# Arcsine distribution

In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function is

${\displaystyle F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {x}}\right)={\frac {\arcsin(2x-1)}{\pi }}+{\frac {1}{2}}}$

for 0 ≤ x ≤ 1, and whose probability density function is

${\displaystyle f(x)={\frac {1}{\pi {\sqrt {x(1-x)}}}}}$

on (0, 1). The standard arcsine distribution is a special case of the beta distribution with α = β = 1/2. That is, if ${\displaystyle X}$ is the standard arcsine distribution then ${\displaystyle X\sim {\rm {Beta}}({\tfrac {1}{2}},{\tfrac {1}{2}})\ }$

The arcsine distribution appears

## Generalization

### Arbitrary bounded support

The distribution can be expanded to include any bounded support from a ≤ x ≤ b by a simple transformation

${\displaystyle F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {\frac {x-a}{b-a}}}\right)}$

for a ≤ x ≤ b, and whose probability density function is

${\displaystyle f(x)={\frac {1}{\pi {\sqrt {(x-a)(b-x)}}}}}$

on (ab).

### Shape factor

The generalized standard arcsine distribution on (0,1) with probability density function

${\displaystyle f(x;\alpha )={\frac {\sin \pi \alpha }{\pi }}x^{-\alpha }(1-x)^{\alpha -1}}$

is also a special case of the beta distribution with parameters ${\displaystyle {\rm {Beta}}(1-\alpha ,\alpha )}$.

Note that when ${\displaystyle \alpha ={\tfrac {1}{2}}}$ the general arcsine distribution reduces to the standard distribution listed above.