Arcsine distribution

Arcsine

Probability density function
Cumulative distribution function
Parameters	none
Support	${\displaystyle x\in [0,1]}$
PDF	${\displaystyle f(x)={\frac {1}{\pi {\sqrt {x(1-x)}}}}}$
CDF	${\displaystyle F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {x}}\right)}$
Mean	${\displaystyle {\frac {1}{2}}}$
Median	${\displaystyle {\frac {1}{2}}}$
Mode	${\displaystyle x\in \{0,1\}}$
Variance	${\displaystyle {\tfrac {1}{8}}}$
Skewness	${\displaystyle 0}$
Ex. kurtosis	${\displaystyle -{\tfrac {3}{2}}}$
Entropy	${\displaystyle \ln {\tfrac {\pi }{4}}}$
MGF	${\displaystyle 1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {2r+1}{2r+2}}\right){\frac {t^{k}}{k!}}}$
CF	${\displaystyle {}_{1}F_{1}({\tfrac {1}{2}};1;i\,t)\ }$

Arcsine – bounded support

Parameters	${\displaystyle -\infty <a<b<\infty \,}$
Support	${\displaystyle x\in [a,b]}$
PDF	${\displaystyle f(x)={\frac {1}{\pi {\sqrt {(x-a)(b-x)}}}}}$
CDF	${\displaystyle F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {\frac {x-a}{b-a}}}\right)}$
Mean	${\displaystyle {\frac {a+b}{2}}}$
Median	${\displaystyle {\frac {a+b}{2}}}$
Mode	${\displaystyle x\in {a,b}}$
Variance	${\displaystyle {\tfrac {1}{8}}(b-a)^{2}}$
Skewness	${\displaystyle 0}$
Ex. kurtosis	${\displaystyle -{\tfrac {3}{2}}}$

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

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

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 ${\displaystyle X\sim {\rm {Beta}}{\bigl (}{\tfrac {1}{2}},{\tfrac {1}{2}}{\bigr )}}$.

The arcsine distribution appears