Arcsine distribution

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	${\displaystyle 0}$
Ex. 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	${}_{1}F_{1}({\tfrac {1}{2}};1;i\,t)\$

In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function is

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

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 the standard arcsine distribution then $X\sim {{\rm {Beta}}}({\tfrac {1}{2}},{\tfrac {1}{2}})\$

The arcsine distribution appears

• in the Lévy arcsine law;
• in the Erdős arcsine law;
• 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	${\displaystyle 0}$
Ex. kurtosis	$-{\tfrac {3}{2}}$

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 arc sine distribution over (-1, 1) has arc sine distribution over (0, 1)
  • If $X\sim {{\rm {Arcsine}}}(-1,1)\ {\text{then }}X^{2}\sim {{\rm {Arcsine}}}(0,1)$

Differential equation

$\left\{2(x-1)xf'(x)+(2x-1)f(x)=0\right\}$

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 )\$

See also

• Arcsine

References

• Rogozin, B.A. (2001), "A/a013160", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4