Arcsine distribution
In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root: F ( x ) = 2 π arcsin ( x ) = arcsin ( 2 x − 1 ) π + 1 2 {\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 f ( x ) = 1 π x ( 1 − x ) {\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.