Grand Unified Distribution

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What is the Grand Unified Distribution?

The Grand Unified distribution (GUD) appears in a visualization of probability relationships proposed by Gavin Crooks in A Field Guide to Continuous Probability Distributions at the Berkeley Institute for Theoretical Science [1].

According to Crooks, most common, continuous distributions, univariate distributions, and unimodal distributions (about a hundred or so) are members of a small number of probability distribution families such as the Pearson family of distributions or Beta distributions, which in turn are special cases of a single “Grand Unified Distribution.” In sum, the Grand Unified Distribution is a generalized family of distributions where all existing distributions are special cases of the GUD [2].

The Grand Unified Distribution of order n satisfies the differential equation (an equation that contains one or more functions and their derivatives):

grand unified distribution differential equation
a, s, a0, a1,…, an, b0, b1,…, bn, β, x in ℝ. β = 1 when a0 = 0.

Although any analytic probability distribution can satisfy this differential equation, Crooks notes that the “most interesting” univariate continuous distributions satisfy it with low order polynomials, with a small number of terms in both the numerator and denominator.

For example, the Extended Pearson distribution is defined as:

extended pearson distribution
a, s, a0, a1, a2, b0, b1, b2 in ℝ.

Grand Unified Distribution visualization

The hierarchy of probability distributions.
The hierarchy of probability distributions.

The Pearson-exponential distributions, which includes the Perks distribution, are all members of the GUD. Crooks also identifies “Greater Grand Unified Distributions,” which includes the Appell Beta distribution and the Laha distribution.

Special and unusual cases of the GUD, that are not depicted in the above image, include:

  • Extended Pearson distribution
  • Inverse Gaussian Distribution
  • Reciprocal inverse Gaussian distribution
  • Halphen distribution
  • Hyperbola distribution
  • Halphen B distribution
  • Inverse Halphen B distribution
  • Sichel distribution
  • Libby-Novick distribution
  • Gauss hypergeometric distribution
  • Confluent hypergeometric distribution
  • Generalized Halphen distribution
  • Generalized Sichel distribution


  1. Crooks. G. (2019). Field Guide to Continuous Probability Distributions. Berkeley Institute for Theoretical Science.
  2. Jayakumar, G. et al. (2019). The Holistic and Generalized (H-G) Family of Continuous Semi-Bounded Distribution. Aligarh Journal of Statistics. Vol. 40 (2020), 53-76

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