeff560.tripod.com/b.html.
2. ↑ Miller, Jeff (2006-12-07). "Gamma distribution". Earliest Known Uses of Some of the Words of Mathematics. http://jeff560.tripod.com/g.html.
3. ↑ Ord J.K. (1972) p. 2
4. ↑ "Guidelines for Determine Flood Flow Frequency". March 1982. https://water.usgs.gov/osw/bulletin17b/dl_flow.pdf.

## Sources

### Primary sources

* 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.

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

* 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. doi:10.1098/rsta.1901.0023. Bibcode: 1901RSPTA.197..443P.

* Pearson, Karl (1902). "On the methematical theory of errors of judgement, with special reference to the personal equation". Philosophical Transactions of the Royal Society A 198 (300–311): 235–299. doi:10.1098/rsta.1902.0005.

* Pearson, Karl (1905). Mathematical contributions to the theory of evolution, XIV: On the general theory of skew correlation and non-linear regression (memoir). Drapers' Company Research Memoirs – Biometric series. II. https://archive.org/details/cu31924003092917.

* 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. doi:10.1098/rsta.1916.0009. Bibcode: 1916RSPTA.216..429P.

* 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. https://zenodo.org/record/1431607.

### Secondary sources

* Milton Abramowitz and Irene A. Stegun (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards.
* Carver, H. C. (1924). "Pearson’s generalized frequency curves". in Rietz, H. L.. Handbook of Mathematical Statistics. Boston: Houghton Mifflin. pp. 103–110. https://archive.org/details/handbookofmathem0000hlri.
* Eric W. Weisstein 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.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Pearson distribution. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Pearson_distribution&oldid=4298589"