Probability Distributions > Pearson Distribution

## What is the Pearson Distribution?

The **Pearson distribution** (sometimes called the *Pearson system of distributions*) is a family of unimodal continuous probability distribution functions that satisfy the following differential equation:

Pearson described **twelve families of distributions** as solutions to the equation. His original paper, (1895, p. 360) identified four (numbered I through IV) as well as type V — now known as the inverse gamma distribution: types VI through XII were identified in later papers. The family now includes the normal distribution.

## Probability Density Function (PDF)

Pearson distribution curves are graphs of p(x) as a function of f. The general Pearson family of curves can show cases of the gamma distribution, log-normal distribution and inverse gamma distribution (Lahcene, 2013) Special cases of the different types give rise to various known distributions. For example:

- Type I: Beta-distribution of the first kind.
- Type II: Uniform distribution.
- Type III: Gamma-distribution and the Chi-squared distribution.
- Type VI: beta-distribution of the second kind (i.e. the beta prime distribution) and the Fisher F-distribution.
- Type VII: Student’s T distribution.
- Type X Exponential distribution.
- Type XI: Pareto distribution.

As such, it is one of the more challenging distributions to find parameters for, finding the PDF and fitting is usually performed with software (such as the PearsonDS package in R) using maximum likelihood and method of moments. The density and range for the twelve types of Pearson distribution are as follows (from Ord, 2006):

Type | Density | Support |

I | (1 + x)^{m1}(1 – x)^{m2} |
(-1 ≤ x ≤ 1) |

II | (1 – x^{2})^{m} |
(-1 ≤ x ≤ 1) |

III | x^{m}exp(-x) |
(0 ≤ x < ∞) |

IV | (1 + x^{2})^{-m}x exp(-υ tan^{-1}(x)) |
(-∞ < x < ∞) |

V | x^{-m}exp(-x^{-1}) |
(0 ≤ x < ∞) |

VI | x^{m2}(1 + x)^{-m1}^{} |
(0 ≤ x < ∞) |

VII | (1 + x^{2})^{-m} |
(-∞ < x < ∞) |

VIII | (1 + x)^{-m} |
(0 ≤ x ≤ 1) |

IX | (1 + x)^{m} |
(0 ≤ x ≤ 1) |

X | e^{-x} |
(0 ≤ x < ∞) |

XI | x^{-m} |
(1 ≤ x < ∞) |

XII | [(g + x)/ (g – x)]^{h} |
(-g ≤ x ≤ g) |

normal | exp(-½x^{2}) |
(-∞ < x < ∞) |

## References

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 107, 1951.

Lahcene, B. On Pearson distribution families and its applications. African Journal of Mathematics and Computer Science Research. Vol. 6(5), pp. 108-117, May 2013. Retrieved July 30, 2017 from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.402.3236&rep=rep1&type=pdf

Ord, J.K. (2006). Oakes’s Test of Concordance to Preference Functions. Retrieved 7/31/2017 from: http://onlinelibrary.wiley.com/doi/10.1002/0471667196.ess1939.pub2/abstract

Pearson K (1895). “Contributions to the mathematical theory of evolution, II: Skew variation in homogeneous material.” Philos. Trans. Royal Soc. London, ARRAY 186:343-414

Pearson system of distributions.