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Rayleigh Distribution

\begin{figure}\begin{center}\BoxedEPSF{RayleighDistribution.epsf scaled 650}\end{center}\end{figure}

The distribution with Probability Function

\begin{displaymath}
P(r) = {re^{-r^2/2s^2}\over s^2}
\end{displaymath} (1)

for $r\in [0,\infty)$. The Moments about 0 are given by
$\displaystyle \mu_m'$ $\textstyle \equiv$ $\displaystyle \int_0^\infty r^m P(r)\,dr = s^{-2} \int_0^\infty r^{m+1}e^{-r^2/2s^2}\,dr$  
  $\textstyle =$ $\displaystyle s^{-2} I_{m+1}\left({1\over 2s^2}\right),$ (2)

where $I(x)$ is a Gaussian Integral. The first few of these are
$\displaystyle I_1(a^{-1})$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}a$ (3)
$\displaystyle I_2(a^{-1})$ $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}a\sqrt{a\pi}$ (4)
$\displaystyle I_3(a^{-1})$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}a^2$ (5)
$\displaystyle I_4(a^{-1})$ $\textstyle =$ $\displaystyle {\textstyle{3\over 8}} a^2\sqrt{a\pi}$ (6)
$\displaystyle I_5(a^{-1})$ $\textstyle =$ $\displaystyle a^3,$ (7)

so
$\displaystyle \mu_0'$ $\textstyle =$ $\displaystyle s^{-2} {\textstyle{1\over 2}}(2s^2)=1$ (8)
$\displaystyle \mu_1'$ $\textstyle =$ $\displaystyle s^{-2} {\textstyle{1\over 4}}(2s^2)\sqrt{2s^2\pi} = {\textstyle{1\over 2}}s\sqrt{2\pi}=s\sqrt{\pi\over 2}$ (9)
$\displaystyle \mu_2'$ $\textstyle =$ $\displaystyle s^{-2} {\textstyle{1\over 2}}(2s^2)^2=2s^2$ (10)
$\displaystyle \mu_3'$ $\textstyle =$ $\displaystyle s^{-2} {\textstyle{3\over 8}}(2s^2)^2\sqrt{2s^2\pi} = {\textstyle{3\over 2}}s^3\sqrt{2\pi} =3s^3\sqrt{\pi\over 2}$ (11)
$\displaystyle \mu_4'$ $\textstyle =$ $\displaystyle s^{-2} (2s^2)^3=8s^4.$ (12)

The Moments about the Mean are
$\displaystyle \mu_2$ $\textstyle =$ $\displaystyle \mu_2'-(\mu_1')^2={4-\pi\over 2} s^2$ (13)
$\displaystyle \mu_3$ $\textstyle =$ $\displaystyle \mu_3'-3\mu_2'\mu_1'+2(\mu_1')^3=\sqrt{\pi\over 2}\,(\pi-3)s^3$ (14)
$\displaystyle \mu_4$ $\textstyle =$ $\displaystyle \mu_4'-4\mu_3'\mu_1'+6\mu_2'(\mu_1')^2-3(\mu-1')^4$  
  $\textstyle =$ $\displaystyle {32-3\pi^2\over 4} s^4,$ (15)

so the Mean, Variance, Skewness, and Kurtosis are
$\displaystyle \mu$ $\textstyle =$ $\displaystyle \mu_1'=s\sqrt{\pi\over 2}$ (16)
$\displaystyle \sigma^2$ $\textstyle =$ $\displaystyle \mu_2={4-\pi\over 2} s^2$ (17)
$\displaystyle \gamma_1$ $\textstyle =$ $\displaystyle {\mu_3\over\sigma^3} = {2(\pi-3)\sqrt{\pi}\over (4-\pi)^{3/2}}$ (18)
$\displaystyle \gamma_2$ $\textstyle =$ $\displaystyle {\mu_4\over\sigma^4}-3 = {2(-3\pi^2+12\pi-8)\over(\pi-4)^2}.$ (19)



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© 1996-9 Eric W. Weisstein
1999-05-25