It is used to model physical phenomena such as those found in medical ultrasound imaging, communications engineering, meteorology, hydrology, multimedia, and seismology.

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

Its cumulative distribution function (CDF) is[1] where P is the regularized (lower) incomplete gamma function.

are[2] and No closed form solution exists for the median of this distribution, although special cases do exist, such as

For practical purposes the median would have to be calculated as the 50th-percentile of the observations.

An alternative way of fitting the distribution is to re-parametrize

from the Nakagami distribution, the likelihood function is Its logarithm is Therefore These derivatives vanish only when and the value of m for which the derivative with respect to m vanishes is found by numerical methods including the Newton–Raphson method.

It can be shown that at the critical point a global maximum is attained, so the critical point is the maximum-likelihood estimate of (m,σ).

Because of the equivariance of maximum-likelihood estimation, a maximum likelihood estimate for Ω is obtained as well.

, and taking the square root of

: Alternatively, the Nakagami distribution

can be generated from the chi distribution with parameter

and then following it by a scaling transformation of random variables.

is generated by a simple scaling transformation on a chi-distributed random variable

For a chi-distribution, the degrees of freedom

can be any real number greater than 1/2.

This is the critical difference and accordingly, Nakagami-m is viewed as a generalization of chi-distribution, similar to a gamma distribution being considered as a generalization of chi-squared distributions.

The Nakagami distribution is relatively new, being first proposed in 1960 by Minoru Nakagami as a mathematical model for small-scale fading in long-distance high-frequency radio wave propagation.

[4] It has been used to model attenuation of wireless signals traversing multiple paths[5] and to study the impact of fading channels on wireless communications.