Arcsine distribution

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 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. 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). 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. Arcsine distribution is closed under translation and scaling by a positive factor If The square of an arcsine distribution over (-1, 1) has arcsine distribution over (0, 1) If The coordinates of points uniformly selected on a circle of radius centered at the origin (0, 0), have an distribution For example, if we select a point uniformly on the circumference, , we have that the point's x coordinate distribution is , and its y coordinate distribution is The characteristic function of the arcsine distribution is a confluent hypergeometric function and given as . 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 If X ~ Cauchy(0, 1) then has a standard arcsine distribution The arcsine distribution has an application to beamforming and pattern synthesis. It is also the classical probability density for the simple harmonic oscillator.