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Arcsine distribution

Type of probability distribution

Summary

Type of probability distribution

name =Arcsine| type =density| pdf_image =[[Image:Arcsin density.svg|350px|Probability density function for the arcsine distribution]]| cdf_image =[[Image:Arcsin cdf.svg|350px|Cumulative distribution function for the arcsine distribution]]| parameters =none| support =x \in (0,1)| pdf =f(x) = \frac{1}{\pi\sqrt{x(1-x)}} | cdf =F(x) = \frac{2}{\pi}\arcsin\left(\sqrt x \right) | quantile =F^{-1}(x) = \sin\left(\frac{\pi x}{2}\right)^{2} | mean =\frac{1}{2} | median =\frac{1}{2} | mode =x \in {0,1} | variance =\tfrac{1}{8} | skewness =0| kurtosis =-\tfrac{3}{2}| entropy =\log \tfrac{\pi}{4} | mgf =1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{2r+1}{2r+2} \right) \frac{t^k}{k!}| char =e^{i\frac{t}{2}}J_0(\frac{t}{2})|

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. 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. 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

name =Arcsine – bounded support| type =density| pdf_image = | cdf_image = | parameters =-\infty | support =x \in (a,b)| pdf =f(x) = \frac{1}{\pi\sqrt{(x-a)(b-x)}} | cdf =F(x) = \frac{2}{\pi}\arcsin\left(\sqrt \frac{x-a}{b-a} \right) | quantile =F^{-1}(x) = \left(b-a\right)\sin\left(\frac{\pi x}{2}\right)^{2}+a | mean =\frac{a+b}{2} | median =\frac{a+b}{2} | mode =x \in {a,b} | variance =\tfrac{1}{8}(b-a)^2 | skewness =0| kurtosis =-\tfrac{3}{2}| entropy =\log\left(\pi\frac{b-a}{4}\right)| mgf = | char = e^{it\frac{b+a}{2}}J_0(\frac{b-a}{2}t)|

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).

References

References

  1. (2017). "Investigation of beamforming patterns from volumetrically distributed phased arrays".
  2. (2020). "Null Beamsteering Using Distributed Arrays and Shared Aperture Distributions". IEEE Transactions on Antennas and Propagation.
  3. Feller, William. (1971). "An Introduction to Probability Theory and Its Applications, Vol. 2". Wiley.
  4. Feller, William. (1968). "An Introduction to Probability Theory and Its Applications". Wiley.
Wikipedia Source

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