# Arcsine distribution

Short description: Type of probability distribution
Parameters Probability density function Cumulative distribution function none $\displaystyle{ x \in [0,1] }$ $\displaystyle{ f(x) = \frac{1}{\pi\sqrt{x(1-x)}} }$ $\displaystyle{ F(x) = \frac{2}{\pi}\arcsin\left(\sqrt x \right) }$ $\displaystyle{ \frac{1}{2} }$ $\displaystyle{ \frac{1}{2} }$ $\displaystyle{ x \in \{0,1\} }$ $\displaystyle{ \tfrac{1}{8} }$ $\displaystyle{ 0 }$ $\displaystyle{ -\tfrac{3}{2} }$ $\displaystyle{ \ln \tfrac{\pi}{4} }$ $\displaystyle{ 1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{2r+1}{2r+2} \right) \frac{t^k}{k!} }$ $\displaystyle{ {}_1F_1(\tfrac{1}{2}; 1; i\,t)\ }$

In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root:

$\displaystyle{ F(x) = \frac{2}{\pi}\arcsin\left(\sqrt x\right)=\frac{\arcsin(2x-1)}{\pi}+\frac{1}{2} }$

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 an arcsine-distributed random variable, 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 in the Lévy arcsine law, in the Erdős arcsine law, and as the Jeffreys prior for the probability of success of a Bernoulli trial.[1] [2]

## Generalization

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

### 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 $\displaystyle{ {}_1F_1(\tfrac{1}{2}; 1; i\,t)\ }$.

## 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 $\displaystyle{ \frac{X-a}{b-a} \sim {\rm Beta}(1-\alpha,\alpha) \ }$
• If X ~ Cauchy(0, 1) then $\displaystyle{ \tfrac{1}{1+X^2} }$ has a standard arcsine distribution

## Application

The arcsine distribution has an application to beamforming and pattern synthesis.[3] It is also the classical probability density for the simple harmonic oscillator.

## References

1. Overturf, Drew; Buchanan, Kristopher; Jensen, Jeffrey; Wheeland, Sara; Huff, Gregory (2017). "Investigation of beamforming patterns from volumetrically distributed phased arrays". MILCOM 2017 - 2017 IEEE Military Communications Conference (MILCOM). pp. 817–822. doi:10.1109/MILCOM.2017.8170756. ISBN:978-1-5386-0595-0. https://ieeexplore.ieee.org/abstract/document/8170756/
2. K. Buchanan, J. Jensen, C. Flores-Molina, S. Wheeland and G. H. Huff, "Null Beamsteering Using Distributed Arrays and Shared Aperture Distributions," in IEEE Transactions on Antennas and Propagation, vol. 68, no. 7, pp. 5353-5364, July 2020, doi: 10.1109/TAP.2020.2978887.
3. Overturf, Drew; Buchanan, Kris; Jensen, Jeff; Flores-Molina, Carlos; Wheeland, Sara; Huff, Gregory H. (2017). "Investigation of beamforming patterns from volumetrically distributed phased arrays". MILCOM 2017 - 2017 IEEE Military Communications Conference (MILCOM). pp. 817–822. doi:10.1109/MILCOM.2017.8170756. ISBN 978-1-5386-0595-0.