Arcsine distribution

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

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	$e^{i{\frac {t}{2}}}J_{0}({\frac {t}{2}})$

$F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {x}}\right)={\frac {\arcsin(2x-1)}{\pi }}+{\frac {1}{2}}$

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.

Generalization

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	$e^{it{\frac {b+a}{2}}}J_{0}({\frac {b-a}{2}}t)$

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 $r$ centered at the origin (0, 0), have an ${\rm {Arcsine}}(-r,r)$ 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

Further reading

• Rogozin, B.A. (2001) [1994], "Arcsine distribution", Encyclopedia of Mathematics, EMS Press