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. ISBN978-1-5386-0595-0.
^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.
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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 ArcsineProbability density functionCumulative distribution functionParametersnoneSupportx 0 1 displaystyle x in 0 1 PDFf x 1 p x 1 x displaystyle f x frac 1 pi sqrt x 1 x CDFF x 2 p arcsin x displaystyle F x frac 2 pi arcsin left sqrt x right Mean1 2 displaystyle frac 1 2 Median1 2 displaystyle frac 1 2 Modex 0 1 displaystyle x in 0 1 Variance1 8 displaystyle tfrac 1 8 Skewness0 displaystyle 0 Ex kurtosis 3 2 displaystyle tfrac 3 2 Entropyln p 4 displaystyle ln tfrac pi 4 MGF1 k 1 r 0 k 1 2 r 1 2 r 2 t k k displaystyle 1 sum k 1 infty left prod r 0 k 1 frac 2r 1 2r 2 right frac t k k CFe i t 2 J 0 t 2 displaystyle e i frac t 2 J 0 frac t 2 F x 2 p arcsin x arcsin 2 x 1 p 1 2 displaystyle F x frac 2 pi arcsin left sqrt x right frac arcsin 2x 1 pi frac 1 2 for 0 x 1 and whose probability density function is f x 1 p x 1 x displaystyle f x frac 1 pi sqrt x 1 x on 0 1 The standard arcsine distribution is a special case of the beta distribution with a b 1 2 That is if X displaystyle X is an arcsine distributed random variable then X B e t a 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 Levy arcsine law in the Erdos arcsine law and as the Jeffreys prior for the probability of success of a Bernoulli trial 1 2 Contents 1 Generalization 1 1 Arbitrary bounded support 1 2 Shape factor 2 Properties 3 Characteristic function 4 Related distributions 5 References 6 Further readingGeneralization editArcsine bounded supportParameters lt a lt b lt displaystyle infty lt a lt b lt infty nbsp Supportx a b displaystyle x in a b nbsp PDFf x 1 p x a b x displaystyle f x frac 1 pi sqrt x a b x nbsp CDFF x 2 p arcsin x a b a displaystyle F x frac 2 pi arcsin left sqrt frac x a b a right nbsp Meana b 2 displaystyle frac a b 2 nbsp Mediana b 2 displaystyle frac a b 2 nbsp Modex a b displaystyle x in a b nbsp Variance1 8 b a 2 displaystyle tfrac 1 8 b a 2 nbsp Skewness0 displaystyle 0 nbsp Ex kurtosis 3 2 displaystyle tfrac 3 2 nbsp CFe i t b a 2 J 0 b a 2 t displaystyle e it frac b a 2 J 0 frac b a 2 t nbsp Arbitrary bounded support edit The distribution can be expanded to include any bounded support from a x b by a simple transformation F x 2 p arcsin x a b a displaystyle F x frac 2 pi arcsin left sqrt frac x a b a right nbsp for a x b and whose probability density function is f x 1 p x a b x displaystyle f x frac 1 pi sqrt x a b x nbsp on a b Shape factor edit The generalized standard arcsine distribution on 0 1 with probability density function f x a sin p a p x a 1 x a 1 displaystyle f x alpha frac sin pi alpha pi x alpha 1 x alpha 1 nbsp is also a special case of the beta distribution with parameters B e t a 1 a a displaystyle rm Beta 1 alpha alpha nbsp Note that when a 1 2 displaystyle alpha tfrac 1 2 nbsp the general arcsine distribution reduces to the standard distribution listed above Properties editArcsine distribution is closed under translation and scaling by a positive factor If X A r c s i n e a b then k X c A r c s i n e a k c b k c displaystyle X sim rm Arcsine a b text then kX c sim rm Arcsine ak c bk c nbsp The square of an arcsine distribution over 1 1 has arcsine distribution over 0 1 If X A r c s i n e 1 1 then X 2 A r c s i n e 0 1 displaystyle X sim rm Arcsine 1 1 text then X 2 sim rm Arcsine 0 1 nbsp The coordinates of points uniformly selected on a circle of radius r displaystyle r nbsp centered at the origin 0 0 have an A r c s i n e r r displaystyle rm Arcsine r r nbsp distribution For example if we select a point uniformly on the circumference U U n i f o r m 0 2 p r displaystyle U sim rm Uniform 0 2 pi r nbsp we have that the point s x coordinate distribution is r cos U A r c s i n e r r displaystyle r cdot cos U sim rm Arcsine r r nbsp and its y coordinate distribution is r sin U A r c s i n e r r textstyle r cdot sin U sim rm Arcsine r r nbsp Characteristic function editThe characteristic function of the generalized arcsine distribution is a zero order Bessel function of the first kind multiplied by a complex exponential given by e i t b a 2 J 0 b a 2 t displaystyle e it frac b a 2 J 0 frac b a 2 t nbsp For the special case of b a displaystyle b a nbsp the characteristic function takes the form of J 0 b t displaystyle J 0 bt nbsp Related distributions editIf U and V are i i d uniform p p random variables then sin U displaystyle sin U nbsp sin 2 U displaystyle sin 2U nbsp cos 2 U displaystyle cos 2U nbsp sin U V displaystyle sin U V nbsp and sin U V displaystyle sin U V nbsp all have an A r c s i n e 1 1 displaystyle rm Arcsine 1 1 nbsp distribution If X displaystyle X nbsp is the generalized arcsine distribution with shape parameter a displaystyle alpha nbsp supported on the finite interval a b then X a b a B e t a 1 a a displaystyle frac X a b a sim rm Beta 1 alpha alpha nbsp If X Cauchy 0 1 then 1 1 X 2 displaystyle tfrac 1 1 X 2 nbsp has a standard arcsine distributionReferences edit 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 Further reading editRogozin B A 2001 1994 Arcsine distribution Encyclopedia of Mathematics EMS Press Retrieved from https en wikipedia org w index php title Arcsine distribution amp oldid 1202905686, wikipedia, wiki, book, books, library,