= Rayleigh moments
{c}
{title2=$\mathbb E[R^q]=\sigma^q2^{q/2}\Gamma(1+q/2),\quad q>-2$}
Using $u=r^2/(2\sigma^2)$ in the defining <expected value> integral gives
$$
\mathbb E[R^q]
=\int_0^\infty r^q\frac r{\sigma^2}e^{-r^2/(2\sigma^2)}\,dr
=\sigma^q2^{q/2}\int_0^\infty u^{q/2}e^{-u}\,du
=\sigma^q2^{q/2}\Gamma(1+q/2),\qquad q>-2.
$$
The last integral is the <gamma function>. Near zero, the integrand before substitution behaves as $r^{q+1}$, explaining the threshold $q>-2$; at infinity the Gaussian tail controls every power. In particular,
$$
\mathbb E[R]=\sigma\sqrt{\frac\pi2},\quad
\mathbb E[R^2]=2\sigma^2,\quad
\operatorname{Var}(R)=\left(2-\frac\pi2\right)\sigma^2.
$$
<Random: Probability, Mathematical Statistics, Stochastic Processes by Kyle Siegrist/The Rayleigh Distribution>[Random: Probability, Mathematical Statistics, Stochastic Processes by Kyle Siegrist; The General Rayleigh Distribution, Moments: mean, variance and nonnegative integer moments; extension to real q greater than -2 follows from the displayed integral]{ref}
For the chosen $\sigma=2\,\mathrm{cm}$, the <mean> <radius> is about $2.5066\,\mathrm{cm}$, its <median> is $2\sqrt{2\log2}\simeq2.3548\,\mathrm{cm}$, and its <mode> is $2\,\mathrm{cm}$. Its <root mean square> is $2\sqrt2\simeq2.8284\,\mathrm{cm}$. Different summaries answer different questions; the component scale $\sigma$ is neither the radial <mean> nor the radial <standard deviation>.
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