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

- NakagamiDistribution is also known as Nakagami-
distribution.
- The probability density for value
is proportional to
for
, and is zero for
.
- NakagamiDistribution allows μ and ω to be any positive real numbers.
- NakagamiDistribution allows ω to be a quantity of any unit dimension and μ to be a dimensionless quantity. »
- NakagamiDistribution can be used with such functions as Mean, CDF, and RandomVariate.
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
open allclose allBasic Examples (4)Summary of the most common use cases

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-ej2m25


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-eiibbh


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-mzshmb

Cumulative distribution function:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-rvlqgp


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-sq5f9a


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-n6hpee


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-hrnyqn


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-cqhec3


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-ixbjjw

Scope (8)Survey of the scope of standard use cases
Generate a sample of pseudorandom numbers from a Nakagami distribution:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-qhtk5j
Compare its histogram to the PDF:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-03mwaz

Distribution parameters estimation:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-45b7g2
Estimate the distribution parameters from sample data:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-epi747

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

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-f8ui5o

Skewness depends only on the first parameter:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-rsntmb


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-w4bp69


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-1pk7yz


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-wmvrea

Kurtosis depends only on the first parameter:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-py4y6j


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-8sb8qq


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-ju03u6


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-dahw2l

Different moments with closed forms as functions of parameters:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-js043h

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-rx074o

Closed form for symbolic order:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-is1ysv


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-pknsqa

Closed form for symbolic order:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-3okr4


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-zg9ct4


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-9gzmth


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-7ycowq


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-hyrtkv


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-64u45g


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-l1vcie


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-npqwev


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-hd8qau

Consistent use of Quantity in parameters yields QuantityDistribution:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-veie5


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-uh98e

Applications (1)Sample problems that can be solved with this function
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:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-icxgyw
Show that SNRdist is a GammaDistribution:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-l9wqpp


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-1soldv


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-nw2wkx

Find the moment-generating function (MGF):

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-m57vzv

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

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-64oxos


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-yetrme


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-um7py8

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-ucnk8y

Properties & Relations (7)Properties of the function, and connections to other functions
Nakagami distribution is closed under scaling by a positive factor:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-ylv4u

Relationships to other distributions:

RayleighDistribution is a special case of Nakagami distribution:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-sao1wa


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-hd4fle


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-wyou1m

HoytDistribution is related to Nakagami distribution:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-20nfi7


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-vzvrs9


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-hcf784

NakagamiDistribution is a special case of GammaDistribution:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-qa07k7


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-11xhvn


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-yhulqh

HalfNormalDistribution is a special case of NakagamiDistribution:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-zyaybn


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-nl3h10


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-zhq42w

A limit of a RiceDistribution is a NakagamiDistribution:

https://wolfram.com/xid/0mfg09eai2uqyqh3kq-6fn9h


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-dpncp8


https://wolfram.com/xid/0mfg09eai2uqyqh3kq-s5eoyi

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).
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.
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
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: 10-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: 10-July-2025
]}