K-distribution
In probability and statistics, the 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: K-distribution is a special case of variance-gamma distribution, which in turn is a special case of generalised hyperbolic distribution. Density. The model is that random variable formula_1 has a gamma distribution with mean formula_2 and shape parameter formula_3, with formula_2 being treated as a random variable having another gamma  distribution, this time with mean formula_5 and shape parameter formula_6. The result is that formula_1 has the following probability density function (pdf) for formula_8: where formula_10 is a modified Bessel function of the second kind. Note that for the modified Bessel function of the second kind, we have formula_11. 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 formula_3, the second having a gamma distribution with mean formula_5 and  shape parameter formula_6. A simpler two parameter formalization of the K-distribution can be obtained by setting formula_15 as where formula_17 is the shape factor, formula_18 is the scale factor, and formula_10 is the modified Bessel function of second kind. 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. 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. Moments. The moment generating function is given by where formula_21 formula_22 formula_23 and formula_24 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 formula_3 and formula_29 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.