Arcsine distribution

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

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
Excess 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 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

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Arcsine – bounded support
Parameters  
Support  
PDF  
CDF  
Mean  
Median  
Mode  
Variance  
Skewness  
Excess kurtosis  
CF  

Arbitrary bounded support

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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

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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

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  • 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  
  • The coordinates of points uniformly selected on a circle of radius   centered at the origin (0, 0), have an   distribution
    • For example, if we select a point uniformly on the circumference,  , we have that the point's x coordinate distribution is  , and its y coordinate distribution is  

Characteristic function

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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  . For the special case of  , the characteristic function takes the form of  .

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  • 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

References

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  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. doi:10.1109/TAP.2020.2978887.

Further reading

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