In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root:
F ( x ) = 2 π arcsin ( x ) = arcsin ( 2 x − 1 ) π + 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 π 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 α = β = 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 Lévy arcsine law, in the Erdős arcsine law, and as the Jeffreys prior for the probability of success of a Bernoulli trial. The arcsine probability density is a distribution that appears in several random-walk fundamental theorems. In a fair coin toss random walk, the probability for the time of the last visit to the origin is distributed as an (U-shaped) arcsine distribution. In a two-player fair-coin-toss game, a player is said to be in the lead if the random walk (that started at the origin) is above the origin. The most probable number of times that a given player will be in the lead, in a game of length 2N, is not N. On the contrary, N is the least likely number of times that the player will be in the lead. The most likely number of times in the lead is 0 or 2N (following the arcsine distribution).
Generalization
Arbitrary bounded support
The distribution can be expanded to include any bounded support from a ≤ x ≤ b by a simple transformation
F ( x ) = 2 π arcsin ( x − a b − a ) {\displaystyle F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {\frac {x-a}{b-a}}}\right)}for a ≤ x ≤ b, and whose probability density function is
f ( x ) = 1 π ( x − a ) ( b − x ) {\displaystyle f(x)={\frac {1}{\pi {\sqrt {(x-a)(b-x)}}}}}on (a, b).
Shape factor
The generalized standard arcsine distribution on (0,1) with probability density function
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}}is also a special case of the beta distribution with parameters B e t a ( 1 − α , α ) {\displaystyle {\rm {Beta}}(1-\alpha ,\alpha )} .
Note that when α = 1 2 {\displaystyle \alpha ={\tfrac {1}{2}}} 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 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)}
- 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)}
- 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
- 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)}
Characteristic function
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 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)} .
Related distributions
- If U and V are i.i.d uniform (−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.
- 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 )\ }
- If X ~ Cauchy(0, 1) then 1 1 + X 2 {\displaystyle {\tfrac {1}{1+X^{2}}}} has a standard arcsine distribution
Further reading
- Rogozin, B.A. (2001) [1994], "Arcsine distribution", Encyclopedia of Mathematics, EMS Press
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 978-1-5386-0595-0. 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. Bibcode:2020ITAP...68.5353B. doi:10.1109/TAP.2020.2978887. /wiki/Bibcode_(identifier) ↩
Feller, William (1971). An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley. ISBN 978-0471257097. 978-0471257097 ↩
Feller, William (1968). An Introduction to Probability Theory and Its Applications. Vol. 1 (3rd ed.). Wiley. ISBN 978-0471257080. 978-0471257080 ↩