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	$0$
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	${}_1F_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	$0$
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

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

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