# Arcsine distribution

In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root:
    $F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {x}}\right)={\frac {\arcsin(2x-1)}{\pi }}+{\frac {1}{2}}$
for 0 ≤ x ≤ 1, and whose probability density function is
    $f(x)={\frac {1}{\pi {\sqrt {x(1-x)}}}}$
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 an arcsine-distributed random variable, then $X\sim {\rm {Beta}}{\bigl{\tfrac {1}{2}},{\tfrac {1}{2}}{\bigr )}$. By extension, the arcsine distribution is a special case of the Pearson type I distribution.
The arcsine distribution appears in the Lévy arcsine law, in the Erdős arcsine law, and as the Jeffreys prior for the probability of success of a Bernoulli trial. The arcsine probability density is a distribution that appears in several random-walk fundamental theorems. In a fair coin toss random walk, the probability for the time of the last visit to the origin is distributed as an (U-shaped) arcsine distribution. In a two-player fair-coin-toss game, a player is said to be in the lead if the random walk (that started at the origin) is above the origin. The most probable number of times that a given player will be in the lead, in a game of length 2N, is not N. On the contrary, N is the least likely number of times that the player will be in the lead. The most likely number of times in the lead is 0 or 2N (following the arcsine distribution).

## Generalization
### 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 arcsine distribution over (-1, 1) has arcsine distribution over (0, 1)
** If $X\sim {\rm {Arcsine}}(-1,1)\ {\text{then }}X^{2}\sim {\rm {Arcsine}}(0,1)$
* The coordinates of points uniformly selected on a circle of radius $r$ centered at the origin (0, 0), have an ${\rm {Arcsine}}(-r,r)$ distribution
** For example, if we select a point uniformly on the circumference, $U\sim {\rm {Uniform}}(0,2\pi r)$, we have that the point's x coordinate distribution is $r\cdot \cos(U)\sim {\rm {Arcsine}}(-r,r)$, and its y coordinate distribution is r⋅sin⁡(U)∼Arcsine(−r, r){\textstyle r\cdot \sin(U)\sim {\rm {Arcsine}}(-r, r)}

## Characteristic function

The characteristic function of the generalized arcsine distribution is a zero order Bessel function of the first kind, multiplied by a complex exponential, given by $e^{it{\frac {b+a}{2}}}J_{0}({\frac {b-a}{2}}t)$. For the special case of $b=-a$, the characteristic function takes the form of $J_{0}(bt)$.

## Related distributions

* 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 )\$
* If X ~ Cauchy(0, 1) then ${\tfrac {1}{1+X^{2}}}$ has a standard arcsine distribution