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# Arcsine distribution

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

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 ${\displaystyle X\sim {\rm {Beta}}{\bigl (}{\tfrac {1}{2}},{\tfrac {1}{2}}{\bigr )}}$. By extension, the arcsine distribution is a special case of the Pearson type I distribution.

The arcsine distribution appears [1]

## Generalization

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

### 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.

## Properties

• Arcsine distribution is closed under translation and scaling by a positive factor
• If ${\displaystyle X\sim {\rm {Arcsine}}(a,b)\ {\text{then }}kX+c\sim {\rm {Arcsine}}(ak+c,bk+c)}$
• The square of an arcsine distribution over (-1, 1) has arcsine distribution over (0, 1)
• If ${\displaystyle X\sim {\rm {Arcsine}}(-1,1)\ {\text{then }}X^{2}\sim {\rm {Arcsine}}(0,1)}$

## Characteristic function

The characteristic function of the arcsine distribution is a confluent hypergeometric function and given as ${\displaystyle {}_{1}F_{1}({\tfrac {1}{2}};1;i\,t)\ }$.

## Related distributions

• If U and V are i.i.d uniform (−π,π) random variables, then ${\displaystyle \sin(U)}$, ${\displaystyle \sin(2U)}$, ${\displaystyle -\cos(2U)}$, ${\displaystyle \sin(U+V)}$ and ${\displaystyle \sin(U-V)}$ all have an ${\displaystyle {\rm {Arcsine}}(-1,1)}$ distribution.
• If ${\displaystyle X}$ is the generalized arcsine distribution with shape parameter ${\displaystyle \alpha }$ supported on the finite interval [a,b] then ${\displaystyle {\frac {X-a}{b-a}}\sim {\rm {Beta}}(1-\alpha ,\alpha )\ }$