Geranium – Pelargonium

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.^[1]^[2]

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 ${\textstyle r\cdot \sin(U)\sim {\rm {Arcsine}}(-r,r)}$

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

References

^ Overturf, Drew; et al. (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.
^ Buchanan, K.; et al. (2020). "Null Beamsteering Using Distributed Arrays and Shared Aperture Distributions". IEEE Transactions on Antennas and Propagation. 68 (7): 5353–5364. doi:10.1109/TAP.2020.2978887.