= Rayleigh distribution
{c}
{title2=$R\sim\operatorname{Rayleigh}(\sigma),\quad\sigma>0$}
{wiki}
The <Rayleigh distribution> is the <probability distribution> of the <Euclidean norm> of a two-dimensional <random vector> with <independent> zero-<mean> <normal random variables> of equal <variance> as components. If $X,Y\sim N(0,\sigma^2)$ are <independent>, then
$$
R=\sqrt{X^2+Y^2}\sim\operatorname{Rayleigh}(\sigma),\qquad
f_R(r)=\begin{cases}\dfrac r{\sigma^2}e^{-r^2/(2\sigma^2)},&r\geq0,\\0,&r<0.\end{cases}
$$
Here $\sigma$ is the <standard deviation> of each component, not of the <radius> $R$. At $\sigma=1$ this gives the unit-scale <probability density function> $r e^{-r^2/2}$.<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, Definition and Distribution Functions: independent equal-scale centered normal components and radial density]{ref}
This geometry turns two signed errors into one nonnegative distance. For example, suppose horizontal and vertical positioning errors are <independent> $N(0,4\,\mathrm{cm}^2)$ <random variables>. Their <means> both vanish, but the distance from the intended point is usually positive. The <Rayleigh distribution> with $\sigma=2\,\mathrm{cm}$ gives the <probability> of landing inside any centered <disk>. These are chosen model assumptions, rather than measurements of a particular instrument.
\Image[/rayleigh-gaussian-area-and-radius.png]
{title=The center maximizes planar density, while a positive <radius> maximizes radial density}
{description=Original calculated model with $\sigma=1$. Left: <joint probability density> per unit <area>, with an <annulus> from $r=0.9$ to $r=1.1$. Right: <probability density function> per unit <radius>; the shaded area is the probability of that same <annulus>. A single point and a single <circle> both have probability zero.}
An apparently paradoxical feature is that the most likely location per unit <area> is the origin, while the radial <mode> is $\sigma$. A narrow <annulus> farther out has more <area>. Near zero that growing <circumference> outweighs the falling planar density; farther out the <exponential function> wins. Distinguishing densities measured against different elements of <area> or length resolves the paradox.
Back to article page
