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

${\displaystyle f(x;\,m,\Omega )={\frac {2m^{m}}{\Gamma (m)\Omega ^{m}}}x^{2m-1}\exp \left(-{\frac {m}{\Omega }}x^{2}\right),\forall x\geq 0.}$ 

where ${\displaystyle m\geq 1/2}$  and ${\displaystyle \Omega >0}$ .

Its cumulative distribution function (CDF) is^[1]

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

where P is the regularized (lower) incomplete gamma function.

The parameters ${\displaystyle m}$  and ${\displaystyle \Omega }$  are^[2]

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

and

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

No closed form solution exists for the median of this distribution, although special cases do exist, such as ${\displaystyle {\sqrt {\Omega \;ln(2)}}}$  when m=1. 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 ${\displaystyle \Omega }$  as σ = Ω/m.^[3]

Given independent observations ${\textstyle X_{1}=x_{1},\ldots ,X_{n}=x_{n}}$  from the Nakagami distribution, the likelihood function is

${\displaystyle L(\sigma ,m)=\left({\frac {2}{\Gamma (m)\sigma ^{m}}}\right)^{n}\left(\prod _{i=1}^{n}x_{i}\right)^{2m-1}\exp \left(-{\frac {\sum _{i=1}^{n}x_{i}^{2}}{\sigma }}\right).}$ 

Its logarithm is

${\displaystyle \ell (\sigma ,m)=\log L(\sigma ,m)=-n\log \Gamma (m)-nm\log \sigma +(2m-1)\sum _{i=1}^{n}\log x_{i}-{\frac {\sum _{i=1}^{n}x_{i}^{2}}{\sigma }}.}$ 

Therefore

${\displaystyle {\begin{aligned}{\frac {\partial \ell }{\partial \sigma }}={\frac {-nm\sigma +\sum _{i=1}^{n}x_{i}^{2}}{\sigma ^{2}}}\quad {\text{and}}\quad {\frac {\partial \ell }{\partial m}}=-n{\frac {\Gamma '(m)}{\Gamma (m)}}-n\log \sigma +2\sum _{i=1}^{n}\log x_{i}.\end{aligned}}}$ 

These derivatives vanish only when

${\displaystyle \sigma ={\frac {\sum _{i=1}^{n}x_{i}^{2}}{nm}}}$ 

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.

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

${\displaystyle X={\sqrt {Y}}.\,}$ 

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

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

For a Chi-distribution, the degrees of freedom ${\displaystyle 2m}$  must be an integer, but for Nakagami the ${\displaystyle m}$  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.^[6]

• Restricting m to the unit interval (q = m; 0 < q < 1)^{[dubious – discuss]} defines the Nakagami-q distribution, also known as Hoyt distribution, first studied by R.S. Hoyt in the 1940s.^[7][8][9] In particular, the radius around the true mean in a bivariate normal random variable, re-written in polar coordinates (radius and angle), follows a Hoyt distribution. Equivalently, the modulus of a complex normal random variable also does.
• With 2m = k, the Nakagami distribution gives a scaled chi distribution.
• With ${\displaystyle m={\tfrac {1}{2}}}$ , the Nakagami distribution gives a scaled half-normal distribution.
• A Nakagami distribution is a particular form of generalized gamma distribution, with p = 2 and d = 2m.

• Gamma distribution
• Modified half-normal distribution
• Sub-Gaussian distribution