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Arcsine distribution
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<h2 id="generalization">Generalization</h2> <h3>Arbitrary bounded support</h3> <p>The distribution can be expanded to include any bounded support from <i>a</i> ≤ <i>x</i> ≤ <i>b</i> by a simple transformation </p> F ( x ) = 2 π arcsin ( x − a b − a ) {\displaystyle F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {\frac {x-a}{b-a}}}\right)} <p>for <i>a</i> ≤ <i>x</i> ≤ <i>b</i>, and whose <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> is </p> f ( x ) = 1 π ( x − a ) ( b − x ) {\displaystyle f(x)={\frac {1}{\pi {\sqrt {(x-a)(b-x)}}}}} <p>on (<i>a</i>, <i>b</i>). </p> <h3>Shape factor</h3> <p>The generalized standard arcsine distribution on (0,1) with probability density function </p> f ( x ; α ) = sin π α π x − α ( 1 − x ) α − 1 {\displaystyle f(x;\alpha )={\frac {\sin \pi \alpha }{\pi }}x^{-\alpha }(1-x)^{\alpha -1}} <p>is also a special case of the <a href="/facts/Beta_distribution/DbJk4eeV">beta distribution</a> with parameters B e t a ( 1 − α , α ) {\displaystyle {\rm {Beta}}(1-\alpha ,\alpha )} . </p><p>Note that when α = 1 2 {\displaystyle \alpha ={\tfrac {1}{2}}} the general arcsine distribution reduces to the standard distribution listed above. </p> <h2 id="properties">Properties</h2> <ul><li>Arcsine distribution is closed under translation and scaling by a positive factor <ul><li>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)} </li></ul></li> <li>The square of an arcsine distribution over (-1, 1) has arcsine distribution over (0, 1) <ul><li>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)} </li></ul></li> <li>The coordinates of points uniformly selected on a circle of radius r {\displaystyle r} centered at the origin (0, 0), have an A r c s i n e ( − r , r ) {\displaystyle {\rm {Arcsine}}(-r,r)} distribution <ul><li>For example, if we select a point uniformly on the circumference, U ∼ U n i f o r m ( 0 , 2 π r ) {\displaystyle U\sim {\rm {Uniform}}(0,2\pi r)} , 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)} , 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)} </li></ul></li></ul> <h2 id="characteristic-function">Characteristic function</h2> <p>The characteristic function of the generalized arcsine distribution is a zero order <a href="/facts/Bessel_function/hSyvFPqC">Bessel function</a> 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)} . For the special case of b = − a {\displaystyle b=-a} , the characteristic function takes the form of J 0 ( b t ) {\displaystyle J_{0}(bt)} . </p> <h2 id="related-distributions">Related distributions</h2> <ul><li>If U and V are <a href="/facts/Independent_and_identically_distributed_random_variables/othIRaWt">i.i.d</a> <a href="/facts/Uniform_distribution_(continuous)/XbnlVljT">uniform</a> (−π,π) random variables, then sin ( U ) {\displaystyle \sin(U)} , sin ( 2 U ) {\displaystyle \sin(2U)} , − cos ( 2 U ) {\displaystyle -\cos(2U)} , sin ( U + V ) {\displaystyle \sin(U+V)} and sin ( U − V ) {\displaystyle \sin(U-V)} all have an A r c s i n e ( − 1 , 1 ) {\displaystyle {\rm {Arcsine}}(-1,1)} distribution.</li> <li>If X {\displaystyle X} is the generalized arcsine distribution with shape parameter α {\displaystyle \alpha } supported on the finite interval [a,b] then X − a b − a ∼ B e t a ( 1 − α , α ) {\displaystyle {\frac {X-a}{b-a}}\sim {\rm {Beta}}(1-\alpha ,\alpha )\ } </li> <li>If <i>X</i> ~ Cauchy(0, 1) then 1 1 + X 2 {\displaystyle {\tfrac {1}{1+X^{2}}}} has a standard arcsine distribution</li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Rogozin, B.A. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Arcsine_distribution">"Arcsine distribution"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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. <a href="978-1-5386-0595-0" target="_blank">978-1-5386-0595-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Buchanan, K.; et al. (2020). "Null Beamsteering Using Distributed Arrays and Shared Aperture Distributions". IEEE Transactions on Antennas and Propagation. 68 (7): 5353–5364. Bibcode:2020ITAP...68.5353B. doi:10.1109/TAP.2020.2978887. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Feller, William (1971). An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley. ISBN 978-0471257097. <a href="978-0471257097" target="_blank">978-0471257097</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Feller, William (1968). An Introduction to Probability Theory and Its Applications. Vol. 1 (3rd ed.). Wiley. ISBN 978-0471257080. <a href="978-0471257080" target="_blank">978-0471257080</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>