In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root: 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 X is an arcsine-distributed random variable, then. By extension, the arcsine distribution is a special case of the Pearson type I distribution. The arcsine distribution appears in the Lévy arcsine law, in the Erdős arcsine law, and as the Jeffreys prior for the probability of success of a Bernoulli trial.

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

Characteristic function

The characteristic function of the generalized arcsine distribution is a zero order Bessel function of the first kind, multiplied by a complex exponential, given by. For the special case of b = -a, the characteristic function takes the form of J_0(b t).

Related distributions