# Kummer distribution

< List of probability distributions < Kummer distribution

The Kummer distribution was derived by Armero and Bayarri [1, 2], who derived it as a posterior distribution of certain basic parameters in a Bayesian analysis of a queuing system. It is named after the inclusion of the Kummer function, aka the confluent hypergeometric function of the second kind, in its density function.

## Properties of the Kummer distribution

A random variable X has a Kummer distribution if its probability density function (PDF) has the form 

where:

• ψ = the confluent hypergeometric function of the second kind
• Γ = the gamma function.

When γ = 0, the distribution becomes the gamma distribution with shape parameter α and scale parameter β.

The Pareto distribution can be derived as a special case of the Kummer distribution when η = 1, ν = 0, and β > η is heavy tailed. The Pareto distribution is a popular for modeling the distribution of wealth, income, and city sizes. It is also used in engineering to model failure rates of components.

## Kummer Distribution Laplace Transform

The Kummer distribution Laplace transform is 

The Laplace transform converts a function of time into a function of complex frequency. It is a powerful tool for solving differential equations, and it has many other applications in engineering, physics, and mathematics.

## References

 C. Armero and M. J. Bayarri, ‘A Bayesian analysis of a queuing system with unlimited service’, J. Statist. Plann. Inference 58 (1997), 241–264

 Konzu, E. (2021). Rate of convergence in total variation for the generalized inverse Gaussian and the Kummer distributions. Open Journal of Mathematical Sciences. Vol. 5 (2021), Issue 1, pp. 182 – 191 DOI: 10.30538/oms2021.0155

 C. Armero and M. J. Bayarri, ‘A Bayesian analysis of a queuing system with unlimited service’, Technical Report No. 93–50, (Department of Statistics, Purdue University, 1993).

 Aalen, O. et al. (2008). Survival and Event History A Process Point of ViewA. Springer.

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