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] 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.[3][4] 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 ), 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).

  • 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

  1. ^ 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.
  2. ^ Buchanan, K. et al. (2020). "Null Beamsteering Using Distributed Arrays and Shared Aperture Distributions". IEEE Transactions on Antennas and Propagation. 68 (7): 5353–5364. Bibcode:2020ITAP...68.5353B. doi:10.1109/TAP.2020.2978887
  3. ^ Feller, William (1971). An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley. ISBN 978-0471257097.
  4. ^ Feller, William (1968). An Introduction to Probability Theory and Its Applications. Vol. 1. 3rd ed. Wiley. ISBN 978-0471257080.