Nakagami distribution

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Nakagami
Probability density function
Cumulative distribution function
Parameters μ > = 0.5 shape (real)
ω > 0 spread (real)
Support x > 0\!
Probability density function (pdf) \frac{2\mu^\mu}{\Gamma(\mu)\omega^\mu} x^{2\mu-1} \exp\left(-\frac{\mu}{\omega}x^2 \right)
Cumulative distribution function (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)
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

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

[edit] Characterization

Its probability density function (pdf) is[1]

p(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]

P\left(\mu, \frac{\mu}{\omega}x^2\right)

where P is the incomplete gamma function (regularized).

[edit] Parameter estimation

The parameters μ and ω are obtained by [1]

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

and

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

[edit] History and applications

The Nakagami distribution is relatively new, being first proposed in 1960.[2] It has been used to model attenuation of wireless signals traversing multiple paths.[3]

[edit] References

  1. ^ a b Laurenson, Dave (1994). Nakagami Distribution. Indoor Radio Channel Propagation Modelling by Ray Tracing Techniques. Retrieved on 2007-08-04.
  2. ^ M. Nakagami. "The m-Distribution, a general formula of intensity of rapid fading". In W. G. Hoffman, editor, Statistical Methods in Radio Wave Propagation: Proceedings of a Symposium held at the University of California, pp 3-36. Permagon Press, 1960.
  3. ^ J. D. Parsons, The Mobile Radio Propagation Channel. New York: Wiley, 1992.
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Probability distributions
 
Continuous univariate supported on a semi-infinite interval