KDistribution

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`KDistribution`

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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 xKDistribution[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 xKDistribution[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)Summary of the most common use cases

Probability density function:

In[1]:=1

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https://wolfram.com/xid/0pva6m56ca-2hwweo

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0pva6m56ca-z996ei

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0pva6m56ca-qnv9at

Out[3]=3

Cumulative distribution function:

In[1]:=1

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https://wolfram.com/xid/0pva6m56ca-4dcu7b

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0pva6m56ca-kipl6p

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0pva6m56ca-n0n30b

Out[3]=3

Mean and variance:

In[1]:=1

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https://wolfram.com/xid/0pva6m56ca-hc9ou5

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0pva6m56ca-wz94td

Out[2]=2

Scope (8)Survey of the scope of standard use cases

Generate a sample of pseudorandom numbers from a K distribution:

In[1]:=1

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https://wolfram.com/xid/0pva6m56ca-zpsmku

Compare its histogram to the PDF:

In[2]:=2

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https://wolfram.com/xid/0pva6m56ca-8hu6rq

Out[2]=2

Distribution parameters estimation:

In[1]:=1

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https://wolfram.com/xid/0pva6m56ca-niohgc

Estimate the distribution parameters from sample data:

In[2]:=2

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https://wolfram.com/xid/0pva6m56ca-14kndr

Out[2]=2

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

In[3]:=3

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https://wolfram.com/xid/0pva6m56ca-npzwb9

Out[3]=3

Skewness depends only on the first parameter:

In[1]:=1

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https://wolfram.com/xid/0pva6m56ca-nuy28c

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0pva6m56ca-n36xfa

Out[2]=2

Limiting values:

In[3]:=3

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https://wolfram.com/xid/0pva6m56ca-3x3npu

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0pva6m56ca-03zfom

Out[4]=4

In[5]:=5

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https://wolfram.com/xid/0pva6m56ca-nl2ks

Out[5]=5

Kurtosis depends only on the first parameter:

In[1]:=1

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https://wolfram.com/xid/0pva6m56ca-jic202

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0pva6m56ca-6wjpva

Out[2]=2

Limiting values:

In[3]:=3

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https://wolfram.com/xid/0pva6m56ca-grv3a

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0pva6m56ca-kye4df

Out[4]=4

In[5]:=5

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https://wolfram.com/xid/0pva6m56ca-lp1ik3

Out[5]=5

Different moments with closed forms as functions of parameters:

In[1]:=1

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https://wolfram.com/xid/0pva6m56ca-pflfx6

Moment:

In[2]:=2

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https://wolfram.com/xid/0pva6m56ca-ufymr5

Out[2]=2

Closed form for symbolic order:

In[3]:=3

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https://wolfram.com/xid/0pva6m56ca-3ea208

Out[3]=3

CentralMoment:

In[4]:=4

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https://wolfram.com/xid/0pva6m56ca-9e5q1c

Out[4]=4

FactorialMoment:

In[5]:=5

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https://wolfram.com/xid/0pva6m56ca-8wx5wx

Out[5]=5

Cumulant:

In[6]:=6

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https://wolfram.com/xid/0pva6m56ca-6z69pr

Out[6]=6

Hazard function:

In[1]:=1

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https://wolfram.com/xid/0pva6m56ca-m6aqz5

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0pva6m56ca-zz27x6

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0pva6m56ca-wfwmkf

Out[3]=3

Quantile function:

In[1]:=1

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https://wolfram.com/xid/0pva6m56ca-n863ev

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0pva6m56ca-wkttq

Out[2]=2

Consistent use of Quantity in parameters yields QuantityDistribution:

In[1]:=1

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https://wolfram.com/xid/0pva6m56ca-u7m9d

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0pva6m56ca-ivhtr7

Out[2]=2

Applications (2)Sample problems that can be solved with this function

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:

In[1]:=1

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https://wolfram.com/xid/0pva6m56ca-45t1gg

The probability density function:

In[2]:=2

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https://wolfram.com/xid/0pva6m56ca-elslj9

Out[2]=2

Find the moment-generating function (MGF):

In[3]:=3

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https://wolfram.com/xid/0pva6m56ca-8oyxbz

Out[3]=3

Find the mean:

In[4]:=4

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https://wolfram.com/xid/0pva6m56ca-kk4vnd

Out[4]=4

Express the MGF in terms of the mean:

In[5]:=5

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https://wolfram.com/xid/0pva6m56ca-qj9jmo

Out[5]=5

Find the amount of fading:

In[6]:=6

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https://wolfram.com/xid/0pva6m56ca-uvfmv9

In[7]:=7

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https://wolfram.com/xid/0pva6m56ca-v7hh1t

Out[7]=7

In[8]:=8

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https://wolfram.com/xid/0pva6m56ca-njq9lk

Out[8]=8

Limiting values:

In[11]:=11

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https://wolfram.com/xid/0pva6m56ca-y5om8o

Out[11]=11

In[10]:=10

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https://wolfram.com/xid/0pva6m56ca-14lf7k

Out[10]=10

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:

In[1]:=1