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 &alpha; = &beta; = 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.

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 \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 $$J_0(b t)$$.

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