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 . 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 arcsine probability density is a distribution that appears in several random-walk fundamental theorems. In a fair coin toss random walk, the probability for the time of the last visit to the origin is distributed as an arcsine distribution. In a two-player fair-coin-toss game, a player is said to be in the lead if the random walk is above the origin. The most probable number of times that a given player will be in the lead, in a game of length 2N, is not N. On the contrary, N is the least likely number of times that the player will be in the lead. The most likely number of times in the lead is 0 or 2N.

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 .

Shape factor

The generalized standard arcsine distribution on 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 arcsine distribution over has arcsine distribution over
  • * If
  • The coordinates of points uniformly selected on a circle of radius centered at the origin, 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

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, the characteristic function takes the form of.

Related distributions

  • If U and V are i.i.d uniform random variables, then,,, and all have an distribution.
  • If is the generalized arcsine distribution with shape parameter supported on the finite interval then
  • If X ~ Cauchy then has a standard arcsine distribution