# Arcsine distribution

Parameters Probability density function Cumulative distribution function none

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

for 0  x  1, and whose probability density function is

on (0, 1). The standard arcsine distribution is a special case of the beta distribution with α = β = 1/2. That is, if is the standard arcsine distribution then

The arcsine distribution appears

## Generalization

### Arbitrary bounded support

The distribution can be expanded to include any bounded support from a  x  b by a simple transformation

for a  x  b, and whose probability density function is

on (a, b).

### Shape factor

The generalized standard arcsine distribution on (0,1) with probability density function

is also a special case of the beta distribution with parameters .

Note that when 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
• The square of an arc sine distribution over (-1, 1) has arc sine distribution over (0, 1)
• If

Differential equation

## Related distributions

• If U and V are i.i.d uniform (−π,π) random variables, then , , , and all have an distribution.
• If is the generalized arcsine distribution with shape parameter supported on the finite interval [a,b] then