Arcsine distribution: Difference between revisions – Wikipedia
==Characteristic function==
==Characteristic function==
The characteristic function of the arcsine distribution is a [[confluent hypergeometric function]] and given as <math>{}_1F_1(\tfrac{1}{2}; 1; i\,t)\ </math>.
The characteristic function of the arcsine distribution is a function given <math>{}(\{}{2} , </math>.
The characteristic function of the (nongeneralized) arcsine distribution by also be viewed as a [[confluent hypergeometric function]], given as <math>{}_1F_1(\tfrac{1}{2}; 1; i\,t)\ </math>.
==Related distributions==
==Related distributions==
Type of probability distribution
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 arcsinedistributed random variable, then
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
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
 The coordinates of points uniformly selected on a circle of radius centered at the origin (0, 0), have an
Characteristic function
The characteristic function of the generalized arcsine distribution is a zero order Bessel function, multiplied by a complex exponential, given by
The characteristic function of the (nongeneralized) arcsine distribution by also be viewed as a confluent hypergeometric function, given as
Related distributions
 If U and V are i.i.d uniform (−π,π) random variables, then , , , and all have an
 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
 ^ 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 9781538605950.
 ^ 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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