Arcsine distribution

From HandWiki
Short description: Type of probability distribution
Arcsine
Probability density function
Probability density function for the arcsine distribution
Cumulative distribution function
Cumulative distribution function for the arcsine distribution
Parameters none
Support [math]\displaystyle{ x \in [0,1] }[/math]
PDF [math]\displaystyle{ f(x) = \frac{1}{\pi\sqrt{x(1-x)}} }[/math]
CDF [math]\displaystyle{ F(x) = \frac{2}{\pi}\arcsin\left(\sqrt x \right) }[/math]
Mean [math]\displaystyle{ \frac{1}{2} }[/math]
Median [math]\displaystyle{ \frac{1}{2} }[/math]
Mode [math]\displaystyle{ x \in \{0,1\} }[/math]
Variance [math]\displaystyle{ \tfrac{1}{8} }[/math]
Skewness [math]\displaystyle{ 0 }[/math]
Ex. kurtosis [math]\displaystyle{ -\tfrac{3}{2} }[/math]
Entropy [math]\displaystyle{ \ln \tfrac{\pi}{4} }[/math]
MGF [math]\displaystyle{ 1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{2r+1}{2r+2} \right) \frac{t^k}{k!} }[/math]
CF [math]\displaystyle{ {}_1F_1(\tfrac{1}{2}; 1; i\,t)\ }[/math]

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

[math]\displaystyle{ F(x) = \frac{2}{\pi}\arcsin\left(\sqrt x\right)=\frac{\arcsin(2x-1)}{\pi}+\frac{1}{2} }[/math]

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

[math]\displaystyle{ f(x) = \frac{1}{\pi\sqrt{x(1-x)}} }[/math]

on (0, 1). The standard arcsine distribution is a special case of the beta distribution with α = β = 1/2. That is, if [math]\displaystyle{ X }[/math] is an arcsine-distributed random variable, then [math]\displaystyle{ X \sim {\rm Beta}\bigl(\tfrac{1}{2},\tfrac{1}{2}\bigr) }[/math]. 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

Arcsine – bounded support
Parameters [math]\displaystyle{ -\infty \lt a \lt b \lt \infty \, }[/math]
Support [math]\displaystyle{ x \in [a,b] }[/math]
PDF [math]\displaystyle{ f(x) = \frac{1}{\pi\sqrt{(x-a)(b-x)}} }[/math]
CDF [math]\displaystyle{ F(x) = \frac{2}{\pi}\arcsin\left(\sqrt \frac{x-a}{b-a} \right) }[/math]
Mean [math]\displaystyle{ \frac{a+b}{2} }[/math]
Median [math]\displaystyle{ \frac{a+b}{2} }[/math]
Mode [math]\displaystyle{ x \in {a,b} }[/math]
Variance [math]\displaystyle{ \tfrac{1}{8}(b-a)^2 }[/math]
Skewness [math]\displaystyle{ 0 }[/math]
Ex. kurtosis [math]\displaystyle{ -\tfrac{3}{2} }[/math]

Arbitrary bounded support

The distribution can be expanded to include any bounded support from a ≤ x ≤ b by a simple transformation

[math]\displaystyle{ F(x) = \frac{2}{\pi}\arcsin\left(\sqrt \frac{x-a}{b-a} \right) }[/math]

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

[math]\displaystyle{ f(x) = \frac{1}{\pi\sqrt{(x-a)(b-x)}} }[/math]

on (ab).

Shape factor

The generalized standard arcsine distribution on (0,1) with probability density function

[math]\displaystyle{ f(x;\alpha) = \frac{\sin \pi\alpha}{\pi}x^{-\alpha}(1-x)^{\alpha-1} }[/math]

is also a special case of the beta distribution with parameters [math]\displaystyle{ {\rm Beta}(1-\alpha,\alpha) }[/math].

Note that when [math]\displaystyle{ \alpha = \tfrac{1}{2} }[/math] 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 [math]\displaystyle{ X \sim {\rm Arcsine}(a,b) \ \text{then } kX+c \sim {\rm Arcsine}(ak+c,bk+c) }[/math]
  • The square of an arcsine distribution over (-1, 1) has arcsine distribution over (0, 1)
    • If [math]\displaystyle{ X \sim {\rm Arcsine}(-1,1) \ \text{then } X^2 \sim {\rm Arcsine}(0,1) }[/math]

Characteristic function

The characteristic function of the arcsine distribution is a confluent hypergeometric function and given as [math]\displaystyle{ {}_1F_1(\tfrac{1}{2}; 1; i\,t)\ }[/math].

Related distributions

  • If U and V are i.i.d uniform (−π,π) random variables, then [math]\displaystyle{ \sin(U) }[/math], [math]\displaystyle{ \sin(2U) }[/math], [math]\displaystyle{ -\cos(2U) }[/math], [math]\displaystyle{ \sin(U+V) }[/math] and [math]\displaystyle{ \sin(U-V) }[/math] all have an [math]\displaystyle{ {\rm Arcsine}(-1,1) }[/math] distribution.
  • If [math]\displaystyle{ X }[/math] is the generalized arcsine distribution with shape parameter [math]\displaystyle{ \alpha }[/math] supported on the finite interval [a,b] then [math]\displaystyle{ \frac{X-a}{b-a} \sim {\rm Beta}(1-\alpha,\alpha) \ }[/math]
  • If X ~ Cauchy(0, 1) then [math]\displaystyle{ \tfrac{1}{1+X^2} }[/math] 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. https://ieeexplore.ieee.org/document/8170756. 

Further reading