digplanet beta 1: Athena
Share digplanet:


Applied sciences






















Probability density function
Nakagami pdf.svg
Cumulative distribution function
Nakagami cdf.svg
Parameters m\ or\ \mu >= 0.5 shape (real)
\Omega\ or\ \omega > 0 spread (real)
Support x > 0\!
PDF \frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1} \exp\left(-\frac{m}{\Omega}x^2 \right)
CDF \frac{\gamma \left(m,\frac{m}{\Omega} x^2\right)}{\Gamma(m)}
Mean \frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\left(\frac{\Omega}{m}\right)^{1/2}
Median \sqrt{\Omega}\!
Mode \frac{\sqrt{2}}{2} \left(\frac{(2m-1)\Omega}{m}\right)^{1/2}
Variance \Omega\left(1-\frac{1}{m}\left(\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\right)^2\right)

The Nakagami distribution or the Nakagami-m distribution is a probability distribution related to the gamma distribution. It has two parameters: a shape parameter m and a second parameter controlling spread, \Omega.


Its probability density function (pdf) is[1]

 f(x;\,m,\Omega) = \frac{2m^m}{\Gamma(m)\Omega^m}x^{2m-1}\exp\left(-\frac{m}{\Omega}x^2\right).

Its cumulative distribution function is[1]

 F(x;\,m,\Omega) = P\left(m, \frac{m}{\Omega}x^2\right)

where P is the incomplete gamma function (regularized).

Differential equation

\left\{x \Omega  f'(x)+f(x) \left(2 m x^2-2 m \Omega +\Omega
   \right)=0,f(1)=\frac{2 m^m e^{-\frac{m}{\Omega }} \Omega ^{-m}}{\Gamma

Parameter estimation[edit]

The parameters m and \Omega are[2]

 m = \frac{\operatorname{E}^2 \left[X^2 \right]}
                   {\operatorname{Var} \left[X^2 \right]},


 \Omega = \operatorname{E} \left[X^2 \right].

An alternative way of fitting the distribution is to re-parametrize  \Omega and m as σ = Ω/m and m.[3] Then, by taking the derivative of log likelihood with respect to each of the new parameters, the following equations are obtained and these can be solved using the Newton-Raphson method:

 \Gamma(m)= \frac{x^{2m}}{\sigma^m},


 \sigma= \frac{x^2}{m}

It is reported by authors[who?] that modelling data with Nakagami distribution and estimating parameters by above mention method results in better performance for low data regime compared to moments based methods.


The Nakagami distribution is related to the gamma distribution. In particular, given a random variable Y \, \sim \textrm{Gamma}(k, \theta), it is possible to obtain a random variable X \, \sim \textrm{Nakagami} (m, \Omega), by setting k=m, \theta=\Omega / m , and taking the square root of Y:

 X = \sqrt{Y} \,.

The Nakagami distribution f(y; \,m,\Omega) can be generated from the chi distribution with parameter k set to 2m and then following it by a scaling transformation of random variables. That is, a Nakagami random variable X is generated by a simple scaling transformation on a Chi-distributed random variable Y \sim \chi(2m) as below:

 X = \sqrt{(\Omega / 2 m)}\, Y.

History and applications[edit]

The Nakagami distribution is relatively new, being first proposed in 1960.[4] It has been used to model attenuation of wireless signals traversing multiple paths.[5]


  1. ^ a b Laurenson, Dave (1994). "Nakagami Distribution". Indoor Radio Channel Propagation Modelling by Ray Tracing Techniques. Retrieved 2007-08-04. 
  2. ^ R. Kolar, R. Jirik, J. Jan (2004) "Estimator Comparison of the Nakagami-m Parameter and Its Application in Echocardiography", Radioengineering, 13 (1), 8–12
  3. ^ Mitra, Rangeet; Mishra, Amit Kumar; Choubisa, Tarun (2012). "Maximum Likelihood Estimate of Parameters of Nakagami-m Distribution". International Conference on Communications, Devices and Intelligent Systems (CODIS), 2012: 9-12. 
  4. ^ Nakagami, M. (1960) "The m-Distribution, a general formula of intensity of rapid fading". In William C. Hoffman, editor, Statistical Methods in Radio Wave Propagation: Proceedings of a Symposium held June 18-20, 1958, pp 3-36. Pergamon Press.
  5. ^ Parsons, J. D. (1992) The Mobile Radio Propagation Channel. New York: Wiley.

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Nakagami_distribution — Please support Wikipedia.
This page uses Creative Commons Licensed content from Wikipedia. A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.
22 videos foundNext > 

Using ultrasound backscattering signals and Nakagami statistical distribution to assess regional cat

Using ultrasound backscattering signals and Nakagami statistical distribution to assess regional cat +91-9994232214,8144199666, ...

Lecture - 20 Mobile Radio Propagation II

Lecture Series on Wireless Communications by Dr.Ranjan Bose, Department of Electrical Engineering, IIT Delhi. For more details on NPTEL, visit ...

How To Fit Distributions Using EasyFit

Download EasyFit from www.mathwave.com and fit distributions to your data in seconds. Supported distributions: Bernoulli, Beta, Binomial, Burr, Cauchy, ...

Mod-01 Lec-02 Wireless Channel and Fading

Advanced 3G and 4G Wireless Mobile Communications by Prof. Aditya K. Jagannatham, Department of Electronics & Communication Engineering, IIT Kanpur.

Mean and Variance of Normal Distribution

Calculus/Probability: We calculate the mean and variance for normal distributions. We also verify the probability density function property using the assumption ...

Cosmic Journals-eDCSECT 2013 (1st International E-Conference)-Priyanka Mehta

e-DCSECT-2013-Performance Analysis of Two-Hop Relayed Transmission in Asymmetric Rayleigh and Nakagami-m Fading Environment.

Using the statistical distribution functions in Excel

How to use some of the basic stats functions in Excel including NORMDIST NORMSDIST NORMINV NORMSINV TDIST and TINV Also includes a drive time ...

Full Movie ♥ Black Karate Kid (2013) † English, Action, Comedy, Drama

Subscribe to the OFFICIAL YouTube Show Page here: FREE MOVIES and Television http://www.YouTube.Com/AntonPictures Anton Pictures PRESENTS a ...

Using Personalization to Improve XML Retrieval

Using Personalization to Improve XML Retrieval +91-9994232214,8144199666, ieeeprojectchennai@gmail.com, www.projectsieee.com, ...

Using Semantic Web Technologies for Exploratory OLAP A Survey

Using Semantic Web Technologies for Exploratory OLAP A Survey +91-9994232214,8144199666, ieeeprojectchennai@gmail.com, www.projectsieee.com, ...

22 videos foundNext > 

We're sorry, but there's no news about "Nakagami distribution" right now.


Oops, we seem to be having trouble contacting Twitter

Support Wikipedia

A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia. Please add your support for Wikipedia!

Searchlight Group

Digplanet also receives support from Searchlight Group. Visit Searchlight