# Arcsine distribution

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

$F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {x}}\right)={\frac {\arcsin(2x-1)}{\pi }}+{\frac {1}{2}}$ Parameters Probability density function Cumulative distribution function none $x\in [0,1]$ $f(x)={\frac {1}{\pi {\sqrt {x(1-x)}}}}$ $F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {x}}\right)$ ${\frac {1}{2}}$ ${\frac {1}{2}}$ $x\in \{0,1\}$ ${\tfrac {1}{8}}$ $0$ $-{\tfrac {3}{2}}$ $\ln {\tfrac {\pi }{4}}$ $1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {2r+1}{2r+2}}\right){\frac {t^{k}}{k!}}$ ${}_{1}F_{1}({\tfrac {1}{2}};1;i\,t)\$ for 0  x  1, and whose probability density function is

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

The arcsine distribution appears

## Generalization

Parameters $-\infty $x\in [a,b]$ $f(x)={\frac {1}{\pi {\sqrt {(x-a)(b-x)}}}}$ $F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {\frac {x-a}{b-a}}}\right)$ ${\frac {a+b}{2}}$ ${\frac {a+b}{2}}$ $x\in {a,b}$ ${\tfrac {1}{8}}(b-a)^{2}$ $0$ $-{\tfrac {3}{2}}$ ### Arbitrary bounded support

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

$F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {\frac {x-a}{b-a}}}\right)$ for a  x  b, and whose probability density function is

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

### Shape factor

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

$f(x;\alpha )={\frac {\sin \pi \alpha }{\pi }}x^{-\alpha }(1-x)^{\alpha -1}$ is also a special case of the beta distribution with parameters ${\rm {Beta}}(1-\alpha ,\alpha )$ .

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

## Properties

• Arcsine distribution is closed under translation and scaling by a positive factor
• If $X\sim {\rm {Arcsine}}(a,b)\ {\text{then }}kX+c\sim {\rm {Arcsine}}(ak+c,bk+c)$ • The square of an arc sine distribution over (-1, 1) has arc sine distribution over (0, 1)
• If $X\sim {\rm {Arcsine}}(-1,1)\ {\text{then }}X^{2}\sim {\rm {Arcsine}}(0,1)$ • If U and V are i.i.d uniform (−π,π) random variables, then $\sin(U)$ , $\sin(2U)$ , $-\cos(2U)$ , $\sin(U+V)$ and $\sin(U-V)$ all have an ${\rm {Arcsine}}(-1,1)$ distribution.
• If $X$ is the generalized arcsine distribution with shape parameter $\alpha$ supported on the finite interval [a,b] then ${\frac {X-a}{b-a}}\sim {\rm {Beta}}(1-\alpha ,\alpha )\$ 