# Lexian Distribution

A Lexian distribution is another name for the binomial distribution (k, p) if p is not constant . One way to interpret the distribution is as a special case of a mixture of binomial distributions . 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 

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) .

## 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 .

References

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

 Suchindran C. M. (1981). A reply to Avery and Hakkert. Population studies35(5), 473–475. https://doi.org/10.1080/00324728.1981.11878519

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

 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

 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