Probability Distributions > Arcsine distribution
The arcsine distribution is a symmetric probability distribution with a minimum at x = 1⁄2 and vertical asymptotes at x = 0 and x = 1. Its name comes from the fact that the cumulative distribution function (CDF) involves the arcsine (inverse sine).
The probability distribution function (pdf) of the arcsine distribution is U-shaped: 
PDF for the Arcsine Distribution
The probability density function (PDF) of the standard arcsine distribution is defined as: 
The PDF is supported for an interval of between 0 and 1 (0 < 1). Outside of this boundary, the density is zero. The distribution can be generalized to include any bounded support between two values a and b, or by using scale parameters and location parameters.
The standard arcsine distribution is a special case of the beta distribution with α = β = ½. This only holds true for support on (0, 1), otherwise tapered tails appear at the top and the bottom of the arcsine distribution, which violates the properties of the beta distribution [1].
CDF for the Arcsine Distribution
The cumulative distribution function (CDF) for the arcsin function is: 
The CDF is valid for x-values from 0 to 1 (0 < x < 1); The CDF is concentrated near the boundary values 0 and 1. As the distribution gets very close to 1, it tends to infinity. It includes the inverse sin (arcsin), which is where the distribution gets its name.
Other properties for the arcsine distribution:
- Mean = ½
- Median = ½
- Mode = the standard arcsin distribution is U shaped and has no mode
- Variance = ⅛
- Skewness = 0
- Kurtosis = -3/2
The standard arcsine probability density function satisfies:
- Symmetry about x = ½.
- Decreasing function to a minimum value of x = ½, then increasing.
- Concave up
- f(x) → ∞ as x ↓ 0 and as x ↑ 1.
Uses
The distribution is used in several areas including:
- Renewal theory (the branch of probability theory that generalizes the Poisson process for arbitrary holding times.).
- Jeffrey’s prior for Bernoulli trial successes.
- The Erdős arcsine law (which states that the prime divisors of a number have a distribution related to the arcsine distribution).
- Random walk fluctuations (such as those seen in stock price fluctuations).
- The generalized arcsine distribution, used in mathematical statistics, is a special case of the beta I distribution, when a = b = ½.
Although the distribution is used in some fairly weighty mathematical areas, a simple example of what the distribution can be used for is as the fraction of time a player can win a coin toss game, assuming the coins are fairly weighted [2].
Arcsine exponentiated-X (ASE-X_
He et al. [3] proposed a new family of distributions called the Arcsine exponentiated-X distributions, with CDF:
Where
- F(x; ξ)= the CDF of the baseline distribution
- ξ = parameter vector of the baseline distribution
- a > 0 = an additional shape parameter.
References
- Chave, A. (2017). Computational Statistics in the Earth Sciences With Applications in MATLAB. Cambridge University Press.
- Rasnick, Rebecca, “Generalizations of the Arcsine Distribution” (2019). Electronic Theses and Dissertations. Paper 3565. https://dc.etsu.edu/etd/3565
- [4] He, W. et al. (2020). The Arcsine Exponentiated-X Family: Validation and Insurance Application. Hindawi. Volume 2020 | Article ID 8394815 | https://doi.org/10.1155/2020/8394815