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. In Probability theory, a Probability distribution is called continuous if its Cumulative distribution function is continuous. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable 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. Karl Pearson FRS ( March 27 1857 &ndash April 27 1936) established the disciplineof 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 Biostatistics (a Portmanteau word made from biology and statistics sometimes referred to as biometry or biometrics) is the application of Statistics

## History

The Pearson system was originally devised in an effort to model visibly skewed observations. In Probability theory and Statistics, skewness is a measure of the asymmetry of the Probability distribution of a real -valued 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. In Probability theory and Statistics, a Random variable X has an Expected value μ = E ( X) and a Variance σ2 In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable In Probability theory, especially as that field is used in Statistics, a location-scale family is a family of Univariate Probability distributions Except in pathological cases, a location-scale family can be made to fit the observed mean (first cumulant) and variance (second cumulant) arbitrarily well. In Mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive 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, Probability distribution, or sample is one measure of 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. In Probability theory and Statistics, skewness is a measure of the asymmetry of the Probability distribution of a real -valued In Probability theory and Statistics, kurtosis (from the Greek word κυρτός kyrtos or kurtos, meaning bulging is a measure of the "peakedness" 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 normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields 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. In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a 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. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields In Probability theory and Statistics, the inverse gamma distribution is a two-parameter family of continuous Probability distributions on the positive 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).

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, skewness is a measure of the asymmetry of the Probability distribution of a real -valued In Probability theory and Statistics, the k th standardized moment of a Probability distribution is \frac{\mu_k}{\sigma^k}\! where The second is the traditional kurtosis, or fourth standardized moment: β2 = γ2 + 3. In Probability theory and Statistics, kurtosis (from the Greek word κυρτός kyrtos or kurtos, meaning bulging is a measure of the "peakedness" (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.

Many of the skewed and/or non-mesokurtic distributions familiar to us today were still unknown in the early 1890s. In Probability theory and Statistics, kurtosis (from the Greek word κυρτός kyrtos or kurtos, meaning bulging is a measure of the "peakedness" 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. In Probability theory and Statistics, the beta distribution is a family of continuous Probability distributions defined on the interval 1 parameterized Thomas Bayes (c 1702 &ndash 17 April 1761) was a British Mathematician and Presbyterian minister known for having formulated The posterior probability of a Random event or an uncertain proposition is the Conditional probability that is assigned after the relevant evidence is taken In Probability theory and Statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete Probability In Probability theory, inverse probability is an obsolete term for the Probability distribution of an unobserved variable 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. In Probability theory and Statistics, the gamma distribution is a two-parameter family of continuous Probability distributions It has a Scale parameter 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. In Probability and Statistics, Student's t -distribution (or simply the t -distribution) is a Probability distribution William Sealy Gosset ( June 13 1876 – October 16 1937) is famous as a Statistician, best known by his pen name Student His 1901 paper introduced the inverse-gamma distribution (type V) and the beta prime distribution (type VI). In Probability theory and Statistics, the inverse gamma distribution is a two-parameter family of continuous Probability distributions on the positive A Beta Prime Distribution is a Probability distribution defined for x>0 with two parameters (of positive real part α and β having the Probability density function

## Definition

A Pearson density p is defined to be any valid solution to the differential equation (cf. In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Pearson 1895, p. 381)

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

with : $b_0=\frac{4 \beta_2-3 \beta_1}{10 \beta_2 -12\beta_1 -18} \mu_2$

$a=b_1=\sqrt{\mu_2 \beta_1}\frac{\beta_2+3}{10 \beta2-12\beta_1 -18}$

$b_2=\frac{2 \beta_2-3 \beta_1 -6}{10 \beta2-12\beta_1 -18}$

The parameter a0 determines a stationary point, and hence under some conditions a mode of the distribution, since

$p'(a_0) = 0 \!$

follows directly from the differential equation. In Mathematics, particularly in Calculus, a stationary point is an input to a function where the Derivative is zero (equivalently the In Statistics, the mode is the value that occurs the most frequently in a Data set or a Probability distribution.

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

$p(x) \propto \exp\left( -\!\int\!\!\frac{x-a}{b_2 x^2 + b_1 x + b_0} \,\mathrm{d}x \right).\!$

The integral in this solution simplifies considerably when certain special cases of the integrand are considered. In Mathematics, a linear differential equation is a Differential equation of the form Ly = f \ where the Differential 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

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

### 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. In Algebra, the discriminant of a Polynomial with real or complex Coefficients is a certain expression in the coefficients of the This article is about the zeros of a function which should not be confused with the value at zero. A quadratic function, in Mathematics, is a Polynomial function of the form f(x=ax^2+bx+c \\! where a \ne 0 \\! 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:

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

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

$\int\frac{y - \frac{2\,b_2\,a + b_1}{2\,b_2}}{y^2 + \alpha^2} \,\mathrm{d}y= \frac{1}{2} \ln(y^2 + \alpha^2)- \frac{2\,b_2\,a + b_1}{2\,b_2\,\alpha} \arctan\left(\frac{y}{\alpha}\right)+ C_0\!$

involves the inverse trigonometic arctan function. Then

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

Finally, let

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

Applying these substitutions, we obtain the parametric function:

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

This unnormalized density has support on the entire real line. In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a It depends on a scale parameter α > 0 and shape parameters m > 1 / 2 and ν. In Probability theory and Statistics, a scale parameter is a special kind of Numerical parameter of a Parametric family of Probability distributions In Probability theory and Statistics, a shape parameter is a kind of Numerical parameter of a parametric family of Probability distributions 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. In Statistics, a location family is a class of probability distributions parametrized by a scalar- or vector-valued parameter μ, which determines the "location" We have thus derived the density of the Pearson type IV distribution:

$p(x) =\frac{\left|\frac{\Gamma\!\left(m+\frac{\nu}{2}i\right)}{\Gamma(m)}\right|^2} {\alpha\,\mathrm{\Beta}\!\left(m-\frac12, \frac12\right)}\left[1 + \left(\frac{x-\lambda}{\alpha}\right)^{\!2\,} \right]^{-m}\exp\left[-\nu \arctan\left(\frac{x-\lambda}{\alpha}\right)\right].\!$

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, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function In Mathematics, the beta function, also called the Euler integral of the first kind is a Special function defined by

#### 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. In Probability theory and Statistics, skewness is a measure of the asymmetry of the Probability distribution of a real -valued 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

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

where B is the Beta function. In Mathematics, the beta function, also called the Euler integral of the first kind is a Special function defined by

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. In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of Now the parameter m only controls the kurtosis of the distribution. In Probability theory and Statistics, kurtosis (from the Greek word κυρτός kyrtos or kurtos, meaning bulging is a measure of the "peakedness" If m approaches infinity as λ and σ are held constant, the normal distribution arises as a special case:

$\lim_{m\to\infty} \frac{1}{\sigma\,\sqrt{2\,m-3}\,\mathrm{\Beta}\!\left(m-\frac12, \frac12\right)}\left[1 + \left(\frac{x-\lambda}{\sigma\,\sqrt{2\,m-3}}\right)^{\!2\,} \right]^{-m}\!$
$= \frac{1}{\sigma\,\sqrt{2}\,\Gamma\!\left(\frac12\right)}\times\lim_{m\to\infty}\frac{\Gamma(m)}{\Gamma\!\left(m-\frac12\right) \sqrt{m-\frac32}}\times\lim_{m\to\infty}\left[1 + \frac{\left(\frac{x-\lambda}{\sigma}\right)^2}{2\,m-3} \right]^{-m}\!$
$= \frac{1}{\sigma\sqrt{2\,\pi}}\times1\times\exp\!\left[-\frac12 \left(\frac{x-\lambda}{\sigma}\right)^{\!2\,} \right]\!$

This is the density of a normal distribution with mean λ and standard deviation σ. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields

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 is a measure of the dispersion of a collection of values In Probability theory and Statistics, skewness is a measure of the asymmetry of the Probability distribution of a real -valued In Probability theory and Statistics, kurtosis (from the Greek word κυρτός kyrtos or kurtos, meaning bulging is a measure of the "peakedness"

#### Student's t-distribution

The Pearson type VII distribution subsumes Student's t-distribution, and hence also the Cauchy distribution. In Probability and Statistics, Student's t -distribution (or simply the t -distribution) is a Probability distribution The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous Probability distribution. Student's t-distribution arises as the result of applying the following substitutions to its original parameterization:

$\lambda = 0, \!$
$\alpha = \sqrt{\nu}, \!$  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

$p(x) =\frac{1}{\sqrt{\nu}\,\mathrm{\Beta}\!\left(\frac{\nu}{2}, \frac12\right)}\left[1 + \frac{x^2}{\nu} \right]^{-\frac{\nu+1}{2}},\!$

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 a1 and a2 (not necessarily distinct):

$a_1 = \frac{-b_1 - \sqrt{b_1^2 - 4 b_2 b_0}}{2 b_2}, \!$
$a_2 = \frac{-b_1 + \sqrt{b_1^2 - 4 b_2 b_0}}{2 b_2}, \!$

One have to define :

$m_1=\frac{a+a_1}{c_2 (a_2-a_1)} \!$
$m_2=-\frac{a+a_2}{c_2 (a_2-a_1)}\!$
$C_1=\frac{c_1}{2 c_2}\!$

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

$p(x) \propto \exp\left( -\frac{1}{b_2} \int\!\!\frac{x-a}{(x - a_1) (x - a_2)} \,\mathrm{d}x \right). \!$

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

$\int\!\!\frac{x-a}{(x - a_1) (x - a_2)} \,\mathrm{d}x= \frac{(a_1-a)\ln(x-a_1) - (a_2-a)\ln(x-a_2)}{a_1-a_2} + C\!$

involves only the logarithm function, and not the arctan function as in the previous case. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce

Using the substitution

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

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

$p(x) \propto(x-r_1)^{-\nu (a_1-a)} (x-a_2)^{\nu (a_2-a)}.\!$

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

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

#### The Pearson type I and type II 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. In Probability theory and Statistics, the beta distribution is a family of continuous Probability distributions defined on the interval 1 parameterized Then the solution p is supported on the interval (r1,r2). Apply the substition

$x = a_1 + y (a_2 - a_1) \qquad \mbox{where}\ 0

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

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

Regrouping constants and parameters, this simplifies to:

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

Thus $\frac{x-\lambda-a_1}{a_2-a_1}\!$ follows a $beta(m_1+1,m_2+1)\!$ with $\lambda=\mu_1- (a_2-a_1) \frac{m_1+1}{m_1+m_2+2}-a_1 \!$

It turns out that $m_1>-1 \land m_2>-1$ 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 [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). In Statistics, Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter \rho 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 a Probability distribution For the table of values, certain values are used as the constants in the previous equation:

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

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

$\lambda= \mu1 + \frac{b_0}{b_1} - (m+1) b_1\!$
$b_0+b_1 (x-\lambda)\!$ follows a :$gamma(m+1,b_1^2)\!$

Pearson type III distribution gamma distribution, chi-square distribution

#### The Pearson type V distribution

$\lambda=\mu_1-\frac{a-C_1} {1-2 b_2}\!$
$x-\lambda\!$ follows a :$inversegamma(\frac{1}{b_2}-1,\frac{a-C_1}{b_2})\!$

Pearson type V distribution inverse-gamma distribution

#### The Pearson type VI distribution

$\lambda=\mu_1 + (a_2-a_1) \frac{m_2+1}{m_2+m_1+2} - a_2\!$
$\frac{x-\lambda-a_2}{a_2-a_1}\!$ follows a :$betaprime(m_2+1,-m_2-m_1-1)\!$

Pearson type VI distribution beta prime distribution, F-distribution

## Relation to other distributions

The Pearson family subsumes the following distributions, among others:

## Applications

These models are used in financial markets, given their ability to be parametrised in a way that has intuitive meaning for market traders. In Probability theory and Statistics, the gamma distribution is a two-parameter family of continuous Probability distributions It has a Scale parameter In Probability theory and Statistics, the chi-square distribution (also chi-squared or \chi^2  distribution) is one In Probability theory and Statistics, the inverse gamma distribution is a two-parameter family of continuous Probability distributions on the positive A Beta Prime Distribution is a Probability distribution defined for x>0 with two parameters (of positive real part α and β having the Probability density function In Probability theory and Statistics, the F -distribution is a continuous Probability distribution. In Probability theory and Statistics, the beta distribution is a family of continuous Probability distributions defined on the interval 1 parameterized A Beta Prime Distribution is a Probability distribution defined for x>0 with two parameters (of positive real part α and β having the Probability density function The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous Probability distribution. In Probability theory and Statistics, the chi-square distribution (also chi-squared or \chi^2  distribution) is one In Probability theory and Statistics, the continuous uniform distribution is a family of Probability distributions such that for each member of the WikipediaWikiProject Probability#Standards for a discussionof standards used for probability distribution articles such as this one In Probability theory and Statistics, the gamma distribution is a two-parameter family of continuous Probability distributions It has a Scale parameter In Probability theory and Statistics, 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 Multiplicative inverse In Probability theory and Statistics, the inverse gamma distribution is a two-parameter family of continuous Probability distributions on the positive The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields In Probability and Statistics, Student's t -distribution (or simply the t -distribution) is a Probability distribution 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.

## Notes

1. ^ Miller, Jeff; et al. (2006-07-09). Year 2006 ( MMVI) was a Common year starting on Sunday of the Gregorian calendar. Events 455 - Roman military commander Avitus is proclaimed Emperor of the Western Roman Empire. 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). Year 2006 ( MMVI) was a Common year starting on Sunday of the Gregorian calendar. Events 43 BC - Marcus Tullius Cicero assassinated 1696 - Connecticut Route 108, one of the oldest highways Gamma distribution. Earliest Known Uses of Some of the Words of Mathematics. Retrieved on December 9, 2006.
3. ^ Ramsey, Philip H. (1989-09-01). Year 1989 ( MCMLXXXIX) was a Common year starting on Sunday (link displays 1989 Gregorian calendar) Events 462 - Possible start of first Byzantine indiction cycle. Critical Values for Spearman's Rank Order Correlation. Retrieved on August 22, 2007.

## Sources

### Secondary sources

• Eric W. Weisstein et al. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld Pearson Type III Distribution. From MathWorld. MathWorld is an online Mathematics reference work created and largely written by Eric W

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