Arcsine distribution

In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function is

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

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 the standard arcsine distribution then . By extension, the arcsine distribution is a special case of the Pearson type I distribution.

The arcsine distribution appears [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 (a, b).

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 .

  • 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

See also

References

  • Rogozin, B.A. (2001) [1994], "Arcsine distribution", Encyclopedia of Mathematics, EMS Press
  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.
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