Arcsine distribution explained

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

F(x) = (2)/(\pi) \arcsin\left(\sqrt x\right) = (\arcsin( 2 x -1 ))/(+ \pi) (1)/(2)

for 0 ≤ x ≤ 1, and whose probability density function is

f(x) = (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} l (\tfrac{ 1 }{ 2 },\tfrac{ 1 }{ 2 } r ) . 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) = (2)/(\pi) \arcsin\left(\sqrt (x - a)/(b - a) \right)

for a ≤ x ≤ b, and whose probability density function is

f(x) = (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) = (\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) 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 ) 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 ⋅ \cos(U) \sim {\rm Arcsine}( - r,r) , and its y coordinate distribution is $r\; \backslash cdot\; \backslash sin(U)\; \backslash sim\; (-r,r)$

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 (b + a)/(2)} J _{0 ( (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( 2 U) , - \cos( 2 U) , \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 (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

Notes and References

1. Overturf . Drew . Buchanan . Kristopher . Jensen . Jeffrey . Wheeland . Sara . Huff . Gregory . 1 . 2017 . Investigation of beamforming patterns from volumetrically distributed phased arrays . MILCOM 2017 - 2017 IEEE Military Communications Conference (MILCOM) . 817–822 . 10.1109/MILCOM.2017.8170756 . 978-1-5386-0595-0 .
2. K. . Buchanan . J. . Jensen . C. . Flores-Molina . S. . Wheeland . G. H. . Huff . 1 . Null Beamsteering Using Distributed Arrays and Shared Aperture Distributions . IEEE Transactions on Antennas and Propagation . 68 . 7 . 5353-5364 . 2020 . 10.1109/TAP.2020.2978887 .