Some interesting properties of lagrangian distributions

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Abstract
Several discrete Lagrangian probability distributions have been generated by Consul and Shanton (1972) by using the Lagrange expansion in y of a probability generating function f(x) under the transformation x = y g(x) where g(x) is another pgf. By using probabilistic arguments the authors show that the transformation x = y g(x) occurs naturally in the distribution of the number of customers served during a busy period which implies that at least one particular family of these Lagraagian distributions must play a basic role in queueing theory. It has also been proved that under one set of conditions all discrete Lagrangian distributions approach to the normal density function while under another set of conditions they approach the inverse Gaussiam density function.

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