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# Arcsine distribution

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### Arcsine distribution

 Parameters Probability density function Cumulative distribution function none x \in [0,1] f(x) = \frac{1}{\pi\sqrt{x(1-x)}} F(x) = \frac{2}{\pi}\arcsin\left(\sqrt x \right) \frac{1}{2} \frac{1}{2} x \in {0,1} \tfrac{1}{8} 0 -\tfrac{3}{2} 1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{2r+1}{2r+2} \right) \frac{t^k}{k!} {}_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

## Contents

• Generalization 1
• Arbitrary bounded support 1.1
• Shape factor 1.2
• Properties 2
• Related distributions 3
• References 5

## Generalization

 Parameters -\infty < a < b < \infty \, x \in [a,b] f(x) = \frac{1}{\pi\sqrt{(x-a)(b-x)}} F(x) = \frac{2}{\pi}\arcsin\left(\sqrt \frac{x-a}{b-a} \right) \frac{a+b}{2} \frac{a+b}{2} x \in {a,b} \tfrac{1}{8}(b-a)^2 0 -\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 (ab).

### 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)

\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 a standard arcsine 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) \

## References

• Rogozin, B.A. (2001), "A/a013160", in Hazewinkel, Michiel,
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