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A **Lexian distribution** is another name for the binomial distribution (*k*, *p*) if *p *is not constant [1]. One way to interpret the distribution is as a special case of a mixture of binomial distributions [2]. The Lexian distribution considers a mixture distribution of subsets of binomials, each of which has its own probability distribution.

The mean of the Lexian variance is [3]

Where

*p-bar*is the average value of the distinct probability distributions

The Lexian variance is

Where

- var(
*p*) is the variance of the average value of the distinct probability distributions.

As a consequence, if mixed binomial variables are treated as pure binomials, the mean would be correct but the variance would be underestimated when using the “binomial estimator” *np*(1- *p*) [4].

## History of the Lexian Distribution

The Lexian distribution is named after German economist Wilhelm Lexis, who published several papers on mixture distributions in 1875-1879. The basis of his work was to test for the structure of a set by comparing its actual variance to one obtained from a theoretical binomial variance through a “Lexis Ratio”: the standard deviation from the data, divided by the theoretical binomial standard deviation [5].

**References**

[1] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.

[2] Suchindran C. M. (1981). A reply to Avery and Hakkert. *Population studies*, *35*(5), 473–475. https://doi.org/10.1080/00324728.1981.11878519

[3] Johnson, N. L. (1969), Discrete distributions, Houghton Mifflin Company, Boston.

[4] Coppens, F. et al. (2007). The performance of credit rating systems in the assessment of collateral used in Eurosystem monetary policy operations. National Bank of Belgium. Online: http://aei.pitt.edu/7612/1/wp118En.pdf

[5] Bensman, S. (2005). Urquhart’s Law: Probability and the Management of Scientific and Technical Journal Collections Part 1. The Law’s Initial Formulation and Statistical Bases. Haworth Press. doi:10.1300/J122v26n01_04