In probability and statistics, the generalized K-distribution is a three-parameter family of continuous probability distributions.

In each case, a re-parametrization of the usual form of the family of gamma distributions is used, such that the parameters are: K-distribution is a special case of variance-gamma distribution, which in turn is a special case of generalised hyperbolic distribution.

Suppose that a random variable

being treated as a random variable having another gamma distribution, this time with mean

has the following probability density function (pdf) for

is a modified Bessel function of the second kind.

Note that for the modified Bessel function of the second kind, we have

In this derivation, the K-distribution is a compound probability distribution.

It is also a product distribution:[1] it is the distribution of the product of two independent random variables, one having a gamma distribution with mean 1 and shape parameter

A simpler two parameter formalization of the K-distribution can be obtained by setting

is the modified Bessel function of second kind.

The above two parameter formalization can also be obtained by setting

, albeit with different physical interpretation of

This two parameter formalization is often referred to as the K-distribution, while the three parameter formalization is referred to as the generalized K-distribution.

This distribution derives from a paper by Eric Jakeman and Peter Pusey (1978) who used it to model microwave sea echo.

[4] Jakeman and Tough (1987) derived the distribution from a biased random walk model.

[5] Keith D. Ward (1981) derived the distribution from the product for two random variables, z = a y, where a has a chi distribution and y a complex Gaussian distribution.

[6] The moment generating function is given by[7] where

The n-th moments of K-distribution is given by[1] So the mean and variance are given by[1] All the properties of the distribution are symmetric in

[1] K-distribution arises as the consequence of a statistical or probabilistic model used in synthetic-aperture radar (SAR) imagery.

The K-distribution is formed by compounding two separate probability distributions, one representing the radar cross-section, and the other representing speckle that is a characteristic of coherent imaging.

It is also used in wireless communication to model composite fast fading and shadowing effects.