NakagamiDistribution

NakagamiDistribution[μ,ω]

represents a Nakagami distribution with shape parameter μ and spread parameter ω.

Details

Background & Context

  • NakagamiDistribution[μ,ω] represents a continuous statistical distribution supported on the interval and parametrized by positive real numbers μ and ω (called a "shape parameter" and a "spread parameter", respectively), which together determine the overall behavior of its probability density function (PDF). Depending on the values of μ and ω, the PDF of a Nakagami distribution may have any of a number of shapes, including unimodal with a single "peak" (i.e. a global maximum) or monotone decreasing with potential singularities approaching the lower boundary of its domain. In addition, the tails of the PDF are "thin" in the sense that the PDF decreases exponentially rather than decreasing algebraically for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The Nakagami distribution is sometimes referred to as the Nakagami -distribution or Nakagami -distribution.
  • The Nakagami distribution was first proposed in a 1960 article by Minoru Nakagami as a mathematical model for small-scale fading in long-distance high-frequency radio wave propagation. In the years since, many applications of the distribution have been wave related. In particular, the Nakagami distribution has been used to model phenomena related to medical ultrasound imaging, communications engineering, and meteorology. It has also been used in various other fields, including hydrology, multimedia, and seismology.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Nakagami distribution. Distributed[x,NakagamiDistribution[μ,ω]], written more concisely as xNakagamiDistribution[μ,ω], can be used to assert that a random variable x is distributed according to a Nakagami 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 Nakagami distributions may be given using PDF[NakagamiDistribution[μ,ω],x] and CDF[NakagamiDistribution[μ,ω],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 Nakagami distribution, EstimatedDistribution to estimate a Nakagami parametric distribution from given data, and FindDistributionParameters to fit data to a Nakagami distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Nakagami distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Nakagami distribution.
  • TransformedDistribution can be used to represent a transformed Nakagami 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 Nakagami distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Nakagami distributions.
  • NakagamiDistribution is related to a number of other distributions. Before the formulation of the Nakagami distribution, the RayleighDistribution and RiceDistribution were commonly used models for wave fading, and the three distributions are qualitatively very similar. In addition, NakagamiDistribution generalizes both RayleighDistribution and HalfNormalDistribution, in the sense that the CDF of NakagamiDistribution[1,2 σ^2] is precisely that of RayleighDistribution[σ], while the PDF of NakagamiDistribution[1/2, π/(2 θ^2)] is exactly that of HalfNormalDistribution[θ]. Moreover, NakagamiDistribution[μ,ω] has the same PDF as both GammaDistribution[μ,Sqrt[ω]/Sqrt[μ],2,0] and as the limit of RiceDistribution[μ,α,Sqrt[ω/2]] as α→0. NakagamiDistribution is also related to HoytDistribution, NormalDistribution, and LogNormalDistribution.

Examples

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Basic Examples  (4)

Probability density function:

Cumulative distribution function:

Mean and variance:

Median:

Scope  (8)

Generate a sample of pseudorandom numbers from a Nakagami distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the 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:

Closed form for symbolic order:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Compute the mean power:

Applications  (1)

In the theory of fading channels, NakagamiDistribution is used to model fading amplitude for land-mobile and indoor-mobile multipath propagation and also in the presence of ionospheric scintillation. Find the distribution of instantaneous signal-to-noise ratio where , is the energy per symbol, and is the spectral density of white noise:

Show that SNRdist is a GammaDistribution:

Find the moment-generating function (MGF):

Find the mean and the MGF in terms of the mean:

Find the amount of fading:

Properties & Relations  (7)

Nakagami distribution is closed under scaling by a positive factor:

Relationships to other distributions:

RayleighDistribution is a special case of Nakagami distribution:

HoytDistribution is related to Nakagami distribution:

NakagamiDistribution is a special case of GammaDistribution:

HalfNormalDistribution is a special case of NakagamiDistribution:

A limit of a RiceDistribution is a NakagamiDistribution:

Neat Examples  (1)

PDFs for different μ values with CDF contours:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_nakagamidistribution, author="Wolfram Research", title="{NakagamiDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/NakagamiDistribution.html}", note=[Accessed: 31-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_nakagamidistribution, organization={Wolfram Research}, title={NakagamiDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/NakagamiDistribution.html}, note=[Accessed: 31-July-2025 ]}