Editing Rayleigh distribution (section)

==Properties==
The [[moment (mathematics)|raw  moments]] are given by:
: <math>\mu_j = \sigma^j2^{j/2}\,\Gamma\left(1 + \frac j 2\right),</math> 
where <math>\Gamma(z)</math> is the [[gamma function]].

The [[mean]] of a  Rayleigh  random variable  is thus <!--(<math>k=1, \Gamma\left(\tfrac32\right) = \tfrac12 \sqrt{\pi}\,</math>)-->:
:<math>\mu(X) = \sigma \sqrt{\frac{\pi}{2}}\ \approx 1.253\ \sigma.</math>

The [[standard deviation]] of a  Rayleigh  random variable  is:

:<math>\operatorname{std}(X) = \sqrt{\left (2-\frac{\pi}{2}\right)} \sigma \approx 0.655\ \sigma</math>

The [[variance]]   of a  Rayleigh  random variable  is :

:<math>\operatorname{var}(X) = \mu_2-\mu_1^2 =  \left(2-\frac{\pi}{2}\right) \sigma^2 \approx 0.429\ \sigma^2</math>

The [[mode (statistics)|mode]] is <math>\sigma,</math> and the maximum pdf is

:<math> f_{\max} = f(\sigma;\sigma) = \frac{1}{\sigma} e^{-1/2} \approx \frac{0.606}{\sigma}.</math>

The [[skewness]] is given by:

:<math>\gamma_1 = \frac{2\sqrt{\pi}(\pi - 3)}{(4 - \pi)^{3/2}} \approx 0.631</math>

The excess [[kurtosis]] is given by:

:<math>\gamma_2 = -\frac{6\pi^2 - 24\pi + 16}{(4 - \pi)^2} \approx 0.245</math>

The [[characteristic function (probability theory)|characteristic function]] is given by:

:<math>\varphi(t) = 1 - \sigma te^{-\frac{1}{2}\sigma^2t^2}\sqrt{\frac{\pi}{2}} \left[\operatorname{erfi}\left(\frac{\sigma t}{\sqrt{2}}\right) - i\right]</math>

where <math>\operatorname{erfi}(z)</math> is the imaginary [[error function]]. The [[moment generating function]] is given by

:<math>
  M(t) = 1 + \sigma t\,e^{\frac{1}{2}\sigma^2t^2}\sqrt{\frac{\pi}{2}}
           \left[\operatorname{erf}\left(\frac{\sigma t}{\sqrt{2}}\right) + 1\right]</math>

where <math>\operatorname{erf}(z)</math> is the [[error function]].

===Differential entropy===

The [[differential entropy]] is given by{{Citation needed|date=April 2013}}

:<math>H = 1 + \ln\left(\frac \sigma {\sqrt{2}}\right) + \frac \gamma 2 </math>

where <math>\gamma</math> is the [[Euler–Mascheroni constant]].