# 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 [3]

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 [4]

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

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

[2] 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

[3] 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).

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

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