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.
Note that when the general arcsine distribution reduces to the standard distribution listed above.
Propertiesedit
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 functionedit
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 .
Related distributionsedit
If U and V are i.i.duniform (−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
Referencesedit
^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.
^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.