KDistribution

KDistribution[ν,w]

represents a K distribution with shape parameters ν and w.

Details

  • The probability density for value in a K distribution is proportional to x^nu TemplateBox[{{nu, -, 1}, {2, , x, , {sqrt(, {nu, /, w}, )}}}, BesselK] for and otherwise.
  • KDistribution allows ν and w to be any positive real numbers.
  • KDistribution allows w to be a quantity of any unit dimension and ν to be any dimensionless quantity. »
  • KDistribution can be used with such functions as Mean, CDF, and RandomVariate.

Background & Context

  • KDistribution[ν,w] represents a statistical distribution supported on the interval and parametrized by the positive real numbers ν and w, known as "shape parameters", that determine the overall behavior of the probability density function (PDF). Depending on the values of ν and w, the PDF of a K distribution may be either unimodal with a single "peak" (i.e. a global maximum) or monotone decreasing with a potential singularity approaching the lower boundary of its domain. In addition, the tails of the PDF are "thin" in the sense that the PDF decreases exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.)
  • The K distribution was developed by Jakemen and Pusey in a paper published in 1978 and was described therein as a modification of so-called Bessel function distributions, which was useful in describing the statistical behavior of scattered radiation. Probabilistically, the K distribution can be derived as a modification of several other probability distributions: For example, it is a compound distribution (in the sense that xKDistribution[ν,w] if and only if is gamma distributed (GammaDistribution) according to parameters which themselves are gamma distributed) as well as a product distribution (in the sense that it models the behavior of the product of two Gamma-distributed random variates). In addition to its theoretical importance, the K distribution has been used to describe a number of phenomena involving radiation and wave displacement.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a K distribution. Distributed[x,KDistribution[ν,w]], written more concisely as xKDistribution[ν,w], can be used to assert that a random variable x is distributed according to a K distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
  • The probability density and cumulative distribution functions for K distributions may be given using PDF[KDistribution[ν,w],x] and CDF[KDistribution[ν,w],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
  • DistributionFitTest can be used to test if a given dataset is consistent with a K distribution, EstimatedDistribution to estimate a K parametric distribution from given data, and FindDistributionParameters to fit data to a K distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic K distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic K distribution.
  • TransformedDistribution can be used to represent a transformed K distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a K distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving K distributions.
  • KDistribution is closely related to a number of other distributions. For example, KDistribution can be realized both as a compound distribution and as a product distribution of random variates distributed according to GammaDistribution. KDistribution can also be obtained by appropriate combinations of GammaDistribution with RayleighDistribution and with ExponentialDistribution, and is closely related to NormalDistribution, PoissonDistribution, GompertzMakehamDistribution, ChiSquareDistribution, MaxwellDistribution, InverseGammaDistribution, PearsonDistribution, ErlangDistribution, BetaDistribution, ExpGammaDistribution, RayleighDistribution, ChiDistribution, WeibullDistribution, and StudentTDistribution.

Examples

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Basic Examples  (3)

Probability density function:

Cumulative distribution function:

Mean and variance:

Scope  (8)

Generate a sample of pseudorandom numbers from a K distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare a density histogram of the sample with the PDF of the estimated distribution:

Skewness depends only on the first parameter:

Limiting values:

Kurtosis depends only on the first parameter:

Limiting values:

Different moments with closed forms as functions of parameters:

Moment:

Closed form for symbolic order:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Applications  (2)

In the theory of fading channels, KDistribution is used to model fading amplitude. Find the distribution of instantaneous signal-to-noise ratio where , is the energy per symbol, and is the spectral density of white noise:

The probability density function:

Find the moment-generating function (MGF):

Find the mean:

Express the MGF in terms of the mean:

Find the amount of fading:

Limiting values:

The displacement distance in a random walk on a plane with the random number of steps from NegativeBinomialDistribution with the large mean converges to KDistribution:

Compare the sample histogram to the PDF of K distribution:

Check the goodness of fit:

Properties & Relations  (3)

K distribution is closed under scaling by a positive factor:

KDistribution can be obtained from ExponentialDistribution and GammaDistribution:

KDistribution can be represented as a parameter mixture of RayleighDistribution and GammaDistribution:

Neat Examples  (1)

PDFs for different w values with CDF contours:

Wolfram Research (2010), KDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/KDistribution.html (updated 2016).

Text

Wolfram Research (2010), KDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/KDistribution.html (updated 2016).

CMS

Wolfram Language. 2010. "KDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/KDistribution.html.

APA

Wolfram Language. (2010). KDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/KDistribution.html

BibTeX

@misc{reference.wolfram_2025_kdistribution, author="Wolfram Research", title="{KDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/KDistribution.html}", note=[Accessed: 26-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_kdistribution, organization={Wolfram Research}, title={KDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/KDistribution.html}, note=[Accessed: 26-April-2025 ]}