Arcsine distribution

Arcsine – bounded support
Parameters
Support
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
CF

Arcsine
	Probability density function
	Cumulative distribution function
Parameters	none
Support
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
MGF
CF

In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root:

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)={\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 $X$ is an arcsine-distributed random variable, then $X\sim {\rm {Beta}}{\bigl (}{\tfrac {1}{2}},{\tfrac {1}{2}}{\bigr )}$ . 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

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)={\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;\alpha )={\frac {\sin \pi \alpha }{\pi }}x^{-\alpha }(1-x)^{\alpha -1}

is also a special case of the beta distribution with parameters ${\rm {Beta}}(1-\alpha ,\alpha )$ .

Note that when $\alpha ={\tfrac {1}{2}}$ 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 $X\sim {\rm {Arcsine}}(a,b)\ {\text{then }}kX+c\sim {\rm {Arcsine}}(ak+c,bk+c)$
The square of an arcsine distribution over (-1, 1) has arcsine distribution over (0, 1)
- If $X\sim {\rm {Arcsine}}(-1,1)\ {\text{then }}X^{2}\sim {\rm {Arcsine}}(0,1)$
The coordinates of points uniformly selected on a circle of radius $r$ $r$ centered at the origin (0, 0), have an ${\rm {Arcsine}}(-r,r)$ ${\rm {Arcsine}}(-r,r)$ distribution
- For example, if we select a point uniformly on the circumference, $U\sim {\rm {Uniform}}(0,2\pi r)$ , we have that the point's x coordinate distribution is $r\cdot \cos(U)\sim {\rm {Arcsine}}(-r,r)$ , and its y coordinate distribution is ${\textstyle r\cdot \sin(U)\sim {\rm {Arcsine}}(-r,r)}$

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

Related distributions

If U and V are i.i.d uniform (−π,π) random variables, then $\sin(U)$ , $\sin(2U)$ , $-\cos(2U)$ , $\sin(U+V)$ and $\sin(U-V)$ all have an ${\rm {Arcsine}}(-1,1)$ distribution.
If $X$ is the generalized arcsine distribution with shape parameter $\alpha$ supported on the finite interval [a,b] then ${\frac {X-a}{b-a}}\sim {\rm {Beta}}(1-\alpha ,\alpha )\$
If X ~ Cauchy(0, 1) then ${\tfrac {1}{1+X^{2}}}$ 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 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.

Arcsine
Probability density function
Cumulative distribution function
Parameters	none
Support	$x\in [0,1]$
PDF	$f(x)={\frac {1}{\pi {\sqrt {x(1-x)}}}}$
CDF	$F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {x}}\right)$
Mean	${\frac {1}{2}}$
Median	${\frac {1}{2}}$
Mode	$x\in \{0,1\}$
Variance	${\tfrac {1}{8}}$
Skewness	$0$
Excess kurtosis	$-{\tfrac {3}{2}}$
Entropy	$\ln {\tfrac {\pi }{4}}$
MGF	$1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {2r+1}{2r+2}}\right){\frac {t^{k}}{k!}}$
CF	$e^{i{\frac {t}{2}}}J_{0}({\frac {t}{2}})$

Arcsine – bounded support
Parameters	$-\infty <a<b<\infty \,$
Support	$x\in [a,b]$
PDF	$f(x)={\frac {1}{\pi {\sqrt {(x-a)(b-x)}}}}$
CDF	$F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {\frac {x-a}{b-a}}}\right)$
Mean	${\frac {a+b}{2}}$
Median	${\frac {a+b}{2}}$
Mode	$x\in {a,b}$
Variance	${\tfrac {1}{8}}(b-a)^{2}$
Skewness	$0$
Excess kurtosis	$-{\tfrac {3}{2}}$
CF	$e^{it{\frac {b+a}{2}}}J_{0}({\frac {b-a}{2}}t)$