# Arcsine distribution

Template:Probability distribution

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

- in the Lévy arcsine law;
- in the Erdős arcsine law;
- as the Jeffreys prior for the probability of success of a Bernoulli trial.

## Generalization

Template:Probability distribution

### 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
- The square of an arc sine distribution over (-1, 1) has arc sine distribution over (0, 1)

## Related distributions

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

## See also

## References

- REDIRECT Template:Probability distributions