< List of probability distributions < *Macdonald distribution*

## Three parameter Macdonald distribution

The three parameter **Macdonald distribution**, introduced by Nagar et al. [1] is defined by the probability density function (PDF):

Where Γ, ν,σ is the extended gamma function, defined as:

For the real part — Re(ν)>0 — and 𝜎 = 0, the extension of the gamma function reduces to the classical gamma function, Γ(ν; 0) = Γ(ν).

## Other “Macdonald distribution”

Occasionally, the term “Macdonald distribution” is used to describe a theory from permutations (for example, see [2]), but this is a different concept from the distribution described by the above PDF.

Kropac [3] discusses distributions involving Macdonald functions (modified Bessel functions of the second kind), defined as:

K_{n}(z)= π(2 sin nπ)^{-1}[I_{-n}(z) – I_{n}(z)]

The modified Bessel function K_{iν}(x) is also sometimes called the Macdonald function.

Johnson et. al [4] also refer to the Macdonald functions as *distributions*. One subclass of these functions containing functions with integer indices *n *are important in applications. The *indices *here refer to parameters of the Macdonald function. Macdonald functions with integer indices have several properties that make them highly suitable for applications such as orthogonal functions (**orthogonal functions** are two functions with an inner product of zero), which enable them to represent many different functions. Macdonald functions with integer indices also have a straightforward Fourier transform (a.k.a. the characteristic function), which simplifies their use in signal processing applications.

## References

[1] D. K. Nagar, A. Roldán-Correa, and A. K. Gupta, “Extended matrix variate gamma and beta functions,” *Journal of Multivariate Analysis*, vol. 122, pp. 53–69, 2013.View at: Publisher Site | Google Scholar | MathSciNet

[2] Young, B. (2014). A Markov growth process for Macdonald’s distribution on reduced words. **arXiv:1409.7714** (math).

[3] Kropac, O. (1982). Some properties and applications of probability distributions based on MacDonald function. Aplikace matematiky Volume: 27, Issue: 4, page 285-302 ISSN: 0862-7940

[4] Johnson, N. et al. Continuous Univariate Distributions. Wiley.