Arcsine distribution

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Arcsine
Probability density function
Probability density function for the arcsine distribution
Cumulative distribution function
Cumulative distribution function for the arcsine distribution
Parameters none
Support
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy
MGF
CF

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

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

on (0, 1). The standard arcsine distribution is a special case of the beta distribution with α = β = 1/2. That is, if is an arcsine-distributed random variable, then . 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
Support
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis

Arbitrary bounded support

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

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

on (ab).

Shape factor

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

is also a special case of the beta distribution with parameters .

Note that when 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
  • The square of an arcsine distribution over (-1, 1) has arcsine distribution over (0, 1)
    • If

Characteristic function

The characteristic function of the arcsine distribution is a confluent hypergeometric function and given as .

Related distributions

  • If U and V are i.i.d uniform (−π,π) random variables, then , , , and all have an distribution.
  • If is the generalized arcsine distribution with shape parameter supported on the finite interval [a,b] then
  • If X ~ Cauchy(0, 1) then 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. S2CID 11591305.

Further reading