Arcsine distribution: Difference between revisions – Wikipedia

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Arcsine distribution: Difference between revisions – Wikipedia


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==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 (non-generalized) 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 arcsine-distributed random variable, then

1 2 , 1 2 )

{\displaystyle X\sim {\rm {Beta}}{\bigl (}{\tfrac {1}{2}},{\tfrac {1}{2}}{\bigr )}} . 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

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

{\displaystyle {\rm {Beta}}(1-\alpha ,\alpha )} .

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

J 0 ( b a 2 t

{\displaystyle e^{it{\frac {b+a}{2}}}J_{0}({\frac {b-a}{2}}t} . For the special case of , the characteristic function takes the form of .

The characteristic function of the (non-generalized) arcsine distribution by also be viewed as a confluent hypergeometric function, given as

1 F 1 ( 1 2 ; 1 ; i t )   {\displaystyle {}_{1}F_{1}({\tfrac {1}{2}};1;i\,t)\ } .

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

  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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