Probability distributions > Unusual Probability Distributions

These unusual probability distributions (or unusual names for distributions) never made it to the mainstream. You won’t find these in your regular textbooks, but I did come across this interesting list put together by Prof. Shakti [1].

## List of Unusual Probability Distributions

- The
**array distribution**refers to the conditional distribution of independent random variables X_{1}(given X_{2}, … ,X_{n}). - The
**Arfwedsen distribution**(the coupon-collecting distribution or Dixie cup) is a discrete probability distribution for an urn sampling problem for drawings without replacement. Specifically, it tackles the problem “An urn contains N numbered balls. We make n drawings replacing the ball into the urn each time. What is the probability of getting v different balls?”[2] **Binomials Added Distribution**: The problem of adding binomials was addressed briefly in [3], referencing a formula for a toric ideal. Otherwise, it seems to have disappeared into obscurity.**Bipolar distribution**: a specific type of bimodal distribution that has two distinct peaks offering contradictory information. In some cases, the term may also refer to the Watson bipolar distribution.- The
**Bravais distribution**is another name for the bivariate normal distribution, (also sometimes called the bivariate Gaussian or Bivariate Laplace–Gauss distribution). **Chi-bar-squared distribution**: a mixture of chi-square distributions, mixed over their degrees of freedom.**Darmois-Koopman distribution**(or Koopman-Darmois) is a member of the*exponential-type*class of probability distributions. This class of probability distributions was recognized by Darmois [3] and Koopman [4] almost simultaneously [1]. Pitman [5] is often credited with contributions to the introduction of the distribution. In both cases, the form of the PDF described a single sufficient statistic for θ, given*n*independent and identically distributed (i.i.d.) random variables. A broad subclass of these distributions was presented by Morris [6]. The distribution remains relatively obscure, although it has made relatively recent appearances in estimation theory [7, 8].- The
**de Moivre distribution**: another name for the normal distribution. The use of “de Moivre distribution” to describe the normal distribution is thought to originate with Freudenthal [4], who advocated the name because De Moivre was the first to define the distribution, in 1733 [5]. - The
**Elfving distribution**is defined as

The distribution is named after Finnish statistician and mathematician Gustav Elfving (1908-1984), who described the distribution in 1947. - The
**exceedance distribution function**(EDF) is defined as 1 minus the cumulative distribution function (CDF) [1]:EDF = 1 – CDF

## References

[1] Shakti, P. (2022). P-Distribution. Retrieved March 15, 2022 from: https://p-distribution.com/probability-distributions-list/

[2] G. Arfwedson, A probability distribution connected with Stirling’s second class numbers. Skand. Aktuarietidskr. 34 (1951), 121–132.

[3] Carlini, E. & Rapallo, F. (2013). Toric ideals with linear components: an algebraic interpretation of clustering the cells of a contingency table.

[3] Freudenthal, H. (1966b). Waarschijnlijkheid en Statistiek [Probability and Statistics]. Haarlem: De Erven F. Bohn.

[4] Daw, R. & Pearson, E. Studies in the History of Probability and Statistics. XXX. Abraham De Moivre’s 1733 Derivation of the Normal Curve: A Bibliographical Note. Biometrika Vol. 59, No. 3 (Dec., 1972), pp. 677-680 (4 pages)

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