Nakagami distribution

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

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

Contents

1 Characterization
2 Parameter estimation
3 Generation
4 History and applications
5 References

Characterization

Its probability density function (pdf) is^[1]

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

Its cumulative distribution function is^[1]

$F(x;\,\mu,\omega) = P\left(\mu, \frac{\mu}{\omega}x^2\right)$

where P is the incomplete gamma function (regularized).

Parameter estimation

The parameters μ and ω are^[2]

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

and

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

Generation

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 $X \, \sim \textrm{Nakagami} (\mu, \omega)$ , by setting $k=\mu$ , $\theta=\omega / \mu$ , and taking the square root of $Y$ :

$X = \sqrt{Y} \,$ .

When 2μ is an integer, the Nakagami distribution $f(y; \,\mu,\omega)$ can be generated from the Chi distribution with parameter k set to 2μ and then following it by the scaling transformation:^{[citation needed]}

$y = \sqrt{(\omega / 2 \mu)} x.$

History and applications

The Nakagami distribution is relatively new, being first proposed in 1960.^[3] It has been used to model attenuation of wireless signals traversing multiple paths.^[4]

References

^ ^a ^b Laurenson, Dave (1994). "Nakagami Distribution". Indoor Radio Channel Propagation Modelling by Ray Tracing Techniques. http://www.see.ed.ac.uk/~dil/thesis_mosaic/section2_19.html. Retrieved 2007-08-04.
^ R. Kolar, R. Jirik, J. Jan (2004) "Estimator Comparison of the Nakagami-m Parameter and Its Application in Echocardiography", Radioengineering, 13 (1), 8–12
^ M. Nakagami. "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. Permagon Press, 1960.
^ J. D. Parsons, The Mobile Radio Propagation Channel. New York: Wiley, 1992.

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf-Mandelbrot

Discrete univariate with infinite support

beta negative binomial Boltzmann Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding-Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Bose–Einstein
Burr
chi-squared
chi
Coxian
Dagum
Davis
Erlang
exponential
F
Fermi–Dirac
folded normal
Fréchet
Gamma
generalized inverse Gaussian
half-logistic
half-normal
Hotelling's T-squared
hyper-exponential
hypoexponential
inverse chi-squared (scaled-inverse-chi-squared)
inverse Gaussian
inverse gamma
Kolmogorov
Lévy
log-Cauchy
log-Laplace
log-logistic
log-normal
Maxwell–Boltzmann
Maxwell speed
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Rayleigh
relativistic Breit–Wigner
Rice
Rosin–Rammler
shifted Gompertz
truncated normal
type-2 Gumbel
Weibull
Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial multivariate Pólya negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse-Wishart matrix normal Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Nakagami distribution