< 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. Today, it is a major player in probability theory, providing solutions to some characterization problems and investigation of the Matsumoto-Yor type independence property.

It is named after the inclusion of one of two Kummer’s functions *U*, aka confluent hypergeometric functions of the second kind, in its density function. This is a standard function in many popular mathematical software packages.

## Properties of the Kummer distribution

A continuous random variable has a Kummer distribution if its probability density function (pdf) has the form [3]

where C, the proportionality constant, is Γ(α)/δ^{α} *U*(α, α + 1 – γ, Β/δ)

- When γ = 0, the distribution becomes the gamma distribution with shape parameter α and scale parameter β [4].
- When δ = 1, the distribution can be viewed as scale-standard.

The Pareto distribution can be derived as a special case of the Kummer distribution.

## 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).
- Nagar, D. & Gupta, A. Matrix-Variate Kummer-Beta Distribution. Journal of the Australian Mathematics Society 73 (2002), 11-25.