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 
- p-bar is the average value of the distinct probability distributions
The Lexian variance is
- 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 .
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