K-distribution
Introduction In probability and statistics, the '''generalized K-distribution''' is a three-parameter family of continuous probability distributions. The distribution arises by compounding two gamma distributions. In each case, a re-parametrization of the usual form of the family of gamma distributions is used, such that the parameters are:* the mean of the distribution,* the usual shape parameter.K-distribution is a special case of variance-gamma distribution, which in turn is a special case of generalised hyperbolic distribution. A simpler special case of the generalized K-distribution is often referred as ''the'' K-distribution.
Density Suppose that a random variable  has gamma distribution with mean  and shape parameter , with  being treated as a random variable having another gamma distribution, this time with mean  and shape parameter . The result is that  has the following probability density function (pdf) for ::where  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: it is the distribution of the product of two independent random variables, one having a gamma distribution with mean 1 and shape parameter , the second having a gamma distribution with mean  and shape parameter .A simpler two parameter formalization of the K-distribution can be obtained by setting  as:where  is the shape factor,  is the scale factor, and  is the modified Bessel function of second kind. The above two parameter formalization can also be obtained by setting , , and , albeit with different physical interpretation of  and  parameters. 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. Jakeman and Tough (1987) derived the distribution from a biased random walk model. 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. The modulus of ''z'', ''|z|'', then has K-distribution.
[[Moment (mathematics)|Moments]] The moment generating function is given by:where    and  is the Whittaker function.The n-th moments of K-distribution is given by:So the mean and variance are given by::
Other properties All the properties of the distribution are symmetric in  and
Applications 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.
Notes
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Further reading * * Ward, Keith D.; Tough, Robert J. A; Watts, Simon (2006) ''Sea Clutter: Scattering, the K Distribution and Radar Performance'', Institution of Engineering and Technology. .