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).

[1][2] 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 (U-shaped) arcsine distribution.

[3][4] In a two-player fair-coin-toss game, a player is said to be in the lead if the random walk (that started at the origin) 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 (following the arcsine distribution).

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

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

, the characteristic function takes the form of