= Arcsine distribution
{title2=$\operatorname{Beta}(\tfrac12,\tfrac12)$}
{wiki}
The standard <arcsine distribution> is a <continuous probability distribution> on $[0,1]$ with <probability density function>
$$
f(x)=\frac1{\pi\sqrt{x(1-x)}},\qquad0<x<1.
$$
It is the <beta distribution> with both parameters equal to $1/2$. The density grows without bound toward both endpoints, yet neither endpoint has any <probability mass>. Its name comes from the <inverse sine> in its <cumulative distribution function>.<Random: Probability, Mathematical Statistics, Stochastic Processes by Kyle Siegrist/The Arcsine Distribution>[Random: Probability, Mathematical Statistics, Stochastic Processes by Kyle Siegrist; The Standard Arcsine Distribution, Distribution Functions and Related Distributions: density, normalization, beta representation and CDF]{ref}
A useful way to see the shape is to choose a point uniformly around a <circle> and project it onto a diameter. Equal angular intervals do not give equal horizontal distances: the projection moves slowly near either end. An ideal <harmonic oscillator> sampled at a uniformly chosen time in one complete period has the same endpoint concentration after rescaling its position. The distribution translates a simple geometric slowdown into a precise <probability> calculation.
\Image[/arcsine-circle-projection-and-density.png]
{title=Uniform angles project unevenly onto a diameter}
{description=Original calculated geometry. The circle marks an equally spaced angle grid to illustrate the uniform-angle model; these are not experimental observations. Horizontal projection gives $Y=\cos\Phi$ on $[-1,1]$. The right curve is its exact <probability density function> $1/(\pi\sqrt{1-y^2})$; the standard variable is $X=(1+Y)/2$.}
For an original comparison, take equal-length regions: the middle half $[1/4,3/4]$, and the two outside quarters together. Then
$$
\boxed{\mathbb P(1/4\leq X\leq3/4)=\frac13,\qquad
\mathbb P(X<1/4\text{ or }X>3/4)=\frac23.}
$$
A <uniform distribution> on $[0,1]$ instead assigns each region <probability> $1/2$. Both distributions have <mean> $1/2$; the <mean> alone hides their different concentration.
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