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:

• 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.

## Density

The model is that random variable $X$ has a gamma distribution with mean $\sigma$ and shape parameter $\alpha$ , with $\sigma$ being treated as a random variable having another gamma distribution, this time with mean $\mu$ and shape parameter $\beta$ . The result is that $X$ has the following probability density function (pdf) for $x>0$ :

$f_{X}(x;\mu ,\alpha ,\beta )={\frac {2}{\Gamma (\alpha )\Gamma (\beta )))\,\left({\frac {\alpha \beta }{\mu ))\right)^{\frac {\alpha +\beta }{2))\,x^((\frac {\alpha +\beta }{2))-1}K_{\alpha -\beta }\left(2{\sqrt {\frac {\alpha \beta x}{\mu ))}\right),$ where $K$ is a modified Bessel function of the second kind. Note that for the modified Bessel function of the second kind, we have $K_{\nu }=K_{-\nu )$ . 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 $\alpha$ , the second having a gamma distribution with mean $\mu$ and shape parameter $\beta$ .

A simpler two parameter formalization of the K-distribution can be obtained by setting $\beta =1$ as

$f_{X}(x;b,v)={\frac {2b}{\Gamma (v)))\left({\sqrt {bx))\right)^{v-1}K_{v-1}(2{\sqrt {bx))),$ where $v=\alpha$ is the shape factor, $b=\alpha \beta /\mu$ is the scale factor, and $K$ 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

$M_{X}(s)=\left({\frac {\xi }{s))\right)^{\beta /2}\exp \left({\frac {\xi }{2s))\right)W_{-\delta /2,\gamma /2}\left({\frac {\xi }{s))\right),$ where $\gamma =\beta -\alpha ,$ $\delta =\alpha +\beta -1,$ $\xi =\alpha \beta /\mu ,$ and $W_{-\delta /2,\gamma /2}(\cdot )$ is the Whittaker function.

The n-th moments of K-distribution is given by

$\mu _{n}=\xi ^{-n}{\frac {\Gamma (\alpha +n)\Gamma (\beta +n)}{\Gamma (\alpha )\Gamma (\beta ))).$ So the mean and variance are given by

$\operatorname {E} (X)=\mu$ $\operatorname {var} (X)=\mu ^{2}{\frac {\alpha +\beta +1}{\alpha \beta )).$ ## Other properties

All the properties of the distribution are symmetric in $\alpha$ and $\beta .$ ## 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.

1. Redding (1999).
2. ^ Long (2001)
3. ^ Bithas (2006).

## Sources

• Redding, Nicholas J. (1999) Estimating the Parameters of the K Distribution in the Intensity Domain. Report DSTO-TR-0839, DSTO Electronics and Surveillance Laboratory, South Australia. p. 60.
• Jakeman, E. and Pusey, P. N. (1978) "Significance of K-Distributions in Scattering Experiments", Physical Review Letters, 40, 546–550, doi:10.1103/PhysRevLett.40.546.
• Jakeman, E. and Tough, R. J. A. (1987) "Generalized K distribution: a statistical model for weak scattering", J. Opt. Soc. Am., 4, (9), pp. 1764–1772.
• Ward, K. D. (1981) "Compound representation of high resolution sea clutter", Electron. Lett., 17, pp. 561–565.
• Long, M. W. (2001) "Radar Reflectivity of Land and Sea", 3rd ed., Artech House, Norwood, MA, 2001.
• Bithas, P. S.; Sagias, N. C.; Mathiopoulos, P. T.; Karagiannidis, G. K.; Rontogiannis, A. A. (2006) "On the performance analysis of digital communications over generalized-K fading channels", IEEE Communications Letters, 10 (5), pp. 353–355.